Datasets:

niceanyh

/

arXiv_index_ds

like 0

cs/9809012

David Karger

David R. Karger

A Fully Polynomial Randomized Approximation Scheme for the All Terminal Network Reliability Problem

To appear in SICOMP

cs.DS

The classic all-terminal network reliability problem posits a graph, each of whose edges fails independently with some given probability.

cs/9809064

Madhav Marathe

Madhav V. Marathe, Harry B. Hunt III, Richard E. Stearns, Venkatesh Radhakrishnan

Approximation Algorithms for PSPACE-Hard Hierarchically and Periodically Specified Problems

5 Figures, 24 pages

SIAM J. Computing, Vol. 27, No 5, Oct. 1998, pp. 1237--1261

cs.CC cs.DS

We study the efficient approximability of basic graph and logic problems in the literature when instances are specified hierarchically as in \cite{Le89} or are specified by 1-dimensional finite narrow periodic specifications as in \cite{Wa93}. We show that, for most of the problems $\Pi$ considered when specified using {\bf k-level-restricted} hierarchical specifications or $k$-narrow periodic specifications the following holds: \item Let $\rho$ be any performance guarantee of a polynomial time approximation algorithm for $\Pi$, when instances are specified using standard specifications. Then $\forall \epsilon > 0$, $ \Pi$ has a polynomial time approximation algorithm with performance guarantee $(1 + \epsilon) \rho$. \item $\Pi$ has a polynomial time approximation scheme when restricted to planar instances. \end{romannum} These are the first polynomial time approximation schemes for PSPACE-hard hierarchically or periodically specified problems. Since several of the problems considered are PSPACE-hard, our results provide the first examples of natural PSPACE-hard optimization problems that have polynomial time approximation schemes. This answers an open question in Condon et. al. \cite{CF+93}.

cs/9809103

Madhav Marathe

Madhav V. Marathe, R. Ravi, Ravi Sundaram, S. S. Ravi, Daniel J. Rosenkrantz, Harry B. Hunt III

Bicriteria Network Design Problems

24 pages 1 figure

J. Algorithms, 28, 142-171, (1998)

cs.CC cs.DS

We study a general class of bicriteria network design problems. A generic problem in this class is as follows: Given an undirected graph and two minimization objectives (under different cost functions), with a budget specified on the first, find a <subgraph \from a given subgraph-class that minimizes the second objective subject to the budget on the first. We consider three different criteria - the total edge cost, the diameter and the maximum degree of the network. Here, we present the first polynomial-time approximation algorithms for a large class of bicriteria network design problems for the above mentioned criteria. The following general types of results are presented. First, we develop a framework for bicriteria problems and their approximations. Second, when the two criteria are the same %(note that the cost functions continue to be different) we present a ``black box'' parametric search technique. This black box takes in as input an (approximation) algorithm for the unicriterion situation and generates an approximation algorithm for the bicriteria case with only a constant factor loss in the performance guarantee. Third, when the two criteria are the diameter and the total edge costs we use a cluster-based approach to devise a approximation algorithms --- the solutions output violate both the criteria by a logarithmic factor. Finally, for the class of treewidth-bounded graphs, we provide pseudopolynomial-time algorithms for a number of bicriteria problems using dynamic programming. We show how these pseudopolynomial-time algorithms can be converted to fully polynomial-time approximation schemes using a scaling technique.

cs/9809122

Osamu Watanabe

Carlos Domingo, Ricard Gavalda, and Osamu Watanabe

Practical algorithms for on-line sampling

To appear in the Proc. of Discovery Science '98, Dec. 1998

C-123

cs.LG cs.DS

One of the core applications of machine learning to knowledge discovery consists on building a function (a hypothesis) from a given amount of data (for instance a decision tree or a neural network) such that we can use it afterwards to predict new instances of the data. In this paper, we focus on a particular situation where we assume that the hypothesis we want to use for prediction is very simple, and thus, the hypotheses class is of feasible size. We study the problem of how to determine which of the hypotheses in the class is almost the best one. We present two on-line sampling algorithms for selecting hypotheses, give theoretical bounds on the number of necessary examples, and analize them exprimentally. We compare them with the simple batch sampling approach commonly used and show that in most of the situations our algorithms use much fewer number of examples.

cs/9810009

Maurizio Pizzonia

Maurizio Pizzonia, Giuseppe Di Battista

Object-Oriented Design of Graph Oriented Data Structures

10 pages, 9 figures, code examples, ALENEX (accepted)

cs.SE cs.CG cs.DS

Applied research in graph algorithms and combinatorial structures needs comprehensive and versatile software libraries. However, the design and the implementation of flexible libraries are challenging activities. Among the other problems involved in such a difficult field, a very special role is played by graph classification issues. We propose new techniques devised to help the designer and the programmer in the development activities. Such techniques are especially suited for dealing with graph classification problems and rely on an extension of the usual object-oriented paradigm. In order to support the usage of our approach, we devised an extension of the C++ programming language and implemented the corresponding pre-compiler.

cs/9811019

Joseph O'Rourke

T. Biedl, E. Demaine, M. Demaine, S. Lazard, A. Lubiw, J. O'Rourke, M. Overmars, S. Robbins, I. Streinu, G. Toussaint, S. Whitesides

Locked and Unlocked Polygonal Chains in 3D

To appear in Proc. 10th ACM-SIAM Sympos. Discrete Algorithms, Jan. 1999

Proc. 10th ACM-SIAM Sympos. Discrete Algorithms, Jan. 1999, pp. S866-7.

cs.CG cs.DS cs.RO

In this paper, we study movements of simple polygonal chains in 3D. We say that an open, simple polygonal chain can be straightened if it can be continuously reconfigured to a straight sequence of segments in such a manner that both the length of each link and the simplicity of the chain are maintained throughout the movement. The analogous concept for closed chains is convexification: reconfiguration to a planar convex polygon. Chains that cannot be straightened or convexified are called locked. While there are open chains in 3D that are locked, we show that if an open chain has a simple orthogonal projection onto some plane, it can be straightened. For closed chains, we show that there are unknotted but locked closed chains, and we provide an algorithm for convexifying a planar simple polygon in 3D with a polynomial number of moves.

cs/9812007

David Karger

David R. Karger

Minimum Cuts in Near-Linear Time

cs.DS

We significantly improve known time bounds for solving the minimum cut problem on undirected graphs. We use a ``semi-duality'' between minimum cuts and maximum spanning tree packings combined with our previously developed random sampling techniques. We give a randomized algorithm that finds a minimum cut in an m-edge, n-vertex graph with high probability in O(m log^3 n) time. We also give a simpler randomized algorithm that finds all minimum cuts with high probability in O(n^2 log n) time. This variant has an optimal RNC parallelization. Both variants improve on the previous best time bound of O(n^2 log^3 n). Other applications of the tree-packing approach are new, nearly tight bounds on the number of near minimum cuts a graph may have and a new data structure for representing them in a space-efficient manner.

cs/9812008

David Karger

David Karger, Rajeev Motwani, and Madhu Sudan

Approximate Graph Coloring by Semidefinite Programming

JACM 45(2), mar. 1998, pp.246--265

cs.DS

We consider the problem of coloring k-colorable graphs with the fewest possible colors. We present a randomized polynomial time algorithm that colors a 3-colorable graph on $n$ vertices with min O(Delta^{1/3} log^{1/2} Delta log n), O(n^{1/4} log^{1/2} n) colors where Delta is the maximum degree of any vertex. Besides giving the best known approximation ratio in terms of n, this marks the first non-trivial approximation result as a function of the maximum degree Delta. This result can be generalized to k-colorable graphs to obtain a coloring using min O(Delta^{1-2/k} log^{1/2} Delta log n), O(n^{1-3/(k+1)} log^{1/2} n) colors. Our results are inspired by the recent work of Goemans and Williamson who used an algorithm for semidefinite optimization problems, which generalize linear programs, to obtain improved approximations for the MAX CUT and MAX 2-SAT problems. An intriguing outcome of our work is a duality relationship established between the value of the optimum solution to our semidefinite program and the Lovasz theta-function. We show lower bounds on the gap between the optimum solution of our semidefinite program and the actual chromatic number; by duality this also demonstrates interesting new facts about the theta-function.

cs/9901004

Vladimir Pestov

Vladimir Pestov

On the geometry of similarity search: dimensionality curse and concentration of measure

7 pages, LaTeX 2e

Information Processing Letters 73 (2000), 47-51.

