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0806.4735

Shai Gutner

Noga Alon and Shai Gutner

Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs

Proc. of 13th COCOON (2007), 394-405

cs.DS cs.DM

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

There is substantial literature dealing with fixed parameter algorithms for the dominating set problem on various families of graphs. In this paper, we give a $k^{O(dk)} n$ time algorithm for finding a dominating set of size at most $k$ in a $d$-degenerated graph with $n$ vertices. This proves that the dominating set problem is fixed-parameter tractable for degenerated graphs. For graphs that do not contain $K_h$ as a topological minor, we give an improved algorithm for the problem with running time $(O(h))^{hk} n$. For graphs which are $K_h$-minor-free, the running time is further reduced to $(O(\log h))^{hk/2} n$. Fixed-parameter tractable algorithms that are linear in the number of vertices of the graph were previously known only for planar graphs. For the families of graphs discussed above, the problem of finding an induced cycle of a given length is also addressed. For every fixed $H$ and $k$, we show that if an $H$-minor-free graph $G$ with $n$ vertices contains an induced cycle of size $k$, then such a cycle can be found in O(n) expected time as well as in $O(n \log n)$ worst-case time. Some results are stated concerning the (im)possibility of establishing linear time algorithms for the more general family of degenerated graphs.

0806.4790

Vladimir Braverman

Vladimir Braverman, Kai-Min Chung, Zhenming Liu, Michael Mitzenmacher, Rafail Ostrovsky

AMS Without 4-Wise Independence on Product Domains

cs.DS

http://creativecommons.org/licenses/by/3.0/

In their seminal work, Alon, Matias, and Szegedy introduced several sketching techniques, including showing that 4-wise independence is sufficient to obtain good approximations of the second frequency moment. In this work, we show that their sketching technique can be extended to product domains $[n]^k$ by using the product of 4-wise independent functions on $[n]$. Our work extends that of Indyk and McGregor, who showed the result for $k = 2$. Their primary motivation was the problem of identifying correlations in data streams. In their model, a stream of pairs $(i,j) \in [n]^2$ arrive, giving a joint distribution $(X,Y)$, and they find approximation algorithms for how close the joint distribution is to the product of the marginal distributions under various metrics, which naturally corresponds to how close $X$ and $Y$ are to being independent. By using our technique, we obtain a new result for the problem of approximating the $\ell_2$ distance between the joint distribution and the product of the marginal distributions for $k$-ary vectors, instead of just pairs, in a single pass. Our analysis gives a randomized algorithm that is a $(1 \pm \epsilon)$ approximation (with probability $1-\delta$) that requires space logarithmic in $n$ and $m$ and proportional to $3^k$.

0806.4899

Mordecai Golin

Mordecai Golin and Yan Zhang

A Dynamic Programming Approach To Length-Limited Huffman Coding

cs.DS cs.IT math.IT

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

The ``state-of-the-art'' in Length Limited Huffman Coding algorithms is the $\Theta(ND)$-time, $\Theta(N)$-space one of Hirschberg and Larmore, where $D\le N$ is the length restriction on the code. This is a very clever, very problem specific, technique. In this note we show that there is a simple Dynamic-Programming (DP) method that solves the problem with the same time and space bounds. The fact that there was an $\Theta(ND)$ time DP algorithm was previously known; it is a straightforward DP with the Monge property (which permits an order of magnitude speedup). It was not interesting, though, because it also required $\Theta(ND)$ space. The main result of this paper is the technique developed for reducing the space. It is quite simple and applicable to many other problems modeled by DPs with the Monge property. We illustrate this with examples from web-proxy design and wireless mobile paging.

0807.0038

Changyong Zhang

Changyong Zhang

A Novel Mathematical Model for the Unique Shortest Path Routing Problem

31 pages, 4 figures

math.OC cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Link weights are the principal parameters of shortest path routing protocols, the most commonly used protocols for IP networks. The problem of optimally setting link weights for unique shortest path routing is addressed. Due to the complexity of the constraints involved, there exist challenges to formulate the problem properly, so that a solution algorithm may be developed which could prove to be more efficient than those already in existence. In this paper, a novel complete formulation with a polynomial number of constraints is first introduced and then mathematically proved to be correct. It is further illustrated that the formulation has advantages over a prior one in terms of both constraint structure and model size for a proposed decomposition method to solve the problem.

0807.0222

Sariel Har-Peled

Sariel Har-Peled, S. Muthukrishnan

Range Medians

To appear in ESA 08

cs.DS cs.OH

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

We study a generalization of the classical median finding problem to batched query case: given an array of unsorted $n$ items and $k$ (not necessarily disjoint) intervals in the array, the goal is to determine the median in {\em each} of the intervals in the array. We give an algorithm that uses $O(n\log n + k\log k \log n)$ comparisons and show a lower bound of $\Omega(n\log k)$ comparisons for this problem. This is optimal for $k=O(n/\log n)$.

0807.0644

Neal E. Young

Christos Koufogiannakis and Neal E. Young

Greedy D-Approximation Algorithm for Covering with Arbitrary Constraints and Submodular Cost

Algorithmica 66(1):113-152 (2013)

10.1007/978-3-642-02927-1_53

cs.DS cs.DC

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

This paper describes a simple greedy D-approximation algorithm for any covering problem whose objective function is submodular and non-decreasing, and whose feasible region can be expressed as the intersection of arbitrary (closed upwards) covering constraints, each of which constrains at most D variables of the problem. (A simple example is Vertex Cover, with D = 2.) The algorithm generalizes previous approximation algorithms for fundamental covering problems and online paging and caching problems.

0807.0807

Mustaq Ahmed

Mustaq Ahmed, Anna Lubiw

Shortest Paths Avoiding Forbidden Subpaths

12 pages, 2 figures. Fixed a few typos, rephrased a few sentences, and used the STACS style

Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS), Freiburg, Germany, 2009, pp. 63-74

cs.DM cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

In this paper we study a variant of the shortest path problem in graphs: given a weighted graph G and vertices s and t, and given a set X of forbidden paths in G, find a shortest s-t path P such that no path in X is a subpath of P. Path P is allowed to repeat vertices and edges. We call each path in X an exception, and our desired path a shortest exception-avoiding path. We formulate a new version of the problem where the algorithm has no a priori knowledge of X, and finds out about an exception x in X only when a path containing x fails. This situation arises in computing shortest paths in optical networks. We give an algorithm that finds a shortest exception avoiding path in time polynomial in |G| and |X|. The main idea is to run Dijkstra's algorithm incrementally after replicating vertices when an exception is discovered.

0807.0928

Denis Charles X.

Denis Charles, Kumar Chellapilla

Bloomier Filters: A second look

13 Pages, 3 figures, to appear in ESA - 2008

cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

A Bloom filter is a space efficient structure for storing static sets, where the space efficiency is gained at the expense of a small probability of false-positives. A Bloomier filter generalizes a Bloom filter to compactly store a function with a static support. In this article we give a simple construction of a Bloomier filter. The construction is linear in space and requires constant time to evaluate. The creation of our Bloomier filter takes linear time which is faster than the existing construction. We show how one can improve the space utilization further at the cost of increasing the time for creating the data structure.

0807.1139

Nitish Korula

Nitish Korula, Martin Pal

Algorithms for Secretary Problems on Graphs and Hypergraphs

15 pages, 2 figures

cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

We examine several online matching problems, with applications to Internet advertising reservation systems. Consider an edge-weighted bipartite graph G, with partite sets L, R. We develop an 8-competitive algorithm for the following secretary problem: Initially given R, and the size of L, the algorithm receives the vertices of L sequentially, in a random order. When a vertex l \in L is seen, all edges incident to l are revealed, together with their weights. The algorithm must immediately either match l to an available vertex of R, or decide that l will remain unmatched. Dimitrov and Plaxton show a 16-competitive algorithm for the transversal matroid secretary problem, which is the special case with weights on vertices, not edges. (Equivalently, one may assume that for each l \in L, the weights on all edges incident to l are identical.) We use a similar algorithm, but simplify and improve the analysis to obtain a better competitive ratio for the more general problem. Perhaps of more interest is the fact that our analysis is easily extended to obtain competitive algorithms for similar problems, such as to find disjoint sets of edges in hypergraphs where edges arrive online. We also introduce secretary problems with adversarially chosen groups. Finally, we give a 2e-competitive algorithm for the secretary problem on graphic matroids, where, with edges appearing online, the goal is to find a maximum-weight acyclic subgraph of a given graph.

0807.1277

David Gamarnik

David Gamarnik and David Goldberg

Randomized greedy algorithms for independent sets and matchings in regular graphs: Exact results and finite girth corrections

24 pages

cs.DM cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

We derive new results for the performance of a simple greedy algorithm for finding large independent sets and matchings in constant degree regular graphs. We show that for $r$-regular graphs with $n$ nodes and girth at least $g$, the algorithm finds an independent set of expected cardinality $f(r)n - O\big(\frac{(r-1)^{\frac{g}{2}}}{\frac{g}{2}!} n\big)$, where $f(r)$ is a function which we explicitly compute. A similar result is established for matchings. Our results imply improved bounds for the size of the largest independent set in these graphs, and provide the first results of this type for matchings. As an implication we show that the greedy algorithm returns a nearly perfect matching when both the degree $r$ and girth $g$ are large. Furthermore, we show that the cardinality of independent sets and matchings produced by the greedy algorithm in \emph{arbitrary} bounded degree graphs is concentrated around the mean. Finally, we analyze the performance of the greedy algorithm for the case of random i.i.d. weighted independent sets and matchings, and obtain a remarkably simple expression for the limiting expected values produced by the algorithm. In fact, all the other results are obtained as straightforward corollaries from the results for the weighted case.

0807.1891

Benjmain Moseley

Chandra Chekuri and Benjamin Moseley

Online Scheduling to Minimize the Maximum Delay Factor

cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

In this paper two scheduling models are addressed. First is the standard model (unicast) where requests (or jobs) are independent. The other is the broadcast model where broadcasting a page can satisfy multiple outstanding requests for that page. We consider online scheduling of requests when they have deadlines. Unlike previous models, which mainly consider the objective of maximizing throughput while respecting deadlines, here we focus on scheduling all the given requests with the goal of minimizing the maximum {\em delay factor}.We prove strong lower bounds on the achievable competitive ratios for delay factor scheduling even with unit-time requests.For the unicast model we give algorithms that are $(1 + \eps)$-speed $O({1 \over \eps})$-competitive in both the single machine and multiple machine settings. In the broadcast model we give an algorithm for similar-sized pages that is $(2+ \eps)$-speed $O({1 \over \eps^2})$-competitive. For arbitrary page sizes we give an algorithm that is $(4+\eps)$-speed $O({1 \over \eps^2})$-competitive.

0807.2120

Robin Moser

Robin A. Moser

Derandomizing the Lovasz Local Lemma more effectively

8 pages; added acknowledgement

cs.DS cs.CC

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

The famous Lovasz Local Lemma [EL75] is a powerful tool to non-constructively prove the existence of combinatorial objects meeting a prescribed collection of criteria. Kratochvil et al. applied this technique to prove that a k-CNF in which each variable appears at most 2^k/(ek) times is always satisfiable [KST93]. In a breakthrough paper, Beck found that if we lower the occurrences to O(2^(k/48)/k), then a deterministic polynomial-time algorithm can find a satisfying assignment to such an instance [Bec91]. Alon randomized the algorithm and required O(2^(k/8)/k) occurrences [Alo91]. In [Mos06], we exhibited a refinement of his method which copes with O(2^(k/6)/k) of them. The hitherto best known randomized algorithm is due to Srinivasan and is capable of solving O(2^(k/4)/k) occurrence instances [Sri08]. Answering two questions asked by Srinivasan, we shall now present an approach that tolerates O(2^(k/2)/k) occurrences per variable and which can most easily be derandomized. The new algorithm bases on an alternative type of witness tree structure and drops a number of limiting aspects common to all previous methods.

0807.2178

Bernd G\"artner

Bernd G\"artner

Ranking Unit Squares with Few Visibilities

4 pages, 2 EPS-figures

cs.CG cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Given a set of n unit squares in the plane, the goal is to rank them in space in such a way that only few squares see each other vertically. We prove that ranking the squares according to the lexicographic order of their centers results in at most 3n-7 pairwise visibilities for n at least 4. We also show that this bound is best possible, by exhibiting a set of n squares with at least 3n-7 pairwise visibilities under any ranking.