RP-99-01, Victoria University of Wellington, NZ

cs.IR cs.CG cs.DB cs.DS

We suggest that the curse of dimensionality affecting the similarity-based search in large datasets is a manifestation of the phenomenon of concentration of measure on high-dimensional structures. We prove that, under certain geometric assumptions on the query domain $\Omega$ and the dataset $X$, if $\Omega$ satisfies the so-called concentration property, then for most query points $x^\ast$ the ball of radius $(1+\e)d_X(x^\ast)$ centred at $x^\ast$ contains either all points of $X$ or else at least $C_1\exp(-C_2\e^2n)$ of them. Here $d_X(x^\ast)$ is the distance from $x^\ast$ to the nearest neighbour in $X$ and $n$ is the dimension of $\Omega$.

cs/9901010

Tao Jiang

Tao Jiang (McMaster U.), Ming Li (U of Waterloo), Paul Vitanyi (CWI and U of Amsterdam)

Average-Case Complexity of Shellsort

11 pages. Submitted to ICALP'99

cs.DS cs.CC

We prove a general lower bound on the average-case complexity of Shellsort: the average number of data-movements (and comparisons) made by a $p$-pass Shellsort for any incremental sequence is $\Omega (pn^{1 + 1/p)$ for all $p \leq \log n$. Using similar arguments, we analyze the average-case complexity of several other sorting algorithms.

cs/9902005

Paul Vitanyi

Harry Buhrman (CWI), Matthew Franklin (Xerox PARC), Juan A. Garay (Bell Labs - Lucent Technologies), Jaap-Henk Hoepman (University Twente), John Tromp (CWI), Paul Vitanyi (CWI and University of Amsterdam)

Mutual Search

18 pages, Latex, 5 figures, J. Assoc. Comp. Mach., To appear

cs.DS cs.CC cs.DB cs.DC cs.DM cs.IR

We introduce a search problem called ``mutual search'' where $k$ \agents, arbitrarily distributed over $n$ sites, are required to locate one another by posing queries of the form ``Anybody at site $i$?''. We ask for the least number of queries that is necessary and sufficient. For the case of two \agents using deterministic protocols we obtain the following worst-case results: In an oblivious setting (where all pre-planned queries are executed) there is no savings: $n-1$ queries are required and are sufficient. In a nonoblivious setting we can exploit the paradigm of ``no news is also news'' to obtain significant savings: in the synchronous case $0.586n$ queries suffice and $0.536n$ queries are required; in the asynchronous case $0.896n$ queries suffice and a fortiori 0.536 queries are required; for $o(\sqrt{n})$ \agents using a deterministic protocol less than $n$ queries suffice; there is a simple randomized protocol for two \agents with worst-case expected $0.5n$ queries and all randomized protocols require at least $0.125n$ worst-case expected queries. The graph-theoretic framework we formulate for expressing and analyzing algorithms for this problem may be of independent interest.

cs/9902006

Paul Vitanyi

Paul Vitanyi

A Discipline of Evolutionary Programming

25 pages, LaTeX source, Theoretical Computer Science, To appear

Theoret. Comp. Sci., 241:1-2 (2000), 3--23.

cs.NE cs.AI cs.CC cs.DS cs.LG cs.MA

Genetic fitness optimization using small populations or small population updates across generations generally suffers from randomly diverging evolutions. We propose a notion of highly probable fitness optimization through feasible evolutionary computing runs on small size populations. Based on rapidly mixing Markov chains, the approach pertains to most types of evolutionary genetic algorithms, genetic programming and the like. We establish that for systems having associated rapidly mixing Markov chains and appropriate stationary distributions the new method finds optimal programs (individuals) with probability almost 1. To make the method useful would require a structured design methodology where the development of the program and the guarantee of the rapidly mixing property go hand in hand. We analyze a simple example to show that the method is implementable. More significant examples require theoretical advances, for example with respect to the Metropolis filter.

cs/9903009

Paul Vitanyi

Harry Buhrman (CWI), Jaap-Henk Hoepman, Paul Vitanyi (CWI and University of Amsterdam)

Space-Efficient Routing Tables for Almost All Networks and the Incompressibility Method

19 pages, Latex, 1 table, 1 figure; SIAM J. Comput., To appear

CWI Tech Report 1997

cs.DC cs.AR cs.CC cs.DS cs.NI

We use the incompressibility method based on Kolmogorov complexity to determine the total number of bits of routing information for almost all network topologies. In most models for routing, for almost all labeled graphs $\Theta (n^2)$ bits are necessary and sufficient for shortest path routing. By `almost all graphs' we mean the Kolmogorov random graphs which constitute a fraction of $1-1/n^c$ of all graphs on $n$ nodes, where $c > 0$ is an arbitrary fixed constant. There is a model for which the average case lower bound rises to $\Omega(n^2 \log n)$ and another model where the average case upper bound drops to $O(n \log^2 n)$. This clearly exposes the sensitivity of such bounds to the model under consideration. If paths have to be short, but need not be shortest (if the stretch factor may be larger than 1), then much less space is needed on average, even in the more demanding models. Full-information routing requires $\Theta (n^3)$ bits on average. For worst-case static networks we prove a $\Omega(n^2 \log n)$ lower bound for shortest path routing and all stretch factors $<2$ in some networks where free relabeling is not allowed.

cs/9903010

Anatoly D. Plotnikov

Anatoly D. Plotnikov (Vinnitsa Institute of Regional Economics and Management)

A class of problems of NP to be worth to search an efficient solving algorithm

9 pages, 1 figures

cs.DS

We examine possibility to design an efficient solving algorithm for problems of the class \np. It is introduced a classification of \np problems by the property that a partial solution of size $k$ can be extended into a partial solution of size $k+1$ in polynomial time. It is defined an unique class problems to be worth to search an efficient solving algorithm. The problems, which are outside of this class, are inherently exponential. We show that the Hamiltonian cycle problem is inherently exponential.

cs/9903011

Stephan Mertens

Stephan Mertens

A complete anytime algorithm for balanced number partitioning

12 pages, 5 figures

cs.DS cond-mat.dis-nn cs.AI

Given a set of numbers, the balanced partioning problem is to divide them into two subsets, so that the sum of the numbers in each subset are as nearly equal as possible, subject to the constraint that the cardinalities of the subsets be within one of each other. We combine the balanced largest differencing method (BLDM) and Korf's complete Karmarkar-Karp algorithm to get a new algorithm that optimally solves the balanced partitioning problem. For numbers with twelve significant digits or less, the algorithm can optimally solve balanced partioning problems of arbitrary size in practice. For numbers with greater precision, it first returns the BLDM solution, then continues to find better solutions as time allows.

cs/9903012

Anatoly D. Plotnikov

Anatoly D. Plotnikov

Formalization of the class of problems solvable by a nondeterministic Turing machine

10 pages, 2 figures

Cybernetics and Systems Analysis. Vol. 33, 5(1997) pp. 635-640

cs.DS cs.CC

The objective of this article is to formalize the definition of NP problems. We construct a mathematical model of discrete problems as independence systems with weighted elements. We introduce two auxiliary sets that characterize the solution of the problem: the adjoint set, which contains the elements from the original set none of which can be adjoined to the already chosen solution elements; and the residual set, in which every element can be adjoined to previously chosen solution elements. In a problem without lookahead, every adjoint set can be generated by the solution algorithm effectively, in polynomial time. The main result of the study is the assertion that the NP class is identical with the class of problems without lookahead. Hence it follows that if we fail to find an effective (polynomial-time) solution algorithm for a given problem, then we need to look for an alternative formulation of the problem in set of problems without lookahead.

cs/9903020

Christoph D\"urr

Christoph Durr, Eric Goles, Ivan Rapaport, Eric Remila

Tiling with bars under tomographic constraints

Updated references

cs.DS cs.CC

We wish to tile a rectangle or a torus with only vertical and horizontal bars of a given length, such that the number of bars in every column and row equals given numbers. We present results for particular instances and for a more general problem, while leaving open the initial problem.

cs/9904002

Vladimir Pestov

Vladimir Pestov

A geometric framework for modelling similarity search

11 pages, LaTeX 2.e

Proc. 10-th Int. Workshop on Database and Expert Systems Applications (DEXA'99), Sept. 1-3, 1999, Florence, Italy, IEEE Comp. Soc., pp. 150-154.

10.1109/DEXA.1999.795158

RP-99-12, School of Math and Comp Sci, Victoria University of Wellington, New Zealand

cs.IR cs.DB cs.DS

The aim of this paper is to propose a geometric framework for modelling similarity search in large and multidimensional data spaces of general nature, which seems to be flexible enough to address such issues as analysis of complexity, indexability, and the `curse of dimensionality.' Such a framework is provided by the concept of the so-called similarity workload, which is a probability metric space $\Omega$ (query domain) with a distinguished finite subspace $X$ (dataset), together with an assembly of concepts, techniques, and results from metric geometry. They include such notions as metric transform, $\e$-entropy, and the phenomenon of concentration of measure on high-dimensional structures. In particular, we discuss the relevance of the latter to understanding the curse of dimensionality. As some of those concepts and techniques are being currently reinvented by the database community, it seems desirable to try and bridge the gap between database research and the relevant work already done in geometry and analysis.

cs/9906008

Paul Vitanyi

Tao Jiang (McMaster U.), Ming Li (U. Waterloo), Paul Vitanyi (CWI & U. Amsterdam)

A Lower Bound on the Average-Case Complexity of Shellsort

Preliminary version 10 pages, 2 figures, Proc ICALP 99, Springer LNCS; final version (given here) LaTeX 5 pages published in J. Assoc. Comp. Mach. as below

T. Jiang, M. Li, and P. Vitanyi, A lower bound on the average-case complexity of Shellsort, J. Assoc. Comp. Mach., 47:5(2000), 905--91

cs.CC cs.DS

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

We prove a general lower bound on the average-case complexity of Shellsort: the average number of data-movements (and comparisons) made by a $p$-pass Shellsort for any incremental sequence is $\Omega (pn^{1 + 1/p})$ for every $p$. The proof method is an incompressibility argument based on Kolmogorov complexity. Using similar techniques, the average-case complexity of several other sorting algorithms is analyzed.

cs/9906018

Christoph Durr

Christoph Durr and Marek Chrobak

Reconstructing Polyatomic Structures from Discrete X-Rays: NP-Completeness Proof for Three Atoms

Proceedings of the 23rd International Symposium on Mathematical Foundations of Computer Science, LNCS vol 1450, 185-193, 1998

cs.DS cs.CC

We address a discrete tomography problem that arises in the study of the atomic structure of crystal lattices. A polyatomic structure T can be defined as an integer lattice in dimension D>=2, whose points may be occupied by $c$ distinct types of atoms. To ``analyze'' T, we conduct ell measurements that we call_discrete X-rays_. A discrete X-ray in direction xi determines the number of atoms of each type on each line parallel to xi. Given ell such non-parallel X-rays, we wish to reconstruct T. The complexity of the problem for c=1 (one atom type) has been completely determined by Gardner, Gritzmann and Prangenberg, who proved that the problem is NP-complete for any dimension D>=2 and ell>=3 non-parallel X-rays, and that it can be solved in polynomial time otherwise. The NP-completeness result above clearly extends to any c>=2, and therefore when studying the polyatomic case we can assume that ell=2. As shown in another article by the same authors, this problem is also NP-complete for c>=6 atoms, even for dimension D=2 and axis-parallel X-rays. They conjecture that the problem remains NP-complete for c=3,4,5, although, as they point out, the proof idea does not seem to extend to c<=5. We resolve the conjecture by proving that the problem is indeed NP-complete for c>=3 in 2D, even for axis-parallel X-rays. Our construction relies heavily on some structure results for the realizations of 0-1 matrices with given row and column sums.

cs/9906021

Christoph D\"urr

Christoph Durr and Marek Chrobak

Reconstructing hv-Convex Polyominoes from Orthogonal Projections

Information Processing Letters, 69, 1999, 283-289

cs.DS

Tomography is the area of reconstructing objects from projections. Here we wish to reconstruct a set of cells in a two dimensional grid, given the number of cells in every row and column. The set is required to be an hv-convex polyomino, that is all its cells must be connected and the cells in every row and column must be consecutive. A simple, polynomial algorithm for reconstructing hv-convex polyominoes is provided, which is several orders of magnitudes faster than the best previously known algorithm from Barcucci et al. In addition, the problem of reconstructing a special class of centered hv-convex polyominoes is addressed. (An object is centered if it contains a row whose length equals the total width of the object). It is shown that in this case the reconstruction problem can be solved in linear time.

cs/9906024

Christoph D\"urr

Christoph Durr, Huong LeThanh and Miklos Santha

A decision procedure for well-formed linear quantum cellular automata

Random Structures and Algorithms 11, 381-394, 1997

cs.DS cs.CC quant-ph

In this paper we introduce a new quantum computation model, the linear quantum cellular automaton. Well-formedness is an essential property for any quantum computing device since it enables us to define the probability of a configuration in an observation as the squared magnitude of its amplitude. We give an efficient algorithm which decides if a linear quantum cellular automaton is well-formed. The complexity of the algorithm is $O(n^2)$ in the algebraic model of computation if the input automaton has continuous neighborhood.

cs/9907001

David Eppstein

David Eppstein

Setting Parameters by Example

13 pages, 3 figures. To be presented at 40th IEEE Symp. Foundations of Computer Science (FOCS '99)

SIAM J. Computing 32(3):643-653, 2003

10.1137/S0097539700370084

cs.DS cs.CG

We introduce a class of "inverse parametric optimization" problems, in which one is given both a parametric optimization problem and a desired optimal solution; the task is to determine parameter values that lead to the given solution. We describe algorithms for solving such problems for minimum spanning trees, shortest paths, and other "optimal subgraph" problems, and discuss applications in multicast routing, vehicle path planning, resource allocation, and board game programming.

cs/9907011

Ming-Yang Kao

Zhi-Zhong Chen and Ming-Yang Kao

Reducing Randomness via Irrational Numbers

SIAM Journal on Computing, 29(4):1247--1256, 2000

cs.DS cs.DM

We propose a general methodology for testing whether a given polynomial with integer coefficients is identically zero. The methodology evaluates the polynomial at efficiently computable approximations of suitable irrational points. In contrast to the classical technique of DeMillo, Lipton, Schwartz, and Zippel, this methodology can decrease the error probability by increasing the precision of the approximations instead of using more random bits. Consequently, randomized algorithms that use the classical technique can generally be improved using the new methodology. To demonstrate the methodology, we discuss two nontrivial applications. The first is to decide whether a graph has a perfect matching in parallel. Our new NC algorithm uses fewer random bits while doing less work than the previously best NC algorithm by Chari, Rohatgi, and Srinivasan. The second application is to test the equality of two multisets of integers. Our new algorithm improves upon the previously best algorithms by Blum and Kannan and can speed up their checking algorithm for sorting programs on a large range of inputs.

cs/9907015

Ming-Yang Kao

Ming-Yang Kao and Jie Wang

Linear-Time Approximation Algorithms for Computing Numerical Summation with Provably Small Errors

SIAM Journal on Computing, 29(5):1568--1576, 2000

cs.DS cs.NA

Given a multiset $X=\{x_1,..., x_n\}$ of real numbers, the {\it floating-point set summation} problem asks for $S_n=x_1+...+x_n$. Let $E^*_n$ denote the minimum worst-case error over all possible orderings of evaluating $S_n$. We prove that if $X$ has both positive and negative numbers, it is NP-hard to compute $S_n$ with the worst-case error equal to $E^*_n$. We then give the first known polynomial-time approximation algorithm that has a provably small error for arbitrary $X$. Our algorithm incurs a worst-case error at most $2(\mix)E^*_n$.\footnote{All logarithms $\log$ in this paper are base 2.} After $X$ is sorted, it runs in O(n) time. For the case where $X$ is either all positive or all negative, we give another approximation algorithm with a worst-case error at most $\lceil\log\log n\rceil E^*_n$. Even for unsorted $X$, this algorithm runs in O(n) time. Previously, the best linear-time approximation algorithm had a worst-case error at most $\lceil\log n\rceil E^*_n$, while $E^*_n$ was known to be attainable in $O(n \log n)$ time using Huffman coding.