0807.2269

Michel Rueher

Alexandre Goldsztejn (LINA), Claude Michel (I3S, Laboratoire I3S), Michel Rueher (I3S, Laboratoire I3S)

An Efficient Algorithm for a Sharp Approximation of Universally Quantified Inequalities

ACM symposium on Applied computing, Fortaleza, Ceara : Br\'esil (2008)

cs.NA cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

This paper introduces a new algorithm for solving a sub-class of quantified constraint satisfaction problems (QCSP) where existential quantifiers precede universally quantified inequalities on continuous domains. This class of QCSPs has numerous applications in engineering and design. We propose here a new generic branch and prune algorithm for solving such continuous QCSPs. Standard pruning operators and solution identification operators are specialized for universally quantified inequalities. Special rules are also proposed for handling the parameters of the constraints. First experimentation show that our algorithm outperforms the state of the art methods.

0807.2330

Antonios Symvonis

Tamara Mchedlidze, Antonios Symvonis

Optimal Acyclic Hamiltonian Path Completion for Outerplanar Triangulated st-Digraphs (with Application to Upward Topological Book Embeddings)

cs.DS cs.DM

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Given an embedded planar acyclic digraph G, we define the problem of "acyclic hamiltonian path completion with crossing minimization (Acyclic-HPCCM)" to be the problem of determining an hamiltonian path completion set of edges such that, when these edges are embedded on G, they create the smallest possible number of edge crossings and turn G to a hamiltonian digraph. Our results include: --We provide a characterization under which a triangulated st-digraph G is hamiltonian. --For an outerplanar triangulated st-digraph G, we define the st-polygon decomposition of G and, based on its properties, we develop a linear-time algorithm that solves the Acyclic-HPCCM problem with at most one crossing per edge of G. --For the class of st-planar digraphs, we establish an equivalence between the Acyclic-HPCCM problem and the problem of determining an upward 2-page topological book embedding with minimum number of spine crossings. We infer (based on this equivalence) for the class of outerplanar triangulated st-digraphs an upward topological 2-page book embedding with minimum number of spine crossings and at most one spine crossing per edge. To the best of our knowledge, it is the first time that edge-crossing minimization is studied in conjunction with the acyclic hamiltonian completion problem and the first time that an optimal algorithm with respect to spine crossing minimization is presented for upward topological book embeddings.

0807.2496

Ashish Goel

Ashish Goel and Kamesh Munagala

Hybrid Keyword Search Auctions

cs.GT cs.DS cs.IR

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Search auctions have become a dominant source of revenue generation on the Internet. Such auctions have typically used per-click bidding and pricing. We propose the use of hybrid auctions where an advertiser can make a per-impression as well as a per-click bid, and the auctioneer then chooses one of the two as the pricing mechanism. We assume that the advertiser and the auctioneer both have separate beliefs (called priors) on the click-probability of an advertisement. We first prove that the hybrid auction is truthful, assuming that the advertisers are risk-neutral. We then show that this auction is superior to the existing per-click auction in multiple ways: 1) It takes into account the risk characteristics of the advertisers. 2) For obscure keywords, the auctioneer is unlikely to have a very sharp prior on the click-probabilities. In such situations, the hybrid auction can result in significantly higher revenue. 3) An advertiser who believes that its click-probability is much higher than the auctioneer's estimate can use per-impression bids to correct the auctioneer's prior without incurring any extra cost. 4) The hybrid auction can allow the advertiser and auctioneer to implement complex dynamic programming strategies. As Internet commerce matures, we need more sophisticated pricing models to exploit all the information held by each of the participants. We believe that hybrid auctions could be an important step in this direction.

0807.2694

Fei Li

Fei Li

Algorithms for Scheduling Weighted Packets with Deadlines in a Bounded Queue

19 pages. Appears in the Proceedings of the 28th IEEE International Conference on Computer Communications (INFOCOM 2009)

cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Motivated by the Quality-of-Service (QoS) buffer management problem, we consider online scheduling of packets with hard deadlines in a finite capacity queue. At any time, a queue can store at most $b \in \mathbb Z^+$ packets. Packets arrive over time. Each packet is associated with a non-negative value and an integer deadline. In each time step, only one packet is allowed to be sent. Our objective is to maximize the total value gained by the packets sent by their deadlines in an online manner. Due to the Internet traffic's chaotic characteristics, no stochastic assumptions are made on the packet input sequences. This model is called a {\em finite-queue model}. We use competitive analysis to measure an online algorithm's performance versus an unrealizable optimal offline algorithm who constructs the worst possible input based on the knowledge of the online algorithm. For the finite-queue model, we first present a deterministic 3-competitive memoryless online algorithm. Then, we give a randomized ($\phi^2 = ((1 + \sqrt{5}) / 2)^2 \approx 2.618$)-competitive memoryless online algorithm. The algorithmic framework and its theoretical analysis include several interesting features. First, our algorithms use (possibly) modified characteristics of packets; these characteristics may not be same as those specified in the input sequence. Second, our analysis method is different from the classical potential function approach.

0807.3006

Enoch Peserico

Enoch Peserico and Luca Pretto

The rank convergence of HITS can be slow

5 pages, 1 figure. Keywords: algorithm analysis, information retrieval, rank convergence

cs.DS cs.IR

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

We prove that HITS, to "get right" h of the top k ranked nodes of an N>=2k node graph, can require h^(Omega(N h/k)) iterations (i.e. a substantial Omega(N h log(h)/k) matrix multiplications even with a "squaring trick"). Our proof requires no algebraic tools and is entirely self-contained.

0807.3026

Ryan Williams

Ryan Williams

Finding paths of length k in O*(2^k) time

7 pages. Revised version to appear in Information Processing Letters

Information Processing Letters, 109(6):315--318, February 2009

cs.DS cs.DM

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

We give a randomized algorithm that determines if a given graph has a simple path of length at least k in O(2^k poly(n,k)) time.

0807.3387

Jeremy Sumner

J. G. Sumner and M. A. Charleston

Phylogenetic estimation with partial likelihood tensors

20 pages, 7 figures, 3 tables

q-bio.QM cs.DS q-bio.PE

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

We present an alternative method for calculating likelihoods in molecular phylogenetics. Our method is based on partial likelihood tensors, which are generalizations of partial likelihood vectors, as used in Felsenstein's approach. Exploiting a lexicographic sorting and partial likelihood tensors, it is possible to obtain significant computational savings. We show this on a range of simulated data by enumerating all numerical calculations that are required by our method and the standard approach.

0807.4234

Stavros Nikolopoulos D.

Kyriaki Ioannidou and Stavros D. Nikolopoulos

Linear Coloring and Linear Graphs

21 pages, 7 figures

cs.DM cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Motivated by the definition of linear coloring on simplicial complexes, recently introduced in the context of algebraic topology \cite{Civan}, and the framework through which it was studied, we introduce the linear coloring on graphs. We provide an upper bound for the chromatic number $\chi(G)$, for any graph $G$, and show that $G$ can be linearly colored in polynomial time by proposing a simple linear coloring algorithm. Based on these results, we define a new class of perfect graphs, which we call co-linear graphs, and study their complement graphs, namely linear graphs. The linear coloring of a graph $G$ is a vertex coloring such that two vertices can be assigned the same color, if their corresponding clique sets are associated by the set inclusion relation (a clique set of a vertex $u$ is the set of all maximal cliques containing $u$); the linear chromatic number $\mathcal{\lambda}(G)$ of $G$ is the least integer $k$ for which $G$ admits a linear coloring with $k$ colors. We show that linear graphs are those graphs $G$ for which the linear chromatic number achieves its theoretical lower bound in every induced subgraph of $G$. We prove inclusion relations between these two classes of graphs and other subclasses of chordal and co-chordal graphs, and also study the structure of the forbidden induced subgraphs of the class of linear graphs.

0807.4368

Dimitris Papamichail

Dimitris Papamichail and Georgios Papamichail

Improved Algorithms for Approximate String Matching (Extended Abstract)

10 pages

cs.DS

http://creativecommons.org/licenses/by-nc-sa/3.0/

The problem of approximate string matching is important in many different areas such as computational biology, text processing and pattern recognition. A great effort has been made to design efficient algorithms addressing several variants of the problem, including comparison of two strings, approximate pattern identification in a string or calculation of the longest common subsequence that two strings share. We designed an output sensitive algorithm solving the edit distance problem between two strings of lengths n and m respectively in time O((s-|n-m|)min(m,n,s)+m+n) and linear space, where s is the edit distance between the two strings. This worst-case time bound sets the quadratic factor of the algorithm independent of the longest string length and improves existing theoretical bounds for this problem. The implementation of our algorithm excels also in practice, especially in cases where the two strings compared differ significantly in length. Source code of our algorithm is available at http://www.cs.miami.edu/\~dimitris/edit_distance

0807.4479

Yago Ascasibar

Yago Ascasibar

FiEstAS sampling -- a Monte Carlo algorithm for multidimensional numerical integration

18 pages, 3 figures, submitted to Comp. Phys. Comm

10.1016/j.cpc.2008.07.011

astro-ph cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

This paper describes a new algorithm for Monte Carlo integration, based on the Field Estimator for Arbitrary Spaces (FiEstAS). The algorithm is discussed in detail, and its performance is evaluated in the context of Bayesian analysis, with emphasis on multimodal distributions with strong parameter degeneracies. Source code is available upon request.

0807.4582

Ashish Goel

Douglas E. Carroll and Ashish Goel

Lower Bounds for Embedding into Distributions over Excluded Minor Graph Families

cs.DS cs.DM

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

It was shown recently by Fakcharoenphol et al that arbitrary finite metrics can be embedded into distributions over tree metrics with distortion O(log n). It is also known that this bound is tight since there are expander graphs which cannot be embedded into distributions over trees with better than Omega(log n) distortion. We show that this same lower bound holds for embeddings into distributions over any minor excluded family. Given a family of graphs F which excludes minor M where |M|=k, we explicitly construct a family of graphs with treewidth-(k+1) which cannot be embedded into a distribution over F with better than Omega(log n) distortion. Thus, while these minor excluded families of graphs are more expressive than trees, they do not provide asymptotically better approximations in general. An important corollary of this is that graphs of treewidth-k cannot be embedded into distributions over graphs of treewidth-(k-3) with distortion less than Omega(log n). We also extend a result of Alon et al by showing that for any k, planar graphs cannot be embedded into distributions over treewidth-k graphs with better than Omega(log n) distortion.

0807.4626

Assaf Naor

Subhash Khot and Assaf Naor

Approximate kernel clustering

cs.DS cs.CC math.FA

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

In the kernel clustering problem we are given a large $n\times n$ positive semi-definite matrix $A=(a_{ij})$ with $\sum_{i,j=1}^na_{ij}=0$ and a small $k\times k$ positive semi-definite matrix $B=(b_{ij})$. The goal is to find a partition $S_1,...,S_k$ of $\{1,... n\}$ which maximizes the quantity $$ \sum_{i,j=1}^k (\sum_{(i,j)\in S_i\times S_j}a_{ij})b_{ij}. $$ We study the computational complexity of this generic clustering problem which originates in the theory of machine learning. We design a constant factor polynomial time approximation algorithm for this problem, answering a question posed by Song, Smola, Gretton and Borgwardt. In some cases we manage to compute the sharp approximation threshold for this problem assuming the Unique Games Conjecture (UGC). In particular, when $B$ is the $3\times 3$ identity matrix the UGC hardness threshold of this problem is exactly $\frac{16\pi}{27}$. We present and study a geometric conjecture of independent interest which we show would imply that the UGC threshold when $B$ is the $k\times k$ identity matrix is $\frac{8\pi}{9}(1-\frac{1}{k})$ for every $k\ge 3$.

0807.5111

Atish Das Sarma

Atish Das Sarma, Amit Deshpande, Ravi Kannan

Finding Dense Subgraphs in G(n,1/2)

6 pages

cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Finding the largest clique is a notoriously hard problem, even on random graphs. It is known that the clique number of a random graph G(n,1/2) is almost surely either k or k+1, where k = 2log n - 2log(log n) - 1. However, a simple greedy algorithm finds a clique of size only (1+o(1))log n, with high probability, and finding larger cliques -- that of size even (1+ epsilon)log n -- in randomized polynomial time has been a long-standing open problem. In this paper, we study the following generalization: given a random graph G(n,1/2), find the largest subgraph with edge density at least (1-delta). We show that a simple modification of the greedy algorithm finds a subset of 2log n vertices whose induced subgraph has edge density at least 0.951, with high probability. To complement this, we show that almost surely there is no subset of 2.784log n vertices whose induced subgraph has edge density 0.951 or more.