cs/9910013

Michelangelo Grigni

Zhi-Zhong Chen, Michelangelo Grigni, Christos Papadimitriou

Map Graphs

46 pages, LaTeX with 41 PS figures; see http://www.mathcs.emory.edu/~mic/mapgraphs/ for hi-res figures

cs.DM cs.DS

We consider a modified notion of planarity, in which two nations of a map are considered adjacent when they share any point of their boundaries (not necessarily an edge, as planarity requires). Such adjacencies define a map graph. We give an NP characterization for such graphs, and a cubic time recognition algorithm for a restricted version: given a graph, decide whether it is realized by adjacencies in a map without holes, in which at most four nations meet at any point.

cs/9911003

David Eppstein

David Eppstein

Subgraph Isomorphism in Planar Graphs and Related Problems

27 pages, 6 figures. A preliminary version of this paper appeared at the 6th ACM-SIAM Symp. Discrete Algorithms, 1995

J. Graph Algorithms & Applications 3(3):1-27, 1999

cs.DS

We solve the subgraph isomorphism problem in planar graphs in linear time, for any pattern of constant size. Our results are based on a technique of partitioning the planar graph into pieces of small tree-width, and applying dynamic programming within each piece. The same methods can be used to solve other planar graph problems including connectivity, diameter, girth, induced subgraph isomorphism, and shortest paths.

cs/9912001

Gabriel Istrate

Gabriel Istrate

The phase transition in random Horn satisfiability and its algorithmic implications

26 pages. Journal version of papers in AIM'98, SODA'99. Submitted to Random Structures and Algorithms

cs.DS cs.CC

Let c>0 be a constant, and $\Phi$ be a random Horn formula with n variables and $m=c\cdot 2^{n}$ clauses, chosen uniformly at random (with repetition) from the set of all nonempty Horn clauses in the given variables. By analyzing \PUR, a natural implementation of positive unit resolution, we show that $\lim_{n\goesto \infty} \PR ({$\Phi$ is satisfiable})= 1-F(e^{-c})$, where $F(x)=(1-x)(1-x^2)(1-x^4)(1-x^8)... $. Our method also yields as a byproduct an average-case analysis of this algorithm.

cs/9912014

David Eppstein

David Eppstein

Fast Hierarchical Clustering and Other Applications of Dynamic Closest Pairs

20 pages, 9 figures. A preliminary version of this paper appeared at the 9th ACM-SIAM Symp. on Discrete Algorithms, San Francisco, 1998, pp. 619-628. For source code and experimental results, see http://www.ics.uci.edu/~eppstein/projects/pairs/

J. Experimental Algorithmics 5(1):1-23, 2000

10.1145/351827.351829

cs.DS

We develop data structures for dynamic closest pair problems with arbitrary distance functions, that do not necessarily come from any geometric structure on the objects. Based on a technique previously used by the author for Euclidean closest pairs, we show how to insert and delete objects from an n-object set, maintaining the closest pair, in O(n log^2 n) time per update and O(n) space. With quadratic space, we can instead use a quadtree-like structure to achieve an optimal time bound, O(n) per update. We apply these data structures to hierarchical clustering, greedy matching, and TSP heuristics, and discuss other potential applications in machine learning, Groebner bases, and local improvement algorithms for partition and placement problems. Experiments show our new methods to be faster in practice than previously used heuristics.

cs/9912020

Vladimir Pestov

Markus Hegland and Vladimir Pestov

Additive models in high dimensions

LaTeX 2e document, 21 pages, 5 figures

Proc. of 12th Computational Techniques and Applications Conference, CTAC-2004 (Rob May and A.J. Roberts, eds.), ANZIAM J. 46 (2005), C1205-C1221.

cs.DS

We discuss some aspects of approximating functions on high-dimensional data sets with additive functions or ANOVA decompositions, that is, sums of functions depending on fewer variables each. It is seen that under appropriate smoothness conditions, the errors of the ANOVA decompositions are of order $O(n^{m/2})$ for approximations using sums of functions of up to $m$ variables under some mild restrictions on the (possibly dependent) predictor variables. Several simulated examples illustrate this behaviour.

math-ph/0701043

Markus Jalsenius

Markus Jalsenius

Strong Spatial Mixing and Rapid Mixing with Five Colours for the Kagome Lattice

34 pages, 11 figures

math-ph cs.DM cs.DS math.MP

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

We consider proper 5-colourings of the kagome lattice. Proper q-colourings correspond to configurations in the zero-temperature q-state anti-ferromagnetic Potts model. Salas and Sokal have given a computer assisted proof of strong spatial mixing on the kagome lattice for q>=6 under any temperature, including zero temperature. It is believed that there is strong spatial mixing for q>=4. Here we give a computer assisted proof of strong spatial mixing for q=5 and zero temperature. It is commonly known that strong spatial mixing implies that there is a unique infinite-volume Gibbs measure and that the Glauber dynamics is rapidly mixing. We give a proof of rapid mixing of the Glauber dynamics on any finite subset of the vertices of the kagome lattice, provided that the boundary is free (not coloured). The Glauber dynamics is not necessarily irreducible if the boundary is chosen arbitrarily for q=5 colours. The Glauber dynamics can be used to uniformly sample proper 5-colourings. Thus, a consequence of rapidly mixing Glauber dynamics is that there is fully polynomial randomised approximation scheme for counting the number of proper 5-colourings.

math/0005235

James Allen Fill

James Allen Fill (Johns Hopkins Univ.), Svante Janson (Uppsala Univ.)

Smoothness and decay properties of the limiting Quicksort density function

11 pages. Refereed article, to apppear in a book edited by D. Gardy and A. Mokkadem and published in 2000 by Birkhauser

601, Department of Mathematical Sciences, The Johns Hopkins University

math.PR cs.DS

Using Fourier analysis, we prove that the limiting distribution of the standardized random number of comparisons used by Quicksort to sort an array of n numbers has an everywhere positive and infinitely differentiable density f, and that each derivative f^{(k)} enjoys superpolynomial decay at plus and minus infinity. In particular, each f^{(k)} is bounded. Our method is sufficiently computational to prove, for example, that f is bounded by 16.

math/0005236

James Allen Fill

James Allen Fill (Johns Hopkins Univ.), Svante Janson (Uppsala Univ.)

A characterization of the set of fixed points of the Quicksort transformation

9 pages. See also http://www.mts.jhu.edu/~fill/ and http://www.math.uu.se/~svante/papers . Submitted for publication in May,2000

606, Department of Mathematical Sciences, The Johns Hopkins University

math.PR cs.DS

The limiting distribution \mu of the normalized number of key comparisons required by the Quicksort sorting algorithm is known to be the unique fixed point of a certain distributional transformation T -- unique, that is, subject to the constraints of zero mean and finite variance. We show that a distribution is a fixed point of T if and only if it is the convolution of \mu with a Cauchy distribution of arbitrary center and scale. In particular, therefore, \mu is the unique fixed point of T having zero mean.

math/0005237

James Allen Fill

Luc Devroye (McGill Univ.), James Allen Fill (Johns Hopkins Univ.), Ralph Neininger (Univ. Freiburg)

Perfect simulation from the Quicksort limit distribution

7 pages. See also http://www.mts.jhu.edu/~fill/, http://www-cgrl.cs.mcgill.ca/~luc/, and http://www.stochastik.uni-freiburg.de/homepages/neininger/ . Submitted for publication in May, 2000

603, Department of Mathematical Sciences, The Johns Hopkins University

math.PR cs.DS

The weak limit of the normalized number of comparisons needed by the Quicksort algorithm to sort n randomly permuted items is known to be determined implicitly by a distributional fixed-point equation. We give an algorithm for perfect random variate generation from this distribution.

math/0012036

Howard Kleiman

Howard Kleiman (Prof. Emer., Queensborough Community Coll. (CUNY))

Hamilton Circuits in Graphs and Directed Graphs

Text in Word 97, Equations in MathType, sent in a PDF file using Adobe Acrobat Writer (4.05), no figures

math.CO cs.DS

We give polynomial-time algorithms for obtaining hamilton circuits in random graphs, G, and random directed graphs, D. If n is finite, we assume that G or D contains a hamilton circuit. If G is an arbitrary graph containing a hamilton circuit, we conjecture that Algorithm G always obtains a hamilton circuit in polynomial time.

math/0111309

Howard Kleiman

Howard Kleiman

The Floyd-Warshall Algorithm, the AP and the TSP

Text in Word 2000, math in MathType 4.0, sent in a PDF file written in Acrobat 5.0, 23 pages

math.CO cs.DS

We use admissible permutations and a variant of the Floyd-Warshall algorithm to obtain an optimal solution to the Assignment Problem. Using another variant of the F-W algorithm, we obtain an approximate solution to the Traveling Salesman Problem. We also give a sufficient condition for the approximate solution to be an optimal solution.

math/0112052

Howard Kleiman

Howard Kleiman

The Floyd-Warshall Algorithm, the AP and the TSP, Part II

Text in Word 2000, math in Math Type 4.0, sent in a PDF file written in Acrobat 5.0, 63 pages

math.CO cs.DS

In math.CO/0111309, we used admissible permutations and a variant of the Floyd-Warshall Algorithm to obtain an optimal solution to the Assignment Problem and an approximate solution to the Traveling Salesman Problem. Here we give a large, detailed illustration of how the algorithms are applied.

math/0207121

Tyll Krueger

Igor Bjelakovic, Tyll Krueger, Rainer Siegmund-Schultze, Arleta Szkola

The Shannon-McMillan Theorem for Ergodic Quantum Lattice Systems

21 pages, extended version concerning subalgebras

math.DS cs.DS cs.IT math-ph math.IT math.MP math.OA quant-ph

We formulate and prove a quantum Shannon-McMillan theorem. The theorem demonstrates the significance of the von Neumann entropy for translation invariant ergodic quantum spin systems on n-dimensional lattices: the entropy gives the logarithm of the essential number of eigenvectors of the system on large boxes. The one-dimensional case covers quantum information sources and is basic for coding theorems.

math/0209316

Thomas Zaslavsky

Konstantin Rybnikov (University of Massachusetts at Lowell and MSRI), Thomas Zaslavsky (Binghamton University)

Cycle and Circle Tests of Balance in Gain Graphs: Forbidden Minors and Their Groups

19 pages, 3 figures. Format: Latex2e. Changes: minor. To appear in Journal of Graph Theory

J. Graph Theory, 51 (2006), no. 1, 1--21.

math.CO cs.DM cs.DS

We examine two criteria for balance of a gain graph, one based on binary cycles and one on circles. The graphs for which each criterion is valid depend on the set of allowed gain groups. The binary cycle test is invalid, except for forests, if any possible gain group has an element of odd order. Assuming all groups are allowed, or all abelian groups, or merely the cyclic group of order 3, we characterize, both constructively and by forbidden minors, the graphs for which the circle test is valid. It turns out that these three classes of groups have the same set of forbidden minors. The exact reason for the importance of the ternary cyclic group is not clear.

math/0210052

Konstantin Rybnikov

Konstantin Rybnikov, Thomas Zaslavsky

Criteria for Balance in Abelian Gain Graphs, with Applications to Piecewise-Linear Geometry

Changes(28 Dec. 2004): revised title and abstract; shortened, mainly by omitting inessentials; minor errors fixed. Changes (16 Jan. 2005): ADDED--Appendix with detailes on some proofs and another counterexample with picture, a few references. Minor typo and notation fixes. To appear in Discrete & Comput. Geometry (without Appendix and extra references)

Discrete and Computational Geometry, 34 (2005), no. 2, 251-268.

math.CO cs.CG cs.DM cs.DS math.AT

A gain graph is a triple (G,h,H), where G is a connected graph with an arbitrary, but fixed, orientation of edges, H is a group, and h is a homomorphism from the free group on the edges of G to H. A gain graph is called balanced if the h-image of each closed walk on G is the identity. Consider a gain graph with abelian gain group having no odd torsion. If there is a basis of the graph's binary cycle space each of whose members can be lifted to a closed walk whose gain is the identity, then the gain graph is balanced, provided that the graph is finite or the group has no nontrivial infinitely 2-divisible elements. We apply this theorem to deduce a result on the projective geometry of piecewise-linear realizations of cell-decompositions of manifolds.

math/0211317

Kamil Kulesza

Kamil Kulesza, Zbigniew Kotulski

On graph coloring check-digit method

7 pages, paper sumitted to Applied Mathematics Letters (Elsevier)

math.CO cs.CR cs.DM cs.DS

We show a method how to convert any graph into the binary number and vice versa. We derive upper bound for maximum number of graphs, that, have fixed number of vertices and can be colored with n colors (n is any given number). Proof for the result is outlined. Next, graph coloring based check-digit scheme is proposed. We use quantitative result derived, to show, that feasibility of the proposed scheme increases with size of the number which digits are checked, and overall probability of digits errors.

math/0302315

Ernie Croot

Ernie Croot

Memory Efficient Arithmetic

Difference between this version and last: Better notation, light corrections, and more explanations

math.NT cs.DS

In this paper we give an algorithm for computing the mth base-b digit (m=1 is the least significant digit) of an integer n (actually, it finds sharp approximations to n/b^m mod 1), where n is defined as the last number in a sequence of integers s1,s2,...,sL=n, where s1=0, s2=1, and each successive si is either the sum, product, or difference of two previous sj's in the sequence. In many cases, the algorithm will find this mth digit using far less memory than it takes to write down all the base-b digits of n, while the number of bit operations will grow only slighly worse than linear in the number of digits. One consequence of this result is that the mth base-10 digit of 2^t can be found using O(t^{2/3} log^C t) bits of storage (for some C>0), and O(t log^C t) bit operations. The algorithm is also highly parallelizable, and an M-fold reduction in running time can be achieved using M processors, although the memory required will then grow by a factor of M.

math/0309285

Jeffrey D. Scargle

Brad Jackson, Jeffrey D. Scargle, David Barnes, Sundararajan Arabhi, Alina Alt, Peter Gioumousis, Elyus Gwin, Paungkaew Sangtrakulcharoen, Linda Tan, and Tun Tao Tsai

An Algorithm for Optimal Partitioning of Data on an Interval

3 pages, 1 figure, submitted to IEEE Signal Processing Letters, revised version with added references

10.1109/LSP.2001.838216

math.NA astro-ph cs.CE cs.DS cs.IT math.CO math.IT

Many signal processing problems can be solved by maximizing the fitness of a segmented model over all possible partitions of the data interval. This letter describes a simple but powerful algorithm that searches the exponentially large space of partitions of $N$ data points in time $O(N^2)$. The algorithm is guaranteed to find the exact global optimum, automatically determines the model order (the number of segments), has a convenient real-time mode, can be extended to higher dimensional data spaces, and solves a surprising variety of problems in signal detection and characterization, density estimation, cluster analysis and classification.

math/0406094

Pr Philippe Chassaing

Philippe Chassaing, Regine Marchand

Merging costs for the additive Marcus-Lushnikov process, and Union-Find algorithms

28 pages, 1 figure

math.PR cs.DS math.CO

Starting with a monodisperse configuration with $n$ size-1 particles, an additive Marcus-Lushnikov process evolves until it reaches its final state (a unique particle with mass $n$). At each of the $n-1$ steps of its evolution, a merging cost is incurred, that depends on the sizes of the two particles involved, and on an independent random factor. This paper deals with the asymptotic behaviour of the cumulated costs up to the $k$th clustering, under various regimes for $(n,k)$, with applications to the study of Union--Find algorithms.