0808.0084

Frederic Magniez

Frederic Magniez, Ashwin Nayak, Peter C. Richter, and Miklos Santha

On the hitting times of quantum versus random walks

quant-ph cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

In this paper we define new Monte Carlo type classical and quantum hitting times, and we prove several relationships among these and the already existing Las Vegas type definitions. In particular, we show that for some marked state the two types of hitting time are of the same order in both the classical and the quantum case. Further, we prove that for any reversible ergodic Markov chain $P$, the quantum hitting time of the quantum analogue of $P$ has the same order as the square root of the classical hitting time of $P$. We also investigate the (im)possibility of achieving a gap greater than quadratic using an alternative quantum walk. Finally, we present new quantum algorithms for the detection and finding problems. The complexities of both algorithms are related to the new, potentially smaller, quantum hitting times. The detection algorithm is based on phase estimation and is particularly simple. The finding algorithm combines a similar phase estimation based procedure with ideas of Tulsi from his recent theorem for the 2D grid. Extending his result, we show that for any state-transitive Markov chain with unique marked state, the quantum hitting time is of the same order for both the detection and finding problems.

0808.0148

James Lee

Punyashloka Biswal, James R. Lee, Satish Rao

Eigenvalue bounds, spectral partitioning, and metrical deformations via flows

Minor revisions

cs.DS cs.CG math.MG math.SP

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

We present a new method for upper bounding the second eigenvalue of the Laplacian of graphs. Our approach uses multi-commodity flows to deform the geometry of the graph; we embed the resulting metric into Euclidean space to recover a bound on the Rayleigh quotient. Using this, we show that every $n$-vertex graph of genus $g$ and maximum degree $d$ satisfies $\lambda_2(G) = O((g+1)^3 d/n)$. This recovers the $O(d/n)$ bound of Spielman and Teng for planar graphs, and compares to Kelner's bound of $O((g+1) poly(d)/n)$, but our proof does not make use of conformal mappings or circle packings. We are thus able to extend this to resolve positively a conjecture of Spielman and Teng, by proving that $\lambda_2(G) = O(d h^6 \log h/n)$ whenever $G$ is $K_h$-minor free. This shows, in particular, that spectral partitioning can be used to recover $O(\sqrt{n})$-sized separators in bounded degree graphs that exclude a fixed minor. We extend this further by obtaining nearly optimal bounds on $\lambda_2$ for graphs which exclude small-depth minors in the sense of Plotkin, Rao, and Smith. Consequently, we show that spectral algorithms find small separators in a general class of geometric graphs. Moreover, while the standard "sweep" algorithm applied to the second eigenvector may fail to find good quotient cuts in graphs of unbounded degree, our approach produces a vector that works for arbitrary graphs. This yields an alternate proof of the result of Alon, Seymour, and Thomas that every excluded-minor family of graphs has $O(\sqrt{n})$-node balanced separators.

0808.0159

Eli Ben-Naim

T. Antal, D. ben-Avraham, E. Ben-Naim, P.L. Krapivsky

Front Propagation with Rejuvenation in Flipping Processes

10 pages, 9 figures, 4 tables

J. Phys. A 41, 465002 (2008)

10.1088/1751-8113/41/46/465002

cond-mat.stat-mech cs.DS math.PR

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

We study a directed flipping process that underlies the performance of the random edge simplex algorithm. In this stochastic process, which takes place on a one-dimensional lattice whose sites may be either occupied or vacant, occupied sites become vacant at a constant rate and simultaneously cause all sites to the right to change their state. This random process exhibits rich phenomenology. First, there is a front, defined by the position of the left-most occupied site, that propagates at a nontrivial velocity. Second, the front involves a depletion zone with an excess of vacant sites. The total excess D_k increases logarithmically, D_k ~ ln k, with the distance k from the front. Third, the front exhibits rejuvenation -- young fronts are vigorous but old fronts are sluggish. We investigate these phenomena using a quasi-static approximation, direct solutions of small systems, and numerical simulations.

0808.0163

Daniel A. Spielman

Joshua Batson, Daniel A. Spielman, Nikhil Srivastava

Twice-Ramanujan Sparsifiers

cs.DS cs.DM

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

We prove that every graph has a spectral sparsifier with a number of edges linear in its number of vertices. As linear-sized spectral sparsifiers of complete graphs are expanders, our sparsifiers of arbitrary graphs can be viewed as generalizations of expander graphs. In particular, we prove that for every $d>1$ and every undirected, weighted graph $G=(V,E,w)$ on $n$ vertices, there exists a weighted graph $H=(V,F,\tilde{w})$ with at most $\ceil{d(n-1)}$ edges such that for every $x \in \R^{V}$, \[ x^{T}L_{G}x \leq x^{T}L_{H}x \leq (\frac{d+1+2\sqrt{d}}{d+1-2\sqrt{d}})\cdot x^{T}L_{G}x \] where $L_{G}$ and $L_{H}$ are the Laplacian matrices of $G$ and $H$, respectively. Thus, $H$ approximates $G$ spectrally at least as well as a Ramanujan expander with $dn/2$ edges approximates the complete graph. We give an elementary deterministic polynomial time algorithm for constructing $H$.

0808.0298

Dmitrii V. Pasechnik

Edith Elkind and Dmitrii V. Pasechnik

Computing the nucleolus of weighted voting games

LaTeX, 12 pages, COMSOC-2008 workshop

Proceedings of SODA 2009, pp. 327-335

10.1145/1496770.1496807

cs.GT cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Weighted voting games (WVG) are coalitional games in which an agent's contribution to a coalition is given by his it weight, and a coalition wins if its total weight meets or exceeds a given quota. These games model decision-making in political bodies as well as collaboration and surplus division in multiagent domains. The computational complexity of various solution concepts for weighted voting games received a lot of attention in recent years. In particular, Elkind et al.(2007) studied the complexity of stability-related solution concepts in WVGs, namely, of the core, the least core, and the nucleolus. While they have completely characterized the algorithmic complexity of the core and the least core, for the nucleolus they have only provided an NP-hardness result. In this paper, we solve an open problem posed by Elkind et al. by showing that the nucleolus of WVGs, and, more generally, k-vector weighted voting games with fixed k, can be computed in pseudopolynomial time, i.e., there exists an algorithm that correctly computes the nucleolus and runs in time polynomial in the number of players and the maximum weight. In doing so, we propose a general framework for computing the nucleolus, which may be applicable to a wider of class of games.

0808.0540

Paul Tarau

Paul Tarau

Executable Set Theory and Arithmetic Encodings in Prolog

Unpublished draft

cs.LO cs.DM cs.DS cs.MS cs.SC

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

The paper is organized as a self-contained literate Prolog program that implements elements of an executable finite set theory with focus on combinatorial generation and arithmetic encodings. The complete Prolog code is available at http://logic.csci.unt.edu/tarau/research/2008/pHFS.zip . First, ranking and unranking functions for some "mathematically elegant" data types in the universe of Hereditarily Finite Sets with Urelements are provided, resulting in arithmetic encodings for powersets, hypergraphs, ordinals and choice functions. After implementing a digraph representation of Hereditarily Finite Sets we define {\em decoration functions} that can recover well-founded sets from encodings of their associated acyclic digraphs. We conclude with an encoding of arbitrary digraphs and discuss a concept of duality induced by the set membership relation. In the process, we uncover the surprising possibility of internally sharing isomorphic objects, independently of their language level types and meanings.

0808.0753

Paul Tarau

Paul Tarau

Ranking Catamorphisms and Unranking Anamorphisms on Hereditarily Finite Datatypes

unpublished draft

cs.SC cs.DM cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Using specializations of unfold and fold on a generic tree data type we derive unranking and ranking functions providing natural number encodings for various Hereditarily Finite datatypes. In this context, we interpret unranking operations as instances of a generic anamorphism and ranking operations as instances of the corresponding catamorphism. Starting with Ackerman's Encoding from Hereditarily Finite Sets to Natural Numbers we define pairings and tuple encodings that provide building blocks for a theory of Hereditarily Finite Functions. The more difficult problem of ranking and unranking Hereditarily Finite Permutations is then tackled using Lehmer codes and factoradics. The self-contained source code of the paper, as generated from a literate Haskell program, is available at \url{http://logic.csci.unt.edu/tarau/research/2008/fFUN.zip}. Keywords: ranking/unranking, pairing/tupling functions, Ackermann encoding, hereditarily finite sets, hereditarily finite functions, permutations and factoradics, computational mathematics, Haskell data representations

0808.0754

Paul Tarau

Paul Tarau

A Functional Hitchhiker's Guide to Hereditarily Finite Sets, Ackermann Encodings and Pairing Functions

unpublished draft

cs.MS cs.DM cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

The paper is organized as a self-contained literate Haskell program that implements elements of an executable finite set theory with focus on combinatorial generation and arithmetic encodings. The code, tested under GHC 6.6.1, is available at http://logic.csci.unt.edu/tarau/research/2008/fSET.zip . We introduce ranking and unranking functions generalizing Ackermann's encoding to the universe of Hereditarily Finite Sets with Urelements. Then we build a lazy enumerator for Hereditarily Finite Sets with Urelements that matches the unranking function provided by the inverse of Ackermann's encoding and we describe functors between them resulting in arithmetic encodings for powersets, hypergraphs, ordinals and choice functions. After implementing a digraph representation of Hereditarily Finite Sets we define {\em decoration functions} that can recover well-founded sets from encodings of their associated acyclic digraphs. We conclude with an encoding of arbitrary digraphs and discuss a concept of duality induced by the set membership relation. Keywords: hereditarily finite sets, ranking and unranking functions, executable set theory, arithmetic encodings, Haskell data representations, functional programming and computational mathematics

0808.0760

Paul Tarau

Paul Tarau

Declarative Combinatorics: Boolean Functions, Circuit Synthesis and BDDs in Haskell

unpublished draft

cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

We describe Haskell implementations of interesting combinatorial generation algorithms with focus on boolean functions and logic circuit representations. First, a complete exact combinational logic circuit synthesizer is described as a combination of catamorphisms and anamorphisms. Using pairing and unpairing functions on natural number representations of truth tables, we derive an encoding for Binary Decision Diagrams (BDDs) with the unique property that its boolean evaluation faithfully mimics its structural conversion to a a natural number through recursive application of a matching pairing function. We then use this result to derive ranking and unranking functions for BDDs and reduced BDDs. Finally, a generalization of the encoding techniques to Multi-Terminal BDDs is provided. The paper is organized as a self-contained literate Haskell program, available at http://logic.csci.unt.edu/tarau/research/2008/fBDD.zip . Keywords: exact combinational logic synthesis, binary decision diagrams, encodings of boolean functions, pairing/unpairing functions, ranking/unranking functions for BDDs and MTBDDs, declarative combinatorics in Haskell

0808.0980

Gregory Gutin

Peter Dankelmann, Gregory Gutin, Eun Jung Kim

On Complexity of Minimum Leaf Out-branching Problem

cs.DS cs.CC

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Given a digraph $D$, the Minimum Leaf Out-Branching problem (MinLOB) is the problem of finding in $D$ an out-branching with the minimum possible number of leaves, i.e., vertices of out-degree 0. Gutin, Razgon and Kim (2008) proved that MinLOB is polynomial time solvable for acyclic digraphs which are exactly the digraphs of directed path-width (DAG-width, directed tree-width, respectively) 0. We investigate how much one can extend this polynomiality result. We prove that already for digraphs of directed path-width (directed tree-width, DAG-width, respectively) 1, MinLOB is NP-hard. On the other hand, we show that for digraphs of restricted directed tree-width (directed path-width, DAG-width, respectively) and a fixed integer $k$, the problem of checking whether there is an out-branching with at most $k$ leaves is polynomial time solvable.

0808.1108

Nicolas Bock

Nicolas Bock and Emanuel H. Rubensson and Pawe{\l} Sa{\l}ek and Anders M. N. Niklasson and Matt Challacombe

Cache oblivious storage and access heuristics for blocked matrix-matrix multiplication

Fixed typos

cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

We investigate effects of ordering in blocked matrix--matrix multiplication. We find that submatrices do not have to be stored contiguously in memory to achieve near optimal performance. Instead it is the choice of execution order of the submatrix multiplications that leads to a speedup of up to four times for small block sizes. This is in contrast to results for single matrix elements showing that contiguous memory allocation quickly becomes irrelevant as the blocksize increases.

0808.1128

Mihai Patrascu

Timothy M. Chan and Mihai Patrascu and Liam Roditty

Dynamic Connectivity: Connecting to Networks and Geometry

Full version of a paper to appear in FOCS 2008

cs.DS cs.CG

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Dynamic connectivity is a well-studied problem, but so far the most compelling progress has been confined to the edge-update model: maintain an understanding of connectivity in an undirected graph, subject to edge insertions and deletions. In this paper, we study two more challenging, yet equally fundamental problems. Subgraph connectivity asks to maintain an understanding of connectivity under vertex updates: updates can turn vertices on and off, and queries refer to the subgraph induced by "on" vertices. (For instance, this is closer to applications in networks of routers, where node faults may occur.) We describe a data structure supporting vertex updates in O (m^{2/3}) amortized time, where m denotes the number of edges in the graph. This greatly improves over the previous result [Chan, STOC'02], which required fast matrix multiplication and had an update time of O(m^0.94). The new data structure is also simpler. Geometric connectivity asks to maintain a dynamic set of n geometric objects, and query connectivity in their intersection graph. (For instance, the intersection graph of balls describes connectivity in a network of sensors with bounded transmission radius.) Previously, nontrivial fully dynamic results were known only for special cases like axis-parallel line segments and rectangles. We provide similarly improved update times, O (n^{2/3}), for these special cases. Moreover, we show how to obtain sublinear update bounds for virtually all families of geometric objects which allow sublinear-time range queries, such as arbitrary 2D line segments, d-dimensional simplices, and d-dimensional balls.