math/0406353

Manor Mendel

Yair Bartal, Nathan Linial, Manor Mendel, Assaf Naor

On metric Ramsey-type phenomena

67 pages, published version

Ann. of Math. (2) 162 (2005), no. 2, 643--709

10.4007/annals.2005.162.643

math.MG cs.DS

The main question studied in this article may be viewed as a nonlinear analogue of Dvoretzky's theorem in Banach space theory or as part of Ramsey theory in combinatorics. Given a finite metric space on n points, we seek its subspace of largest cardinality which can be embedded with a given distortion in Hilbert space. We provide nearly tight upper and lower bounds on the cardinality of this subspace in terms of n and the desired distortion. Our main theorem states that for any epsilon>0, every n point metric space contains a subset of size at least n^{1-\epsilon} which is embeddable in Hilbert space with O(\frac{\log(1/\epsilon)}{\epsilon}) distortion. The bound on the distortion is tight up to the log(1/\epsilon) factor. We further include a comprehensive study of various other aspects of this problem.

math/0410593

Henrik B\"a\"arnhielm

Henrik B\"a\"arnhielm

The Schreier-Sims algorithm for matrix groups

The author's MSc thesis. Uses AMS-LaTeX and algorithm2e.sty. For the associated source code, see http://matrixss.sourceforge.net/

math.GR cs.DS

This is the report of a project with the aim to make a new implementation of the Schreier-Sims algorithm in GAP, specialized for matrix groups. The standard Schreier-Sims algorithm is described in some detail, followed by descriptions of the probabilistic Schreier-Sims algorithm and the Schreier-Todd-Coxeter-Sims algorithm. Then we discuss our implementation and some optimisations, and finally we report on the performance of our implementation, as compared to the existing implementation in GAP, and we give benchmark results. The conclusion is that our implementation in some cases is faster and consumes much less memory.

math/0411128

Cyril Banderier

Cyril Banderier (LIPN), Sylviane Schwer (LIPN)

Why Delannoy numbers?

Presented to the conference "Lattice Paths Combinatorics and Discrete Distributions" (Athens, June 5-7, 2002) and to appear in the Journal of Statistical Planning and Inferences

Journal of Statistical Planning and Inference 135, 1 (11/2005) 40-54

10.1016/j.jspi.2005.02.004

math.CO cs.DS cs.GT math.HO math.PR math.ST q-bio.GN stat.TH

This article is not a research paper, but a little note on the history of combinatorics: We present here a tentative short biography of Henri Delannoy, and a survey of his most notable works. This answers to the question raised in the title, as these works are related to lattice paths enumeration, to the so-called Delannoy numbers, and were the first general way to solve Ballot-like problems. These numbers appear in probabilistic game theory, alignments of DNA sequences, tiling problems, temporal representation models, analysis of algorithms and combinatorial structures.

math/0411138

Cyril Banderier

Cyril Banderier (LIPN), Jean-Marie Le Bars (LIPN, GREYC), Vlady Ravelomanana (LIPN)

Generating Functions For Kernels of Digraphs (Enumeration & Asymptotics for Nim Games)

Presented (as a poster) to the conference Formal Power Series and Algebraic Combinatorics (Vancouver, 2004), electronic proceedings

Proceedings of FPSAC'04 (2004) 91-105

math.CO cs.DM cs.DS cs.GT math.PR

In this article, we study directed graphs (digraphs) with a coloring constraint due to Von Neumann and related to Nim-type games. This is equivalent to the notion of kernels of digraphs, which appears in numerous fields of research such as game theory, complexity theory, artificial intelligence (default logic, argumentation in multi-agent systems), 0-1 laws in monadic second order logic, combinatorics (perfect graphs)... Kernels of digraphs lead to numerous difficult questions (in the sense of NP-completeness, #P-completeness). However, we show here that it is possible to use a generating function approach to get new informations: we use technique of symbolic and analytic combinatorics (generating functions and their singularities) in order to get exact and asymptotic results, e.g. for the existence of a kernel in a circuit or in a unicircuit digraph. This is a first step toward a generatingfunctionology treatment of kernels, while using, e.g., an approach "a la Wright". Our method could be applied to more general "local coloring constraints" in decomposable combinatorial structures.

math/0411250

Cyril Banderier

Cyril Banderier (LIPN, ALGO UR-R), Philippe Flajolet (ALGO UR-R), Daniele Gardy (PRISM), Mireille Bousquet-Melou (LABRI), Alain Denise (LRI), Dominique Gouyou-Beauchamps (LRI)

Generating functions for generating trees

This article corresponds, up to minor typo corrections, to the article submitted to Discrete Mathematics (Elsevier) in Nov. 1999, and published in its vol. 246(1-3), March 2002, pp. 29-55

Discrete Mathematics 246 (1-3) (2002) 29-55

10.1016/S0012-365X(01)00250-3

math.CO cs.DM cs.DS

Certain families of combinatorial objects admit recursive descriptions in terms of generating trees: each node of the tree corresponds to an object, and the branch leading to the node encodes the choices made in the construction of the object. Generating trees lead to a fast computation of enumeration sequences (sometimes, to explicit formulae as well) and provide efficient random generation algorithms. We investigate the links between the structural properties of the rewriting rules defining such trees and the rationality, algebraicity, or transcendence of the corresponding generating function.

math/0502232

Svante Janson

Svante Janson

Individual displacements in hashing with coalesced chains

17 pages

U.U.D.M. 2005:4

math.PR cs.DS

We study the asymptotic distribution of the displacements in hashing with coalesced chains, for both late-insertion and early-insertion. Asymptotic formulas for means and variances follow. The method uses Poissonization and some stochastic calculus.

math/0508183

Pavel Chebotarev

P. Yu. Chebotarev and E. V. Shamis

On a Duality between Metrics and $\Sigma$-Proximities

5 pages

Automation and Remote Control 59 (1998) 608--612

math.MG cs.DS math.CO

: In studies of discrete structures, functions are frequently used that express proximity, but are not metrics. We consider a class of such functions that is characterized by a normalization condition and an inequality that plays the same role as the triangle inequality does for metrics. We show that the introduced functions, named $\Sigma$-proximities, are in a definite sense dual to metrics: there exists a natural one-to-one correspondence between metrics and $\Sigma$-proximities defined on the same finite set; in contrast to metrics, $\Sigma$-proximities measure {\it comparative} proximity; the closer the objects, the greater the $\Sigma$-proximity; diagonal entries of the $\Sigma$-proximity matrix characterize the ``centrality'' of elements. The results are extended to arbitrary infinite sets.

math/0508199

Pavel Chebotarev

Pavel Chebotarev

Extending Utility Representations of Partial Orders

15 pages

In: Constructing and Applying Objective Functions. Lecture Notes in Economics and Math. Systems, Vol.510, Springer, 2002, P. 63-74

10.1007/978-3-642-56038-5_4

math.OC cs.DS math.FA

The problem is considered as to whether a monotone function defined on a subset P of a Euclidean space can be strictly monotonically extended to the whole space. It is proved that this is the case if and only if the function is {\em separably increasing}. Explicit formulas are given for a class of extensions which involves an arbitrary bounded increasing function. Similar results are obtained for monotone functions that represent strict partial orders on arbitrary abstract sets X. The special case where P is a Pareto subset is considered.

math/0508212

Howard Kleiman

Howard Kleiman

The Symmetric Traveling Salesman Problem

A new theorem has been added

math.CO cs.DS

Let M be an nXn symetric matrix, n, even, T, an upper bound for T_OPT, an optimal tour, sigma_T, the smaller-valued perfect matching obtained from alternate edges of T expressed as a product of 2-cycles. Applying the modified Floyd-Warshall algorithm to (sigma_T)^-1M^-, we construct acceptable and 2-circuit cycles some sets of which may yield circuits that can be patched into tours. We obtain necessary and sufficient conditions for a set, S, of cycles to yield circuits that may be patched into a tour.Assume that the following (Condition A)is valid: If (sigma_T)s = T*, |T*|<T, then all cycles of s have values less than |T| - |sigma_T|.Let SFWOPT),S(OPT)be the respective sets of cycles yielding T_FWOPT, T_OPT. Given Condition(A), using F-W, we can always obtain S(FWOPT). Using Condition A but not F-W, S_OPT is always obtainable from a subset of the cycles obtained.

math/0509575

Elchanan Mossel

Constantinos Daskalakis, Elchanan Mossel, Sebastien Roch

Evolutionary Trees and the Ising Model on the Bethe Lattice: a Proof of Steel's Conjecture