0808.1246

Mugurel Ionut Andreica

Mugurel Ionut Andreica, Romulus Andreica, Angela Andreica

Minimum Dissatisfaction Personnel Scheduling

Some of the algorithmic techniques presented in this paper were later used by the first author for developing solutions to several algorithmic contest tasks (see the attached zip archive for some examples)

ARA Congress, Boston : United States (2008)

cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

In this paper we consider two problems regarding the scheduling of available personnel in order to perform a given quantity of work, which can be arbitrarily decomposed into a sequence of activities. We are interested in schedules which minimize the overall dissatisfaction, where each employee's dissatisfaction is modeled as a time-dependent linear function. For the two situations considered we provide a detailed mathematical analysis, as well as efficient algorithms for determining optimal schedules.

0808.1671

Shuchi Chawla

Shuchi Chawla and Jason Hartline and Robert Kleinberg

Algorithmic Pricing via Virtual Valuations

A preliminary version of this work appeared at the ACM EC'07 conference. The current version contains improved results and simpler algorithms

cs.GT cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Algorithmic pricing is the computational problem that sellers (e.g., in supermarkets) face when trying to set prices for their items to maximize their profit in the presence of a known demand. Guruswami et al. (2005) propose this problem and give logarithmic approximations (in the number of consumers) when each consumer's values for bundles are known precisely. Subsequently several versions of the problem have been shown to have poly-logarithmic inapproximability. This problem has direct ties to the important open question of better understanding the Bayesian optimal mechanism in multi-parameter settings; however, logarithmic approximations are inadequate for this purpose. It is therefore of vital interest to consider special cases where constant approximations are possible. We consider the unit-demand variant of this problem. Here a consumer has a valuation for each different item and their value for a set of items is simply the maximum value they have for any item in the set. We assume that the preferences of the consumers are drawn from a distribution, the standard assumption in economics; furthermore, the setting of a specific set of customers with known preferences, which is employed in all prior work in algorithmic pricing, is a special case of this general problem, where there is a discrete Bayesian distribution for preferences specified by picking one consumer uniformly from the given set of consumers. Our work complements these existing works by considering the case where the consumer's valuations for the different items are independent random variables. Our main result is a constant approximation that makes use of an interesting connection between this problem and the concept of virtual valuations from the single-parameter Bayesian optimal mechanism design literature.

0808.1766

Ping Li

Ping Li

The Optimal Quantile Estimator for Compressed Counting

cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Compressed Counting (CC) was recently proposed for very efficiently computing the (approximate) $\alpha$th frequency moments of data streams, where $0<\alpha <= 2$. Several estimators were reported including the geometric mean estimator, the harmonic mean estimator, the optimal power estimator, etc. The geometric mean estimator is particularly interesting for theoretical purposes. For example, when $\alpha -> 1$, the complexity of CC (using the geometric mean estimator) is $O(1/\epsilon)$, breaking the well-known large-deviation bound $O(1/\epsilon^2)$. The case $\alpha\approx 1$ has important applications, for example, computing entropy of data streams. For practical purposes, this study proposes the optimal quantile estimator. Compared with previous estimators, this estimator is computationally more efficient and is also more accurate when $\alpha> 1$.

0808.1771

Ping Li

Ping Li

A Very Efficient Scheme for Estimating Entropy of Data Streams Using Compressed Counting

cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Compressed Counting (CC)} was recently proposed for approximating the $\alpha$th frequency moments of data streams, for $0<\alpha \leq 2$. Under the relaxed strict-Turnstile model, CC dramatically improves the standard algorithm based on symmetric stable random projections}, especially as $\alpha\to 1$. A direct application of CC is to estimate the entropy, which is an important summary statistic in Web/network measurement and often serves a crucial "feature" for data mining. The R\'enyi entropy and the Tsallis entropy are functions of the $\alpha$th frequency moments; and both approach the Shannon entropy as $\alpha\to 1$. A recent theoretical work suggested using the $\alpha$th frequency moment to approximate the Shannon entropy with $\alpha=1+\delta$ and very small $|\delta|$ (e.g., $<10^{-4}$). In this study, we experiment using CC to estimate frequency moments, R\'enyi entropy, Tsallis entropy, and Shannon entropy, on real Web crawl data. We demonstrate the variance-bias trade-off in estimating Shannon entropy and provide practical recommendations. In particular, our experiments enable us to draw some important conclusions: (1) As $\alpha\to 1$, CC dramatically improves {\em symmetric stable random projections} in estimating frequency moments, R\'enyi entropy, Tsallis entropy, and Shannon entropy. The improvements appear to approach "infinity." (2) Using {\em symmetric stable random projections} and $\alpha = 1+\delta$ with very small $|\delta|$ does not provide a practical algorithm because the required sample size is enormous.

0808.1787

Arnab Bhattacharyya

Arnab Bhattacharyya, Elena Grigorescu, Kyomin Jung, Sofya Raskhodnikova, David P. Woodruff

Transitive-Closure Spanners

Extended abstract with appendices

cs.DS cs.CC

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Given a directed graph G = (V,E) and an integer k>=1, a k-transitive-closure-spanner (k-TC-spanner) of G is a directed graph H = (V, E_H) that has (1) the same transitive-closure as G and (2) diameter at most k. These spanners were implicitly studied in access control, data structures, and property testing, and properties of these spanners have been rediscovered over the span of 20 years. The main goal in each of these applications is to obtain the sparsest k-TC-spanners. We bring these diverse areas under the unifying framework of TC-spanners. We initiate the study of approximability of the size of the sparsest k-TC-spanner for a given directed graph. We completely resolve the approximability of 2-TC-spanners, showing that it is Theta(log n) unless P = NP. For k>2, we present a polynomial-time algorithm that finds a k-TC-spanner with size within O((n log n)^{1-1/k}) of the optimum. Our algorithmic techniques also yield algorithms with the best-known approximation ratio for well-studied problems on directed spanners when k>3: DIRECTED k-SPANNER, CLIENT/SERVER DIRECTED k-SPANNER, and k-DIAMETER SPANNING SUBGRAPH. For constant k>=3, we show that the size of the sparsest k-TC-spanner is hard to approximate with 2^{log^{1-eps} n} ratio unless NP \subseteq DTIME(n^{polylog n}}). Finally, we study the size of the sparsest k-TC-spanners for H-minor-free graph families. Combining our constructions with our insight that 2-TC-spanners can be used for designing property testers, we obtain a monotonicity tester with O(log^2 n /eps) queries for any poset whose transitive reduction is an H-minor free digraph, improving the Theta(sqrt(n) log n/eps)-queries required of the tester due to Fischer et al (2002).

0808.2017

Ittai Abraham

Ittai Abraham, Yair Bartal, and Ofer Neiman

Nearly Tight Low Stretch Spanning Trees

cs.DS cs.DM

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

We prove that any graph $G$ with $n$ points has a distribution $\mathcal{T}$ over spanning trees such that for any edge $(u,v)$ the expected stretch $E_{T \sim \mathcal{T}}[d_T(u,v)/d_G(u,v)]$ is bounded by $\tilde{O}(\log n)$. Our result is obtained via a new approach of building ``highways'' between portals and a new strong diameter probabilistic decomposition theorem.

0808.2222

Andrew McGregor

Alexandr Andoni, Andrew McGregor, Krzysztof Onak, Rina Panigrahy

Better Bounds for Frequency Moments in Random-Order Streams

4 pages

cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Estimating frequency moments of data streams is a very well studied problem and tight bounds are known on the amount of space that is necessary and sufficient when the stream is adversarially ordered. Recently, motivated by various practical considerations and applications in learning and statistics, there has been growing interest into studying streams that are randomly ordered. In the paper we improve the previous lower bounds on the space required to estimate the frequency moments of a randomly ordered streams.

0808.2953

Paul Tarau

Paul Tarau

Declarative Combinatorics: Isomorphisms, Hylomorphisms and Hereditarily Finite Data Types in Haskell

unpublished draft, revision 3, added various new encodings, with focus on primes and multisets, now 104 pages

cs.PL cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

This paper is an exploration in a functional programming framework of {\em isomorphisms} between elementary data types (natural numbers, sets, multisets, finite functions, permutations binary decision diagrams, graphs, hypergraphs, parenthesis languages, dyadic rationals, primes, DNA sequences etc.) and their extension to hereditarily finite universes through {\em hylomorphisms} derived from {\em ranking/unranking} and {\em pairing/unpairing} operations. An embedded higher order {\em combinator language} provides any-to-any encodings automatically. Besides applications to experimental mathematics, a few examples of ``free algorithms'' obtained by transferring operations between data types are shown. Other applications range from stream iterators on combinatorial objects to self-delimiting codes, succinct data representations and generation of random instances. The paper covers 59 data types and, through the use of the embedded combinator language, provides 3540 distinct bijective transformations between them. The self-contained source code of the paper, as generated from a literate Haskell program, is available at \url{http://logic.csci.unt.edu/tarau/research/2008/fISO.zip}. {\bf Keywords}: Haskell data representations, data type isomorphisms, declarative combinatorics, computational mathematics, Ackermann encoding, G\"{o}del numberings, arithmetization, ranking/unranking, hereditarily finite sets, functions and permutations, encodings of binary decision diagrams, dyadic rationals, DNA encodings

0808.3197

Ming-Zhe Chen

Ming-Zhe Chen

On the Monotonicity of Work Function in k-Server Conjecture

3 pages, corrected typos

cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

This paper presents a mistake in work function algorithm of k-server conjecture. That is, the monotonicity of the work function is not always true.

0808.3331

Gregory Karagiorgos

Gregory Karagiorgos, Dimitrios Poulakis

Efficient algorithms for the basis of finite Abelian groups

11 pages

cs.DS cs.CC

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Let $G$ be a finite abelian group $G$ with $N$ elements. In this paper we give a O(N) time algorithm for computing a basis of $G$. Furthermore, we obtain an algorithm for computing a basis from a generating system of $G$ with $M$ elements having time complexity $O(M\sum_{p|N} e(p)\lceil p^{1/2}\rceil^{\mu(p)})$, where $p$ runs over all the prime divisors of $N$, and $p^{e(p)}$, $\mu(p)$ are the exponent and the number of cyclic groups which are direct factors of the $p$-primary component of $G$, respectively. In case where $G$ is a cyclic group having a generating system with $M$ elements, a $O(MN^{\epsilon})$ time algorithm for the computation of a basis of $G$ is obtained.

0808.3881

Paul Bonsma

Paul Bonsma, Felix Breuer

Counting Hexagonal Patches and Independent Sets in Circle Graphs

cs.DM cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

A hexagonal patch is a plane graph in which inner faces have length 6, inner vertices have degree 3, and boundary vertices have degree 2 or 3. We consider the following counting problem: given a sequence of twos and threes, how many hexagonal patches exist with this degree sequence along the outer face? This problem is motivated by the study of benzenoid hydrocarbons and fullerenes in computational chemistry. We give the first polynomial time algorithm for this problem. We show that it can be reduced to counting maximum independent sets in circle graphs, and give a simple and fast algorithm for this problem.

0808.4134

Daniel A. Spielman

Daniel A. Spielman and Shang-Hua Teng

Spectral Sparsification of Graphs

This revision addresses comments of the referees. In particular, we have completely re-written the proof of the main graph partitioning theorem in section 8

cs.DS cs.DM

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

We introduce a new notion of graph sparsificaiton based on spectral similarity of graph Laplacians: spectral sparsification requires that the Laplacian quadratic form of the sparsifier approximate that of the original. This is equivalent to saying that the Laplacian of the sparsifier is a good preconditioner for the Laplacian of the original. We prove that every graph has a spectral sparsifier of nearly linear size. Moreover, we present an algorithm that produces spectral sparsifiers in time $\softO{m}$, where $m$ is the number of edges in the original graph. This construction is a key component of a nearly-linear time algorithm for solving linear equations in diagonally-dominant matrcies. Our sparsification algorithm makes use of a nearly-linear time algorithm for graph partitioning that satisfies a strong guarantee: if the partition it outputs is very unbalanced, then the larger part is contained in a subgraph of high conductance.