Second major revision. Updated proofs and statements

math.PR cs.CE cs.DS math.CA math.CO math.ST q-bio.PE stat.TH

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

A major task of evolutionary biology is the reconstruction of phylogenetic trees from molecular data. The evolutionary model is given by a Markov chain on a tree. Given samples from the leaves of the Markov chain, the goal is to reconstruct the leaf-labelled tree. It is well known that in order to reconstruct a tree on $n$ leaves, sample sequences of length $\Omega(\log n)$ are needed. It was conjectured by M. Steel that for the CFN/Ising evolutionary model, if the mutation probability on all edges of the tree is less than $p^{\ast} = (\sqrt{2}-1)/2^{3/2}$, then the tree can be recovered from sequences of length $O(\log n)$. The value $p^{\ast}$ is given by the transition point for the extremality of the free Gibbs measure for the Ising model on the binary tree. Steel's conjecture was proven by the second author in the special case where the tree is "balanced." The second author also proved that if all edges have mutation probability larger than $p^{\ast}$ then the length needed is $n^{\Omega(1)}$. Here we show that Steel's conjecture holds true for general trees by giving a reconstruction algorithm that recovers the tree from $O(\log n)$-length sequences when the mutation probabilities are discretized and less than $p^\ast$. Our proof and results demonstrate that extremality of the free Gibbs measure on the infinite binary tree, which has been studied before in probability, statistical physics and computer science, determines how distinguishable are Gibbs measures on finite binary trees.

math/0510573

Hossein Zare

Shmuel Friedland, Mostafa Kaveh, Amir Niknejad, Hossein Zare

Fast Monte-Carlo Low Rank Approximations for Matrices

math.NA cs.DS

In many applications, it is of interest to approximate data, given by mxn matrix A, by a matrix B of at most rank k, which is much smaller than m and n. The best approximation is given by singular value decomposition, which is too time consuming for very large m and n. We present here a Monte Carlo algorithm for iteratively computing a k-rank approximation to the data consisting of mxn matrix A. Each iteration involves the reading of O(k) of columns or rows of A. The complexity of our algorithm is O(kmn). Our algorithm, distinguished from other known algorithms, guarantees that each iteration is a better k-rank approximation than the previous iteration. We believe that this algorithm will have many applications in data mining, data storage and data analysis.

math/0602059

Pavel Chebotarev

Rafig Agaev and Pavel Chebotarev

The Matrix of Maximum Out Forests of a Digraph and Its Applications

27 pages, 3 figures

Automation and Remote Control 61 (2000) 1424--1450

math.CO cs.DS math.AG

We study the maximum out forests of a (weighted) digraph and the matrix of maximum out forests. A maximum out forest of a digraph G is a spanning subgraph of G that consists of disjoint diverging trees and has the maximum possible number of arcs. If a digraph contains any out arborescences, then maximum out forests coincide with them. We provide a new proof to the Markov chain tree theorem saying that the matrix of Ces`aro limiting probabilities of an arbitrary stationary finite Markov chain coincides with the normalized matrix of maximum out forests of the weighted digraph that corresponds to the Markov chain. We discuss the applications of the matrix of maximum out forests and its transposition, the matrix of limiting accessibilities of a digraph, to the problems of preference aggregation, measuring the vertex proximity, and uncovering the structure of a digraph.

math/0602073

Pavel Chebotarev

Pavel Chebotarev and Elena Shamis

On Proximity Measures for Graph Vertices

17 pages, 3 figures

Automation and Remote Control 59 (1998), No. 10, Part 2 1443-1459. Erratum: 60 (1999), No. 2, Part 2 297

math.CO cs.DS cs.NI math.MG

We study the properties of several proximity measures for the vertices of weighted multigraphs and multidigraphs. Unlike the classical distance for the vertices of connected graphs, these proximity measures are applicable to weighted structures and take into account not only the shortest, but also all other connections, which is desirable in many applications. To apply these proximity measures to unweighted structures, every edge should be assigned the same weight which determines the proportion of taking account of two routes, from which one is one edge longer than the other. Among the proximity measures we consider path accessibility, route accessibility, relative forest accessibility along with its components, accessibility via dense forests, and connection reliability. A number of characteristic conditions is introduced and employed to characterize the proximity measures. A topological interpretation is obtained for the Moore-Penrose generalized inverse of the Laplacian matrix of a weighted multigraph.

math/0603207

Olga Holtz

James Demmel, Ioana Dumitriu, Olga Holtz, Robert Kleinberg

Fast matrix multiplication is stable

19 pages; final version, expanded and updated to reflect referees' remarks; to appear in Numerische Mathematik

Numer. Math. 106 (2007), no. 2, 199-224

10.1007/s00211-007-0061-6

math.NA cs.CC cs.DS math.GR

We perform forward error analysis for a large class of recursive matrix multiplication algorithms in the spirit of [D. Bini and G. Lotti, Stability of fast algorithms for matrix multiplication, Numer. Math. 36 (1980), 63--72]. As a consequence of our analysis, we show that the exponent of matrix multiplication (the optimal running time) can be achieved by numerically stable algorithms. We also show that new group-theoretic algorithms proposed in [H. Cohn, and C. Umans, A group-theoretic approach to fast matrix multiplication, FOCS 2003, 438--449] and [H. Cohn, R. Kleinberg, B. Szegedy and C. Umans, Group-theoretic algorithms for matrix multiplication, FOCS 2005, 379--388] are all included in the class of algorithms to which our analysis applies, and are therefore numerically stable. We perform detailed error analysis for three specific fast group-theoretic algorithms.

math/0604331

Jean-Francois Marckert

Ali Akhavi (LIAFA), Jean-Fran\c{c}ois Marckert (LaBRI), Alain Rouault (LM-Versailles)

On the reduction of a random basis

math.PR cs.DS

For $g < n$, let $b\_1,...,b\_{n-g}$ be $n - g$ independent vectors in $\mathbb{R}^n$ with a common distribution invariant by rotation. Considering these vectors as a basis for the Euclidean lattice they generate, the aim of this paper is to provide asymptotic results when $n\to +\infty$ concerning the property that such a random basis is reduced in the sense of {\sc Lenstra, Lenstra & Lov\'asz}. The proof passes by the study of the process $(r\_{g+1}^{(n)},r\_{g+2}^{(n)},...,r\_{n-1}^{(n)})$ where $r\_j^{(n)}$ is the ratio of lengths of two consecutive vectors $b^*\_{n-j+1}$ and $b^*\_{n-j}$ built from $(b\_1,...,b\_{n-g})$ by the Gram--Schmidt orthogonalization procedure, which we believe to be interesting in its own. We show that, as $n\to+\infty$, the process $(r\_j^{(n)}-1)\_j$ tends in distribution in some sense to an explicit process $({\mathcal R}\_j -1)\_j$; some properties of this latter are provided.

math/0604367

Sebastian Roch

Shankar Bhamidi, Ram Rajagopal, Sebastien Roch

Network Delay Inference from Additive Metrics

math.PR cs.DS cs.NI math.ST stat.TH

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

We demonstrate the use of computational phylogenetic techniques to solve a central problem in inferential network monitoring. More precisely, we design a novel algorithm for multicast-based delay inference, i.e. the problem of reconstructing the topology and delay characteristics of a network from end-to-end delay measurements on network paths. Our inference algorithm is based on additive metric techniques widely used in phylogenetics. It runs in polynomial time and requires a sample of size only $\poly(\log n)$.

math/0605472

Nicolas Pouyanne

Nicolas Pouyanne (LM-Versailles)

An algebraic approach to Polya processes

Annales de l'IHP - Probabilit\'es et Statistiques (2008) Vol. 44, No. 2, 293-323

10.1214/07-AIHP130

math.CO cs.DM cs.DS math.PR

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

P\'olya processes are natural generalization of P\'olya-Eggenberger urn models. This article presents a new approach of their asymptotic behaviour {\it via} moments, based on the spectral decomposition of a suitable finite difference operator on polynomial functions. Especially, it provides new results for {\it large} processes (a P\'olya process is called {\it small} when 1 is simple eigenvalue of its replacement matrix and when any other eigenvalue has a real part $\leq 1/2$; otherwise, it is called large).