0809.0188

Marek Karpinski

Piotr Berman, Bhaskar DasGupta and Marek Karpinski

Approximating Transitivity in Directed Networks

cs.CC cs.DM cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

We study the problem of computing a minimum equivalent digraph (also known as the problem of computing a strong transitive reduction) and its maximum objective function variant, with two types of extensions. First, we allow to declare a set $D\subset E$ and require that a valid solution $A$ satisfies $D\subset A$ (it is sometimes called transitive reduction problem). In the second extension (called $p$-ary transitive reduction), we have integer edge labeling and we view two paths as equivalent if they have the same beginning, ending and the sum of labels modulo $p$. A solution $A\subseteq E$ is valid if it gives an equivalent path for every original path. For all problems we establish the following: polynomial time minimization of $|A|$ within ratio 1.5, maximization of $|E-A|$ within ratio 2, MAX-SNP hardness even of the length of simple cycles is limited to 5. Furthermore, we believe that the combinatorial technique behind the approximation algorithm for the minimization version might be of interest to other graph connectivity problems as well.

0809.0400

Xuan Cai

Xuan Cai

Canonical Coin Systems for Change-Making Problems

7 pages

cs.DS cs.DM

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

The Change-Making Problem is to represent a given value with the fewest coins under a given coin system. As a variation of the knapsack problem, it is known to be NP-hard. Nevertheless, in most real money systems, the greedy algorithm yields optimal solutions. In this paper, we study what type of coin systems that guarantee the optimality of the greedy algorithm. We provide new proofs for a sufficient and necessary condition for the so-called \emph{canonical} coin systems with four or five types of coins, and a sufficient condition for non-canonical coin systems, respectively. Moreover, we present an $O(m^2)$ algorithm that decides whether a tight coin system is canonical.

0809.0460

Shipra Agrawal

Shipra Agrawal, Amin Saberi, Yinyu Ye

Stochastic Combinatorial Optimization under Probabilistic Constraints

cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

In this paper, we present approximation algorithms for combinatorial optimization problems under probabilistic constraints. Specifically, we focus on stochastic variants of two important combinatorial optimization problems: the k-center problem and the set cover problem, with uncertainty characterized by a probability distribution over set of points or elements to be covered. We consider these problems under adaptive and non-adaptive settings, and present efficient approximation algorithms for the case when underlying distribution is a product distribution. In contrast to the expected cost model prevalent in stochastic optimization literature, our problem definitions support restrictions on the probability distributions of the total costs, via incorporating constraints that bound the probability with which the incurred costs may exceed a given threshold.

0809.0949

Michael Baer

Michael B. Baer

Efficient Implementation of the Generalized Tunstall Code Generation Algorithm

5 pages, 5 figures, accepted to ISIT 2009

cs.IT cs.DS math.IT

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

A method is presented for constructing a Tunstall code that is linear time in the number of output items. This is an improvement on the state of the art for non-Bernoulli sources, including Markov sources, which require a (suboptimal) generalization of Tunstall's algorithm proposed by Savari and analytically examined by Tabus and Rissanen. In general, if n is the total number of output leaves across all Tunstall trees, s is the number of trees (states), and D is the number of leaves of each internal node, then this method takes O((1+(log s)/D) n) time and O(n) space.

0809.1171

Cheng-Wei Luo

Cheng-Wei Luo, Hsiao-Fei Liu, Peng-An Chen, and Kun-Mao Chao

Minkowski Sum Selection and Finding

23 pages, 10 figures, accepted by ISAAC 2008

cs.DS cs.CG

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

For the \textsc{Minkowski Sum Selection} problem with linear objective functions, we obtain the following results: (1) optimal $O(n\log n)$ time algorithms for $\lambda=1$; (2) $O(n\log^2 n)$ time deterministic algorithms and expected $O(n\log n)$ time randomized algorithms for any fixed $\lambda>1$. For the \textsc{Minkowski Sum Finding} problem with linear objective functions or objective functions of the form $f(x,y)=\frac{by}{ax}$, we construct optimal $O(n\log n)$ time algorithms for any fixed $\lambda\geq 1$.

0809.1681

Rafael Laufer

Rafael Laufer and Leonard Kleinrock

Multirate Anypath Routing in Wireless Mesh Networks

13 pages, 8 figures

IEEE INFOCOM 2009

10.1109/INFCOM.2009.5061904

UCLA-CSD-TR080025

cs.NI cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

In this paper, we present a new routing paradigm that generalizes opportunistic routing in wireless mesh networks. In multirate anypath routing, each node uses both a set of next hops and a selected transmission rate to reach a destination. Using this rate, a packet is broadcast to the nodes in the set and one of them forwards the packet on to the destination. To date, there is no theory capable of jointly optimizing both the set of next hops and the transmission rate used by each node. We bridge this gap by introducing a polynomial-time algorithm to this problem and provide the proof of its optimality. The proposed algorithm runs in the same running time as regular shortest-path algorithms and is therefore suitable for deployment in link-state routing protocols. We conducted experiments in a 802.11b testbed network, and our results show that multirate anypath routing performs on average 80% and up to 6.4 times better than anypath routing with a fixed rate of 11 Mbps. If the rate is fixed at 1 Mbps instead, performance improves by up to one order of magnitude.

0809.1715

Bodo Manthey

Bodo Manthey and Heiko R\"oglin

Improved Smoothed Analysis of the k-Means Method

To be presented at the 20th ACM-SIAM Symposium on Discrete Algorithms (SODA 2009)

cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

The k-means method is a widely used clustering algorithm. One of its distinguished features is its speed in practice. Its worst-case running-time, however, is exponential, leaving a gap between practical and theoretical performance. Arthur and Vassilvitskii (FOCS 2006) aimed at closing this gap, and they proved a bound of $\poly(n^k, \sigma^{-1})$ on the smoothed running-time of the k-means method, where n is the number of data points and $\sigma$ is the standard deviation of the Gaussian perturbation. This bound, though better than the worst-case bound, is still much larger than the running-time observed in practice. We improve the smoothed analysis of the k-means method by showing two upper bounds on the expected running-time of k-means. First, we prove that the expected running-time is bounded by a polynomial in $n^{\sqrt k}$ and $\sigma^{-1}$. Second, we prove an upper bound of $k^{kd} \cdot \poly(n, \sigma^{-1})$, where d is the dimension of the data space. The polynomial is independent of k and d, and we obtain a polynomial bound for the expected running-time for $k, d \in O(\sqrt{\log n/\log \log n})$. Finally, we show that k-means runs in smoothed polynomial time for one-dimensional instances.

0809.1810

Felipe Cruz

Felipe A. Cruz, L. A. Barba

Characterization of the errors of the FMM in particle simulations

34 pages, 38 images

Int. J. Num. Meth. Engrg., 79(13):1577-1604 (2009)

10.1002/nme.2611

cs.DS physics.comp-ph

http://creativecommons.org/licenses/by/3.0/

The Fast Multipole Method (FMM) offers an acceleration for pairwise interaction calculation, known as $N$-body problems, from $\mathcal{O}(N^2)$ to $\mathcal{O}(N)$ with $N$ particles. This has brought dramatic increase in the capability of particle simulations in many application areas, such as electrostatics, particle formulations of fluid mechanics, and others. Although the literature on the subject provides theoretical error bounds for the FMM approximation, there are not many reports of the measured errors in a suite of computational experiments. We have performed such an experimental investigation, and summarized the results of about 1000 calculations using the FMM algorithm, to characterize the accuracy of the method in relation with the different parameters available to the user. In addition to the more standard diagnostic of the maximum error, we supply illustrations of the spatial distribution of the errors, which offers visual evidence of all the contributing factors to the overall approximation accuracy: multipole expansion, local expansion, hierarchical spatial decomposition (interaction lists, local domain, far domain). This presentation is a contribution to any researcher wishing to incorporate the FMM acceleration to their application code, as it aids in understanding where accuracy is gained or compromised.

0809.1895

Benjamin Birnbaum

Yossi Azar, Benjamin Birnbaum, Anna R. Karlin, and C. Thach Nguyen

Thinking Twice about Second-Price Ad Auctions

cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Recent work has addressed the algorithmic problem of allocating advertisement space for keywords in sponsored search auctions so as to maximize revenue, most of which assume that pricing is done via a first-price auction. This does not realistically model the Generalized Second Price (GSP) auction used in practice, in which bidders pay the next-highest bid for keywords that they are allocated. Towards the goal of more realistically modeling these auctions, we introduce the Second-Price Ad Auctions problem, in which bidders' payments are determined by the GSP mechanism. We show that the complexity of the Second-Price Ad Auctions problem is quite different than that of the more studied First-Price Ad Auctions problem. First, unlike the first-price variant, for which small constant-factor approximations are known, it is NP-hard to approximate the Second-Price Ad Auctions problem to any non-trivial factor, even when the bids are small compared to the budgets. Second, this discrepancy extends even to the 0-1 special case that we call the Second-Price Matching problem (2PM). Offline 2PM is APX-hard, and for online 2PM there is no deterministic algorithm achieving a non-trivial competitive ratio and no randomized algorithm achieving a competitive ratio better than 2. This contrasts with the results for the analogous special case in the first-price model, the standard bipartite matching problem, which is solvable in polynomial time and which has deterministic and randomized online algorithms achieving better competitive ratios. On the positive side, we provide a 2-approximation for offline 2PM and a 5.083-competitive randomized algorithm for online 2PM. The latter result makes use of a new generalization of a result on the performance of the "Ranking" algorithm for online bipartite matching.

0809.1902

Manor Mendel

Manor Mendel, Chaya Schwob

Fast C-K-R Partitions of Sparse Graphs

15 pages, title changed, a small error in the running time was fixed. Many errors in English were eliminated

Chicago J. Theoretical Comp. Sci., 2009(2), 2009

cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

We present fast algorithms for constructing probabilistic embeddings and approximate distance oracles in sparse graphs. The main ingredient is a fast algorithm for sampling the probabilistic partitions of Calinescu, Karloff, and Rabani in sparse graphs.

0809.1906

Shiva Kintali

Shiva Kintali

Betweenness Centrality : Algorithms and Lower Bounds

cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

One of the most fundamental problems in large scale network analysis is to determine the importance of a particular node in a network. Betweenness centrality is the most widely used metric to measure the importance of a node in a network. In this paper, we present a randomized parallel algorithm and an algebraic method for computing betweenness centrality of all nodes in a network. We prove that any path-comparison based algorithm cannot compute betweenness in less than O(nm) time.

0809.2075

Jittat Fakcharoenphol

Jittat Fakcharoenphol, Boonserm Kijsirikul

Low congestion online routing and an improved mistake bound for online prediction of graph labeling

5 pages

cs.DS cs.DM cs.LG

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

In this paper, we show a connection between a certain online low-congestion routing problem and an online prediction of graph labeling. More specifically, we prove that if there exists a routing scheme that guarantees a congestion of $\alpha$ on any edge, there exists an online prediction algorithm with mistake bound $\alpha$ times the cut size, which is the size of the cut induced by the label partitioning of graph vertices. With previous known bound of $O(\log n)$ for $\alpha$ for the routing problem on trees with $n$ vertices, we obtain an improved prediction algorithm for graphs with high effective resistance. In contrast to previous approaches that move the graph problem into problems in vector space using graph Laplacian and rely on the analysis of the perceptron algorithm, our proof are purely combinatorial. Further more, our approach directly generalizes to the case where labels are not binary.

0809.2097

Hsiao-Fei Liu

Hsiao-Fei Liu, Peng-An Chen, and Kun-Mao Chao

Algorithms for Locating Constrained Optimal Intervals

An earlier version of the second part of this work appeared in Proceedings of the 18th International Symposium on Algorithms and Computation, Japan, 2007

cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

In this work, we obtain the following new results. 1. Given a sequence $D=((h_1,s_1), (h_2,s_2) ..., (h_n,s_n))$ of number pairs, where $s_i>0$ for all $i$, and a number $L_h$, we propose an O(n)-time algorithm for finding an index interval $[i,j]$ that maximizes $\frac{\sum_{k=i}^{j} h_k}{\sum_{k=i}^{j} s_k}$ subject to $\sum_{k=i}^{j} h_k \geq L_h$. 2. Given a sequence $D=((h_1,s_1), (h_2,s_2) ..., (h_n,s_n))$ of number pairs, where $s_i=1$ for all $i$, and an integer $L_s$ with $1\leq L_s\leq n$, we propose an $O(n\frac{T(L_s^{1/2})}{L_s^{1/2}})$-time algorithm for finding an index interval $[i,j]$ that maximizes $\frac{\sum_{k=i}^{j} h_k}{\sqrt{\sum_{k=i}^{j} s_k}}$ subject to $\sum_{k=i}^{j} s_k \geq L_s$, where $T(n')$ is the time required to solve the all-pairs shortest paths problem on a graph of $n'$ nodes. By the latest result of Chan \cite{Chan}, $T(n')=O(n'^3 \frac{(\log\log n')^3}{(\log n')^2})$, so our algorithm runs in subquadratic time $O(nL_s\frac{(\log\log L_s)^3}{(\log L_s)^2})$.

0809.2489

Petteri Kaski

Andreas Bj\"orklund, Thore Husfeldt, Petteri Kaski, Mikko Koivisto

The fast intersection transform with applications to counting paths

11 pages

cs.DS cs.DM

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

We present an algorithm for evaluating a linear ``intersection transform'' of a function defined on the lattice of subsets of an $n$-element set. In particular, the algorithm constructs an arithmetic circuit for evaluating the transform in ``down-closure time'' relative to the support of the function and the evaluation domain. As an application, we develop an algorithm that, given as input a digraph with $n$ vertices and bounded integer weights at the edges, counts paths by weight and given length $0\leq\ell\leq n-1$ in time $O^*(\exp(n\cdot H(\ell/(2n))))$, where $H(p)=-p\log p-(1-p)\log(1-p)$, and the notation $O^*(\cdot)$ suppresses a factor polynomial in $n$.

0809.2554

Kanat Tangwongsan

Anupam Gupta and Kanat Tangwongsan

Simpler Analyses of Local Search Algorithms for Facility Location

cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

We study local search algorithms for metric instances of facility location problems: the uncapacitated facility location problem (UFL), as well as uncapacitated versions of the $k$-median, $k$-center and $k$-means problems. All these problems admit natural local search heuristics: for example, in the UFL problem the natural moves are to open a new facility, close an existing facility, and to swap a closed facility for an open one; in $k$-medians, we are allowed only swap moves. The local-search algorithm for $k$-median was analyzed by Arya et al. (SIAM J. Comput. 33(3):544-562, 2004), who used a clever ``coupling'' argument to show that local optima had cost at most constant times the global optimum. They also used this argument to show that the local search algorithm for UFL was 3-approximation; their techniques have since been applied to other facility location problems. In this paper, we give a proof of the $k$-median result which avoids this coupling argument. These arguments can be used in other settings where the Arya et al. arguments have been used. We also show that for the problem of opening $k$ facilities $F$ to minimize the objective function $\Phi_p(F) = \big(\sum_{j \in V} d(j, F)^p\big)^{1/p}$, the natural swap-based local-search algorithm is a $\Theta(p)$-approximation. This implies constant-factor approximations for $k$-medians (when $p=1$), and $k$-means (when $p = 2$), and an $O(\log n)$-approximation algorithm for the $k$-center problem (which is essentially $p = \log n$).

0809.2858

Christophe Paul

Stephane Bessy and Christophe Paul and Anthony Perez

Polynomial kernels for 3-leaf power graph modification problems

Submitted

10.1007/978-3-642-10217-2_10

cs.DM cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

A graph G=(V,E) is a 3-leaf power iff there exists a tree T whose leaves are V and such that (u,v) is an edge iff u and v are at distance at most 3 in T. The 3-leaf power graph edge modification problems, i.e. edition (also known as the closest 3-leaf power), completion and edge-deletion, are FTP when parameterized by the size of the edge set modification. However polynomial kernel was known for none of these three problems. For each of them, we provide cubic kernels that can be computed in linear time for each of these problems. We thereby answer an open problem first mentioned by Dom, Guo, Huffner and Niedermeier (2005).

0809.2957

Sergi Elizalde

Sergi Elizalde, Peter Winkler

Sorting by Placement and Shift

13 pages, 4 figures, Proceedings of SODA 2009

math.CO cs.DM cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

In sorting situations where the final destination of each item is known, it is natural to repeatedly choose items and place them where they belong, allowing the intervening items to shift by one to make room. (In fact, a special case of this algorithm is commonly used to hand-sort files.) However, it is not obvious that this algorithm necessarily terminates. We show that in fact the algorithm terminates after at most $2^{n-1}-1$ steps in the worst case (confirming a conjecture of L. Larson), and that there are super-exponentially many permutations for which this exact bound can be achieved. The proof involves a curious symmetrical binary representation.

0809.2970

Raphael Yuster

Raphael Yuster

Single source shortest paths in $H$-minor free graphs

cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

We present an algorithm for the Single Source Shortest Paths (SSSP) problem in \emph{$H$-minor free} graphs. For every fixed $H$, if $G$ is a graph with $n$ vertices having integer edge lengths and $s$ is a designated source vertex of $G$, the algorithm runs in $\tilde{O}(n^{\sqrt{11.5}-2} \log L) \le O(n^{1.392} \log L)$ time, where $L$ is the absolute value of the smallest edge length. The algorithm computes shortest paths and the distances from $s$ to all vertices of the graph, or else provides a certificate that $G$ is not $H$-minor free. Our result improves an earlier $O(n^{1.5} \log L)$ time algorithm for this problem, which follows from a general SSSP algorithm of Goldberg.

0809.3232

Daniel A. Spielman

Daniel A. Spielman, Shang-Hua Teng

A Local Clustering Algorithm for Massive Graphs and its Application to Nearly-Linear Time Graph Partitioning

cs.DS cs.DM

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

We study the design of local algorithms for massive graphs. A local algorithm is one that finds a solution containing or near a given vertex without looking at the whole graph. We present a local clustering algorithm. Our algorithm finds a good cluster--a subset of vertices whose internal connections are significantly richer than its external connections--near a given vertex. The running time of our algorithm, when it finds a non-empty local cluster, is nearly linear in the size of the cluster it outputs. Our clustering algorithm could be a useful primitive for handling massive graphs, such as social networks and web-graphs. As an application of this clustering algorithm, we present a partitioning algorithm that finds an approximate sparsest cut with nearly optimal balance. Our algorithm takes time nearly linear in the number edges of the graph. Using the partitioning algorithm of this paper, we have designed a nearly-linear time algorithm for constructing spectral sparsifiers of graphs, which we in turn use in a nearly-linear time algorithm for solving linear systems in symmetric, diagonally-dominant matrices. The linear system solver also leads to a nearly linear-time algorithm for approximating the second-smallest eigenvalue and corresponding eigenvector of the Laplacian matrix of a graph. These other results are presented in two companion papers.

0809.3527

Mugurel Ionut Andreica

Mugurel Ionut Andreica, Angela Andreica, Romulus Andreica

Inferring Company Structure from Limited Available Information

Some of the algorithmic techniques presented in this paper were used as part of the solutions for some of the tasks proposed in several programming contests in which the first author participated (see the related materials for several such tasks and their solutions)

International Symposium on Social Development and Economic Performance, Satu Mare : Romania (2008)

cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

In this paper we present several algorithmic techniques for inferring the structure of a company when only a limited amount of information is available. We consider problems with two types of inputs: the number of pairs of employees with a given property and restricted information about the hierarchical structure of the company. We provide dynamic programming and greedy algorithms for these problems.

0809.3528

Mugurel Ionut Andreica

Mugurel Ionut Andreica, Cristina Teodora Andreica, Madalina Ecaterina Andreica

Locating Restricted Facilities on Binary Maps

The algorithmic techniques presented in this paper were used by the first author in the implementation of solutions to various algorithmic contest tasks (see the attached zip archive for some examples)

International Symposium on Social Development and Economic Performance, Satu Mare : Romania (2008)

cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

In this paper we consider several facility location problems with applications to cost and social welfare optimization, when the area map is encoded as a binary (0,1) mxn matrix. We present algorithmic solutions for all the problems. Some cases are too particular to be used in practical situations, but they are at least a starting point for more generic solutions.

0809.3577

Philippe Robert

Han\`ene Mohamed, Philippe Robert

Dynamic tree algorithms

Published in at http://dx.doi.org/10.1214/09-AAP617 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Annals of Applied Probability 2010, Vol. 20, No. 1, 26-51

10.1214/09-AAP617

IMS-AAP-AAP617

math.PR cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

In this paper, a general tree algorithm processing a random flow of arrivals is analyzed. Capetanakis--Tsybakov--Mikhailov's protocol in the context of communication networks with random access is an example of such an algorithm. In computer science, this corresponds to a trie structure with a dynamic input. Mathematically, it is related to a stopped branching process with exogeneous arrivals (immigration). Under quite general assumptions on the distribution of the number of arrivals and on the branching procedure, it is shown that there exists a positive constant $\lambda_c$ so that if the arrival rate is smaller than $\lambda_c$, then the algorithm is stable under the flow of requests, that is, that the total size of an associated tree is integrable. At the same time, a gap in the earlier proofs of stability in the literature is fixed. When the arrivals are Poisson, an explicit characterization of $\lambda_c$ is given. Under the stability condition, the asymptotic behavior of the average size of a tree starting with a large number of individuals is analyzed. The results are obtained with the help of a probabilistic rewriting of the functional equations describing the dynamics of the system. The proofs use extensively this stochastic background throughout the paper. In this analysis, two basic limit theorems play a key role: the renewal theorem and the convergence to equilibrium of an auto-regressive process with a moving average.

0809.3646

Petr Golovach

Fedor V. Fomin, Petr A. Golovach and Dimitrios M. Thilikos

Approximating acyclicity parameters of sparse hypergraphs

cs.DS cs.CC

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

The notions of hypertree width and generalized hypertree width were introduced by Gottlob, Leone, and Scarcello in order to extend the concept of hypergraph acyclicity. These notions were further generalized by Grohe and Marx, who introduced the fractional hypertree width of a hypergraph. All these width parameters on hypergraphs are useful for extending tractability of many problems in database theory and artificial intelligence. In this paper, we study the approximability of (generalized, fractional) hyper treewidth of sparse hypergraphs where the criterion of sparsity reflects the sparsity of their incidence graphs. Our first step is to prove that the (generalized, fractional) hypertree width of a hypergraph H is constant-factor sandwiched by the treewidth of its incidence graph, when the incidence graph belongs to some apex-minor-free graph class. This determines the combinatorial borderline above which the notion of (generalized, fractional) hypertree width becomes essentially more general than treewidth, justifying that way its functionality as a hypergraph acyclicity measure. While for more general sparse families of hypergraphs treewidth of incidence graphs and all hypertree width parameters may differ arbitrarily, there are sparse families where a constant factor approximation algorithm is possible. In particular, we give a constant factor approximation polynomial time algorithm for (generalized, fractional) hypertree width on hypergraphs whose incidence graphs belong to some H-minor-free graph class.

0809.4398

Philipp Schuetz

Philipp Schuetz, Amedeo Caflisch

Multistep greedy algorithm identifies community structure in real-world and computer-generated networks

17 pages, 2 figures

Phys. Rev. E 78, 026112 (2008)

10.1103/PhysRevE.78.026112

cs.DS cond-mat.dis-nn physics.soc-ph q-bio.MN q-bio.QM

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

We have recently introduced a multistep extension of the greedy algorithm for modularity optimization. The extension is based on the idea that merging l pairs of communities (l>1) at each iteration prevents premature condensation into few large communities. Here, an empirical formula is presented for the choice of the step width l that generates partitions with (close to) optimal modularity for 17 real-world and 1100 computer-generated networks. Furthermore, an in-depth analysis of the communities of two real-world networks (the metabolic network of the bacterium E. coli and the graph of coappearing words in the titles of papers coauthored by Martin Karplus) provides evidence that the partition obtained by the multistep greedy algorithm is superior to the one generated by the original greedy algorithm not only with respect to modularity but also according to objective criteria. In other words, the multistep extension of the greedy algorithm reduces the danger of getting trapped in local optima of modularity and generates more reasonable partitions.

0809.4577

Jiajin Yu

Mordecai Golin, Xiaoming Xu and Jiajin Yu

A Generic Top-Down Dynamic-Programming Approach to Prefix-Free Coding

cs.DS cs.IT math.IT

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Given a probability distribution over a set of n words to be transmitted, the Huffman Coding problem is to find a minimal-cost prefix free code for transmitting those words. The basic Huffman coding problem can be solved in O(n log n) time but variations are more difficult. One of the standard techniques for solving these variations utilizes a top-down dynamic programming approach. In this paper we show that this approach is amenable to dynamic programming speedup techniques, permitting a speedup of an order of magnitude for many algorithms in the literature for such variations as mixed radix, reserved length and one-ended coding. These speedups are immediate implications of a general structural property that permits batching together the calculation of many DP entries.

0809.4743

Boris Ryabko

Boris Ryabko

The Imaginary Sliding Window As a New Data Structure for Adaptive Algorithms

Published in: Problems of information transmission,1996,v.32,#2

cs.IT cs.DS math.IT

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

The scheme of the sliding window is known in Information Theory, Computer Science, the problem of predicting and in stastistics. Let a source with unknown statistics generate some word $... x_{-1}x_{0}x_{1}x_{2}...$ in some alphabet $A$. For every moment $t, t=... $ $-1, 0, 1, ...$, one stores the word ("window") $ x_{t-w} x_{t-w+1}... x_{t-1}$ where $w$,$w \geq 1$, is called "window length". In the theory of universal coding, the code of the $x_{t}$ depends on source ststistics estimated by the window, in the problem of predicting, each letter $x_{t}$ is predicted using information of the window, etc. After that the letter $x_{t}$ is included in the window on the right, while $x_{t-w}$ is removed from the window. It is the sliding window scheme. This scheme has two merits: it allows one i) to estimate the source statistics quite precisely and ii) to adapt the code in case of a change in the source' statistics. However this scheme has a defect, namely, the necessity to store the window (i.e. the word $x_{t-w}... x_{t-1})$ which needs a large memory size for large $w$. A new scheme named "the Imaginary Sliding Window (ISW)" is constructed. The gist of this scheme is that not the last element $x_{t-w}$ but rather a random one is removed from the window. This allows one to retain both merits of the sliding window as well as the possibility of not storing the window and thus significantly decreasing the memory size.

0809.4794

Adam Smith

Adam Smith

Efficient, Differentially Private Point Estimators

9 pages

cs.CR cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Differential privacy is a recent notion of privacy for statistical databases that provides rigorous, meaningful confidentiality guarantees, even in the presence of an attacker with access to arbitrary side information. We show that for a large class of parametric probability models, one can construct a differentially private estimator whose distribution converges to that of the maximum likelihood estimator. In particular, it is efficient and asymptotically unbiased. This result provides (further) compelling evidence that rigorous notions of privacy in statistical databases can be consistent with statistically valid inference.

0809.4882

Aleksandrs Slivkins

Robert Kleinberg, Aleksandrs Slivkins and Eli Upfal

Multi-Armed Bandits in Metric Spaces

16 pages, 0 figures

cs.DS cs.LG

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

In a multi-armed bandit problem, an online algorithm chooses from a set of strategies in a sequence of trials so as to maximize the total payoff of the chosen strategies. While the performance of bandit algorithms with a small finite strategy set is quite well understood, bandit problems with large strategy sets are still a topic of very active investigation, motivated by practical applications such as online auctions and web advertisement. The goal of such research is to identify broad and natural classes of strategy sets and payoff functions which enable the design of efficient solutions. In this work we study a very general setting for the multi-armed bandit problem in which the strategies form a metric space, and the payoff function satisfies a Lipschitz condition with respect to the metric. We refer to this problem as the "Lipschitz MAB problem". We present a complete solution for the multi-armed problem in this setting. That is, for every metric space (L,X) we define an isometry invariant which bounds from below the performance of Lipschitz MAB algorithms for X, and we present an algorithm which comes arbitrarily close to meeting this bound. Furthermore, our technique gives even better results for benign payoff functions.

0810.0052

Martin Ziegler

Matthias Fischer, Matthias Hilbig, Claudius J\"ahn, Friedhelm Meyer auf der Heide and Martin Ziegler

Planar Visibility Counting

added Section 4: Implementation and Empirical Evaluation

cs.CG cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

For a fixed virtual scene (=collection of simplices) S and given observer position p, how many elements of S are weakly visible (i.e. not fully occluded by others) from p? The present work explores the trade-off between query time and preprocessing space for these quantities in 2D: exactly, in the approximate deterministic, and in the probabilistic sense. We deduce the EXISTENCE of an O(m^2/n^2) space data structure for S that, given p and time O(log n), allows to approximate the ratio of occluded segments up to arbitrary constant absolute error; here m denotes the size of the Visibility Graph--which may be quadratic, but typically is just linear in the size n of the scene S. On the other hand, we present a data structure CONSTRUCTIBLE in O(n*log(n)+m^2*polylog(n)/k) preprocessing time and space with similar approximation properties and query time O(k*polylog n), where k<n is an arbitrary parameter. We describe an implementation of this approach and demonstrate the practical benefit of the parameter k to trade memory for query time in an empirical evaluation on three classes of benchmark scenes.

0810.0075

Alvaro Salas Humberto

Alvaro Salas (Universidad de Caldas, Universidad Nacional de Colombia, sede Manizales)

Acerca del Algoritmo de Dijkstra

In spanish, the paper contains illustrations

cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

In this paper we prove the correctness of Dijkstra's algorithm. We also discuss it and at the end we show an application.

0810.0264

David Musser

David R. Musser, Gor V. Nishanov

A Fast Generic Sequence Matching Algorithm

cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

A string matching -- and more generally, sequence matching -- algorithm is presented that has a linear worst-case computing time bound, a low worst-case bound on the number of comparisons (2n), and sublinear average-case behavior that is better than that of the fastest versions of the Boyer-Moore algorithm. The algorithm retains its efficiency advantages in a wide variety of sequence matching problems of practical interest, including traditional string matching; large-alphabet problems (as in Unicode strings); and small-alphabet, long-pattern problems (as in DNA searches). Since it is expressed as a generic algorithm for searching in sequences over an arbitrary type T, it is well suited for use in generic software libraries such as the C++ Standard Template Library. The algorithm was obtained by adding to the Knuth-Morris-Pratt algorithm one of the pattern-shifting techniques from the Boyer-Moore algorithm, with provision for use of hashing in this technique. In situations in which a hash function or random access to the sequences is not available, the algorithm falls back to an optimized version of the Knuth-Morris-Pratt algorithm.

0810.0558

Ashish Goel

Ashish Goel, Sanjeev Khanna, Brad Null

The Ratio Index for Budgeted Learning, with Applications

This paper has a substantial bug that we are trying to fix. Many thanks to Joe Halpern for pointing this bug out. Please do not cite in the meantime. Please let us know if you would like to understand and/or try to fix the bug

cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

In the budgeted learning problem, we are allowed to experiment on a set of alternatives (given a fixed experimentation budget) with the goal of picking a single alternative with the largest possible expected payoff. Approximation algorithms for this problem were developed by Guha and Munagala by rounding a linear program that couples the various alternatives together. In this paper we present an index for this problem, which we call the ratio index, which also guarantees a constant factor approximation. Index-based policies have the advantage that a single number (i.e. the index) can be computed for each alternative irrespective of all other alternatives, and the alternative with the highest index is experimented upon. This is analogous to the famous Gittins index for the discounted multi-armed bandit problem. The ratio index has several interesting structural properties. First, we show that it can be computed in strongly polynomial time. Second, we show that with the appropriate discount factor, the Gittins index and our ratio index are constant factor approximations of each other, and hence the Gittins index also gives a constant factor approximation to the budgeted learning problem. Finally, we show that the ratio index can be used to create an index-based policy that achieves an O(1)-approximation for the finite horizon version of the multi-armed bandit problem. Moreover, the policy does not require any knowledge of the horizon (whereas we compare its performance against an optimal strategy that is aware of the horizon). This yields the following surprising result: there is an index-based policy that achieves an O(1)-approximation for the multi-armed bandit problem, oblivious to the underlying discount factor.

0810.0674

Shuchi Chawla

Siddharth Barman and Shuchi Chawla

Packing multiway cuts in capacitated graphs

The conference version of this paper is to appear at SODA 2009. This is the full version

cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

We consider the following "multiway cut packing" problem in undirected graphs: we are given a graph G=(V,E) and k commodities, each corresponding to a set of terminals located at different vertices in the graph; our goal is to produce a collection of cuts {E_1,...,E_k} such that E_i is a multiway cut for commodity i and the maximum load on any edge is minimized. The load on an edge is defined to be the number of cuts in the solution crossing the edge. In the capacitated version of the problem the goal is to minimize the maximum relative load on any edge--the ratio of the edge's load to its capacity. Multiway cut packing arises in the context of graph labeling problems where we are given a partial labeling of a set of items and a neighborhood structure over them, and, informally, the goal is to complete the labeling in the most consistent way. This problem was introduced by Rabani, Schulman, and Swamy (SODA'08), who developed an O(log n/log log n) approximation for it in general graphs, as well as an improved O(log^2 k) approximation in trees. Here n is the number of nodes in the graph. We present the first constant factor approximation for this problem in arbitrary undirected graphs. Our approach is based on the observation that every instance of the problem admits a near-optimal laminar solution (that is, one in which no pair of cuts cross each other).

0810.0906

Hirotaka Ono

Toru Hasunuma, Toshimasa Ishii, Hirotaka Ono, Yushi Uno

A linear time algorithm for L(2,1)-labeling of trees

23 pages, 3 figures

cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

An L(2,1)-labeling of a graph $G$ is an assignment $f$ from the vertex set $V(G)$ to the set of nonnegative integers such that $|f(x)-f(y)|\ge 2$ if $x$ and $y$ are adjacent and $|f(x)-f(y)|\ge 1$ if $x$ and $y$ are at distance 2, for all $x$ and $y$ in $V(G)$. A $k$-L(2,1)-labeling is an assignment $f:V(G)\to\{0,..., k\}$, and the L(2,1)-labeling problem asks the minimum $k$, which we denote by $\lambda(G)$, among all possible assignments. It is known that this problem is NP-hard even for graphs of treewidth 2, and tree is one of a very few classes for which the problem is polynomially solvable. The running time of the best known algorithm for trees had been $\mO(\Delta^{4.5} n)$ for more than a decade, however, an $\mO(n^{1.75})$-time algorithm has been proposed recently, which substantially improved the previous one, where $\Delta$ is the maximum degree of $T$ and $n=|V(T)|$. In this paper, we finally establish a linear time algorithm for L(2,1)-labeling of trees.

0810.1355

Michael Mahoney

Jure Leskovec, Kevin J. Lang, Anirban Dasgupta, and Michael W. Mahoney

Community Structure in Large Networks: Natural Cluster Sizes and the Absence of Large Well-Defined Clusters

66 pages, a much expanded version of our WWW 2008 paper

cs.DS physics.data-an physics.soc-ph

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

A large body of work has been devoted to defining and identifying clusters or communities in social and information networks. We explore from a novel perspective several questions related to identifying meaningful communities in large social and information networks, and we come to several striking conclusions. We employ approximation algorithms for the graph partitioning problem to characterize as a function of size the statistical and structural properties of partitions of graphs that could plausibly be interpreted as communities. In particular, we define the network community profile plot, which characterizes the "best" possible community--according to the conductance measure--over a wide range of size scales. We study over 100 large real-world social and information networks. Our results suggest a significantly more refined picture of community structure in large networks than has been appreciated previously. In particular, we observe tight communities that are barely connected to the rest of the network at very small size scales; and communities of larger size scales gradually "blend into" the expander-like core of the network and thus become less "community-like." This behavior is not explained, even at a qualitative level, by any of the commonly-used network generation models. Moreover, it is exactly the opposite of what one would expect based on intuition from expander graphs, low-dimensional or manifold-like graphs, and from small social networks that have served as testbeds of community detection algorithms. We have found that a generative graph model, in which new edges are added via an iterative "forest fire" burning process, is able to produce graphs exhibiting a network community profile plot similar to what we observe in our network datasets.

0810.1499

Lenka Zdeborova

Lenka Zdeborov\'a and Marc M\'ezard

Constraint satisfaction problems with isolated solutions are hard

19 pages, 12 figures

J. Stat. Mech. (2008) P12004

10.1088/1742-5468/2008/12/P12004

cond-mat.dis-nn cond-mat.stat-mech cs.CC cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

We study the phase diagram and the algorithmic hardness of the random `locked' constraint satisfaction problems, and compare them to the commonly studied 'non-locked' problems like satisfiability of boolean formulas or graph coloring. The special property of the locked problems is that clusters of solutions are isolated points. This simplifies significantly the determination of the phase diagram, which makes the locked problems particularly appealing from the mathematical point of view. On the other hand we show empirically that the clustered phase of these problems is extremely hard from the algorithmic point of view: the best known algorithms all fail to find solutions. Our results suggest that the easy/hard transition (for currently known algorithms) in the locked problems coincides with the clustering transition. These should thus be regarded as new benchmarks of really hard constraint satisfaction problems.

0810.1639

Gabriel Istrate

Gabriel Istrate

Identifying almost sorted permutations from TCP buffer dynamics

cs.DS cs.DM math.CO

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Associate to each sequence $A$ of integers (intending to represent packet IDs) a sequence of positive integers of the same length ${\mathcal M}(A)$. The $i$'th entry of ${\mathcal M}(A)$ is the size (at time $i$) of the smallest buffer needed to hold out-of-order packets, where space is accounted for unreceived packets as well. Call two sequences $A$, $B$ {\em equivalent} (written $A\equiv_{FB} B$) if ${\mathcal M}(A)={\mathcal M}(B)$. We prove the following result: any two permutations $A,B$ of the same length with $SUS(A)$, $SUS(B)\leq 3$ (where SUS is the {\em shuffled-up-sequences} reordering measure), and such that $A\equiv_{FB} B$ are identical. The result (which is no longer valid if we replace the upper bound 3 by 4) was motivated by RESTORED, a receiver-oriented model of network traffic we have previously introduced.

0810.1756

Milan Bradonjic

Milan Bradonjic, Eddie Kohler, Rafail Ostrovsky

Near-Optimal Radio Use For Wireless Network Synchronization

23 pages

10.1016/j.tcs.2011.09.026

cs.DS cs.DM

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

We consider the model of communication where wireless devices can either switch their radios off to save energy, or switch their radios on and engage in communication. We distill a clean theoretical formulation of this problem of minimizing radio use and present near-optimal solutions. Our base model ignores issues of communication interference, although we also extend the model to handle this requirement. We assume that nodes intend to communicate periodically, or according to some time-based schedule. Clearly, perfectly synchronized devices could switch their radios on for exactly the minimum periods required by their joint schedules. The main challenge in the deployment of wireless networks is to synchronize the devices' schedules, given that their initial schedules may be offset relative to one another (even if their clocks run at the same speed). We significantly improve previous results, and show optimal use of the radio for two processors and near-optimal use of the radio for synchronization of an arbitrary number of processors. In particular, for two processors we prove deterministically matching $\Theta(\sqrt{n})$ upper and lower bounds on the number of times the radio has to be on, where $n$ is the discretized uncertainty period of the clock shift between the two processors. (In contrast, all previous results for two processors are randomized.) For $m=n^\beta$ processors (for any $\beta < 1$) we prove $\Omega(n^{(1-\beta)/2})$ is the lower bound on the number of times the radio has to be switched on (per processor), and show a nearly matching (in terms of the radio use) $\~{O}(n^{(1-\beta)/2})$ randomized upper bound per processor, with failure probability exponentially close to 0. For $\beta \geq 1$ our algorithm runs with at most $poly-log(n)$ radio invocations per processor. Our bounds also hold in a radio-broadcast model where interference must be taken into account.

0810.1823

Christophe Paul

Emeric Gioan and Christophe Paul

Split decomposition and graph-labelled trees: characterizations and fully-dynamic algorithms for totally decomposable graphs

extended abstract appeared in ISAAC 2007: Dynamic distance hereditary graphs using split decompositon. In International Symposium on Algorithms and Computation - ISAAC. Number 4835 in Lecture Notes, pages 41-51, 2007

cs.DM cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

In this paper, we revisit the split decomposition of graphs and give new combinatorial and algorithmic results for the class of totally decomposable graphs, also known as the distance hereditary graphs, and for two non-trivial subclasses, namely the cographs and the 3-leaf power graphs. Precisely, we give strutural and incremental characterizations, leading to optimal fully-dynamic recognition algorithms for vertex and edge modifications, for each of these classes. These results rely on a new framework to represent the split decomposition, namely the graph-labelled trees, which also captures the modular decomposition of graphs and thereby unify these two decompositions techniques. The point of the paper is to use bijections between these graph classes and trees whose nodes are labelled by cliques and stars. Doing so, we are also able to derive an intersection model for distance hereditary graphs, which answers an open problem.

0810.1851

Marek Karpinski

Piotr Berman, Marek Karpinski, Alex Zelikovsky

1.25 Approximation Algorithm for the Steiner Tree Problem with Distances One and Two

cs.CC cs.DM cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

We give a 1.25 approximation algorithm for the Steiner Tree Problem with distances one and two, improving on the best known bound for that problem.

0810.2150

Christine Task

Christine Task, Arun Chauhan

A Model for Communication in Clusters of Multi-core Machines

This paper has been withdrawn by the author because it was basically a short-hand write-up of an incompletely formed idea from her Masters, and she'd like to start using her ArXiv account for her formal PhD research

cs.DC cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

A common paradigm for scientific computing is distributed message-passing systems, and a common approach to these systems is to implement them across clusters of high-performance workstations. As multi-core architectures become increasingly mainstream, these clusters are very likely to include multi-core machines. However, the theoretical models which are currently used to develop communication algorithms across these systems do not take into account the unique properties of processes running on shared-memory architectures, including shared external network connections and communication via shared memory locations. Because of this, existing algorithms are far from optimal for modern clusters. Additionally, recent attempts to adapt these algorithms to multicore systems have proceeded without the introduction of a more accurate formal model and have generally neglected to capitalize on the full power these systems offer. We propose a new model which simply and effectively captures the strengths of multi-core machines in collective communications patterns and suggest how it could be used to properly optimize these patterns.

0810.2390

Simone Faro

Simone Faro and Thierry Lecroq

Efficient Pattern Matching on Binary Strings

12 pages

cs.DS cs.IR

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

The binary string matching problem consists in finding all the occurrences of a pattern in a text where both strings are built on a binary alphabet. This is an interesting problem in computer science, since binary data are omnipresent in telecom and computer network applications. Moreover the problem finds applications also in the field of image processing and in pattern matching on compressed texts. Recently it has been shown that adaptations of classical exact string matching algorithms are not very efficient on binary data. In this paper we present two efficient algorithms for the problem adapted to completely avoid any reference to bits allowing to process pattern and text byte by byte. Experimental results show that the new algorithms outperform existing solutions in most cases.

0810.3058

Dimitrios Tsolis Dr

Dimitrios K. Tsolis, Spyros Sioutas and Theodore S. Papatheodorou

Watermarking Digital Images Based on a Content Based Image Retrieval Technique

18 pages, 4 figures, 4 tables, submitted to Multimedia Tools and Applications Journal, Springer

cs.DS cs.CR

http://creativecommons.org/licenses/by/3.0/

The current work is focusing on the implementation of a robust watermarking algorithm for digital images, which is based on an innovative spread spectrum analysis algorithm for watermark embedding and on a content-based image retrieval technique for watermark detection. The highly robust watermark algorithms are applying "detectable watermarks" for which a detection mechanism checks if the watermark exists or no (a Boolean decision) based on a watermarking key. The problem is that the detection of a watermark in a digital image library containing thousands of images means that the watermark detection algorithm is necessary to apply all the keys to the digital images. This application is non-efficient for very large image databases. On the other hand "readable" watermarks may prove weaker but easier to detect as only the detection mechanism is required. The proposed watermarking algorithm combine's the advantages of both "detectable" and "readable" watermarks. The result is a fast and robust watermarking algorithm.

0810.3203

David Harvey

David Harvey

A cache-friendly truncated FFT

14 pages, 11 figures, uses algorithm2e package

cs.SC cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

We describe a cache-friendly version of van der Hoeven's truncated FFT and inverse truncated FFT, focusing on the case of `large' coefficients, such as those arising in the Schonhage--Strassen algorithm for multiplication in Z[x]. We describe two implementations and examine their performance.

0810.3227

Oliver Kennedy

Oliver Kennedy, Christoph Koch, Al Demers

Dynamic Approaches to In-Network Aggregation

cs.DC cs.DB cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Collaboration between small-scale wireless devices hinges on their ability to infer properties shared across multiple nearby nodes. Wireless-enabled mobile devices in particular create a highly dynamic environment not conducive to distributed reasoning about such global properties. This paper addresses a specific instance of this problem: distributed aggregation. We present extensions to existing unstructured aggregation protocols that enable estimation of count, sum, and average aggregates in highly dynamic environments. With the modified protocols, devices with only limited connectivity can maintain estimates of the aggregate, despite \textit{unexpected} peer departures and arrivals. Our analysis of these aggregate maintenance extensions demonstrates their effectiveness in unstructured environments despite high levels of node mobility.

0810.3438

Amit Bhosle

Amit M Bhosle and Teofilo F Gonzalez

Efficient Algorithms and Routing Protocols for Handling Transient Single Node Failures

6 pages, 2 columns, 3 figures. To appear in: Proceedings of the 20th IASTED International Conference on Parallel and Distributed Computing and Systems (Nov 16-18, 2008, Orlando, FL, USA)

cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Single node failures represent more than 85% of all node failures in the today's large communication networks such as the Internet. Also, these node failures are usually transient. Consequently, having the routing paths globally recomputed does not pay off since the failed nodes recover fairly quickly, and the recomputed routing paths need to be discarded. Instead, we develop algorithms and protocols for dealing with such transient single node failures by suppressing the failure (instead of advertising it across the network), and routing messages to the destination via alternate paths that do not use the failed node. We compare our solution to that of Ref. [11] wherein the authors have presented a "Failure Insensitive Routing" protocol as a proactive recovery scheme for handling transient node failures. We show that our algorithms are faster by an order of magnitude while our paths are equally good. We show via simulation results that our paths are usually within 15% of the optimal for randomly generated graph with 100-1000 nodes.

0810.4002

Laetitia Omer

Julien Allali (LaBRI), Marie-France Sagot (ENS Lyon / Insa Lyon / INRIA Grenoble Rh\^one-Alpes)

A new distance for high level RNA secondary structure comparison

IEEE/ACM Transactions on Computational Biology and Bioinformatics 2 (2005) 3--14

cs.DS q-bio.QM

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

We describe an algorithm for comparing two RNA secondary structures coded in the form of trees that introduces two new operations, called node fusion and edge fusion, besides the tree edit operations of deletion, insertion, and relabeling classically used in the literature. This allows us to address some serious limitations of the more traditional tree edit operations when the trees represent RNAs and what is searched for is a common structural core of two RNAs. Although the algorithm complexity has an exponential term, this term depends only on the number of successive fusions that may be applied to a same node, not on the total number of fusions. The algorithm remains therefore efficient in practice and is used for illustrative purposes on ribosomal as well as on other types of RNAs.

0810.4061

Satu Elisa Schaeffer

Pekka Orponen, Satu Elisa Schaeffer, Vanesa Avalos Gayt\'an

Locally computable approximations for spectral clustering and absorption times of random walks

21 pages, 8 figures

cs.DM cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

We address the problem of determining a natural local neighbourhood or "cluster" associated to a given seed vertex in an undirected graph. We formulate the task in terms of absorption times of random walks from other vertices to the vertex of interest, and observe that these times are well approximated by the components of the principal eigenvector of the corresponding fundamental matrix of the graph's adjacency matrix. We further present a locally computable gradient-descent method to estimate this Dirichlet-Fiedler vector, based on minimising the respective Rayleigh quotient. Experimental evaluation shows that the approximations behave well and yield well-defined local clusters.

0810.4423

Mugurel Ionut Andreica

Mugurel Ionut Andreica

Efficient Algorithmic Techniques for Several Multidimensional Geometric Data Management and Analysis Problems

The algorithmic techniques presented in this paper were later used by the author in developing solutions for algorithmic tasks in several contests in which the author participated (see the attached zip archive for some examples of task statements and solutions). Knowledge Management - Projects, Systems and Technologies, Bucharest : Romania (2008)

cs.CG cs.DM cs.DS

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

In this paper I present several novel, efficient, algorithmic techniques for solving some multidimensional geometric data management and analysis problems. The techniques are based on several data structures from computational geometry (e.g. segment tree and range tree) and on the well-known sweep-line method.

0810.4796

Daniel Raible

Henning Fernau, Fedor V. Fomin, Daniel Lokshtanov, Daniel Raible, Saket Saurabh, Yngve Villanger

Kernel(s) for Problems With no Kernel: On Out-Trees With Many Leaves

cs.DS cs.CC

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

The {\sc $k$-Leaf Out-Branching} problem is to find an out-branching (i.e. a rooted oriented spanning tree) with at least $k$ leaves in a given digraph. The problem has recently received much attention from the viewpoint of parameterized algorithms {alonLNCS4596,AlonFGKS07fsttcs,BoDo2,KnLaRo}. In this paper we step aside and take a kernelization based approach to the {\sc $k$-Leaf-Out-Branching} problem. We give the first polynomial kernel for {\sc Rooted $k$-Leaf-Out-Branching}, a variant of {\sc $k$-Leaf-Out-Branching} where the root of the tree searched for is also a part of the input. Our kernel has cubic size and is obtained using extremal combinatorics. For the {\sc $k$-Leaf-Out-Branching} problem we show that no polynomial kernel is possible unless polynomial hierarchy collapses to third level %$PH=\Sigma_p^3$ by applying a recent breakthrough result by Bodlaender et al. {BDFH08} in a non-trivial fashion. However our positive results for {\sc Rooted $k$-Leaf-Out-Branching} immediately imply that the seemingly intractable the {\sc $k$-Leaf-Out-Branching} problem admits a data reduction to $n$ independent $O(k^3)$ kernels. These two results, tractability and intractability side by side, are the first separating {\it many-to-one kernelization} from {\it Turing kernelization}. This answers affirmatively an open problem regarding "cheat kernelization" raised in {IWPECOPEN08}.