math/0606122

George Bell

George I. Bell

Diagonal Peg Solitaire

20 pages, 11 figures

INTEGERS: Electronic Journal of Combinatorial Number Theory 7 (2007) #G01

math.CO cs.DM cs.DS

We study the classical game of peg solitaire when diagonal jumps are allowed. We prove that on many boards, one can begin from a full board with one peg missing, and finish with one peg anywhere on the board. We then consider the problem of finding solutions that minimize the number of moves (where a move is one or more jumps by the same peg), and find the shortest solution to the "central game", which begins and ends at the center. In some cases we can prove analytically that our solutions are the shortest possible, in other cases we apply A* or bidirectional search heuristics.

math/0607411

Jean-Gabriel Luque

G\'erard Duchamp (LIPN), Eric Laugerotte (LIFAR EA2655), Jean-Gabriel Luque (IGM-LabInfo)

Extending the scalars of minimizations

SCI, \'{E}tats-Unis d'Am\'{e}rique (2001)

math.CO cs.DS cs.SC

In the classical theory of formal languages, finite state automata allow to recognize the words of a rational subset of $\Sigma^*$ where $\Sigma$ is a set of symbols (or the alphabet). Now, given a semiring $(\K,+,.)$, one can construct $\K$-subsets of $\Sigma^*$ in the sense of Eilenberg, that are alternatively called noncommutative formal power series for which a framework very similar to language theory has been constructed Particular noncommutative formal power series, which are called rational series, are the behaviour of a family of weighted automata (or $\K$-automata). In order to get an efficient encoding, it may be interesting to point out one of them with the smallest number of states. Minimization processes of $\K$-automata already exist for $\K$ being: {\bf a)} a field, {\bf b)} a noncommutative field, {\bf c)} a PID . When $\K$ is the bolean semiring, such a minimization process (with isomorphisms of minimal objects) is known within the category of deterministic automata. Minimal automata have been proved to be isomorphic in cases {\bf (a)} and {\bf (b)}. But the proof given for (b) is not constructive. In fact, it lays on the existence of a basis for a submodule of $\K^n$. Here we give an independent algorithm which reproves this fact and an example of a pair of nonisomorphic minimal automata. Moreover, we examine the possibility of extending {\bf (c)}. To this end, we provide an {\em Effective Minimization Process} (or {\em EMP}) which can be used for more general sets of coefficients.

math/0608210

Henrik B\"a\"arnhielm

Henrik B\"a\"arnhielm

Recognising the Suzuki groups in their natural representations

J. Algebra 300 (1), 171-198, 2006

10.1016/j.jalgebra.2006.02.010

math.GR cs.DS

Under the assumption of a certain conjecture, for which there exists strong experimental evidence, we produce an efficient algorithm for constructive membership testing in the Suzuki groups Sz(q), where q = 2^{2m + 1} for some m > 0, in their natural representations of degree 4. It is a Las Vegas algorithm with running time O{log(q)} field operations, and a preprocessing step with running time O{log(q) loglog(q)} field operations. The latter step needs an oracle for the discrete logarithm problem in GF(q). We also produce a recognition algorithm for Sz(q) = <X>. This is a Las Vegas algorithm with running time O{|X|^2} field operations. Finally, we give a Las Vegas algorithm that, given <X>^h = Sz(q) for some h in GL(4, q), finds some g such that <X>^g = Sz(q). The running time is O{log(q) loglog(q) + |X|} field operations. Implementations of the algorithms are available for the computer system MAGMA.

math/0611679

Dominique Rossin

Dominique Rossin (LIAFA), Mathilde Bouvel (LIAFA)

Longest Common Pattern between two Permutations

Algebr. Geom. Topol. 7 (2007) 829-843

10.2140/agt.2007.7.829

math.CO cs.DM cs.DS

In this paper, we give a polynomial (O(n^8)) algorithm for finding a longest common pattern between two permutations of size n given that one is separable. We also give an algorithm for general permutations whose complexity depends on the length of the longest simple permutation involved in one of our permutations.

math/0612264

Olga Holtz

James Demmel, Ioana Dumitriu, Olga Holtz

Fast linear algebra is stable

26 pages; final version; to appear in Numerische Mathematik

Numer. Math. 108 (2007), no. 1, 59-91

10.1007/s00211-007-0114-x

math.NA cs.CC cs.DS

In an earlier paper, we showed that a large class of fast recursive matrix multiplication algorithms is stable in a normwise sense, and that in fact if multiplication of $n$-by-$n$ matrices can be done by any algorithm in $O(n^{\omega + \eta})$ operations for any $\eta > 0$, then it can be done stably in $O(n^{\omega + \eta})$ operations for any $\eta > 0$. Here we extend this result to show that essentially all standard linear algebra operations, including LU decomposition, QR decomposition, linear equation solving, matrix inversion, solving least squares problems, (generalized) eigenvalue problems and the singular value decomposition can also be done stably (in a normwise sense) in $O(n^{\omega + \eta})$ operations.

math/0702325

Oskar Sandberg

Oskar Sandberg

Neighbor selection and hitting probability in small-world graphs

Published in at http://dx.doi.org/10.1214/07-AAP499 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Annals of Applied Probability 2008, Vol. 18, No. 5, 1771-1793

10.1214/07-AAP499

IMS-AAP-AAP499

math.PR cs.DS

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

Small-world graphs, which combine randomized and structured elements, are seen as prevalent in nature. Jon Kleinberg showed that in some graphs of this type it is possible to route, or navigate, between vertices in few steps even with very little knowledge of the graph itself. In an attempt to understand how such graphs arise we introduce a different criterion for graphs to be navigable in this sense, relating the neighbor selection of a vertex to the hitting probability of routed walks. In several models starting from both discrete and continuous settings, this can be shown to lead to graphs with the desired properties. It also leads directly to an evolutionary model for the creation of similar graphs by the stepwise rewiring of the edges, and we conjecture, supported by simulations, that these too are navigable.

math/0702744

Mark Jerrum

Martin Dyer, Leslie Ann Goldberg, Mark Jerrum

Matrix norms and rapid mixing for spin systems

Published in at http://dx.doi.org/10.1214/08-AAP532 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Annals of Applied Probability 2009, Vol. 19, No. 1, 71-107

10.1214/08-AAP532

IMS-AAP-AAP532

math.PR cs.DS

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

We give a systematic development of the application of matrix norms to rapid mixing in spin systems. We show that rapid mixing of both random update Glauber dynamics and systematic scan Glauber dynamics occurs if any matrix norm of the associated dependency matrix is less than 1. We give improved analysis for the case in which the diagonal of the dependency matrix is $\mathbf{0}$ (as in heat bath dynamics). We apply the matrix norm methods to random update and systematic scan Glauber dynamics for coloring various classes of graphs. We give a general method for estimating a norm of a symmetric nonregular matrix. This leads to improved mixing times for any class of graphs which is hereditary and sufficiently sparse including several classes of degree-bounded graphs such as nonregular graphs, trees, planar graphs and graphs with given tree-width and genus.

math/0703921

Louis Theran

Ileana Streinu and Louis Theran

Sparse Hypergraphs and Pebble Game Algorithms

math.CO cs.DS

A hypergraph $G=(V,E)$ is $(k,\ell)$-sparse if no subset $V'\subset V$ spans more than $k|V'|-\ell$ hyperedges. We characterize $(k,\ell)$-sparse hypergraphs in terms of graph theoretic, matroidal and algorithmic properties. We extend several well-known theorems of Haas, Lov{\'{a}}sz, Nash-Williams, Tutte, and White and Whiteley, linking arboricity of graphs to certain counts on the number of edges. We also address the problem of finding lower-dimensional representations of sparse hypergraphs, and identify a critical behaviour in terms of the sparsity parameters $k$ and $\ell$. Our constructions extend the pebble games of Lee and Streinu from graphs to hypergraphs.

math/0703927

Christine Cheng

V. Arvind, Christine T. Cheng, Nikhil R. Devanur

On Computing the Distinguishing Numbers of Planar Graphs and Beyond: a Counting Approach

27 pages

math.CO cs.DS

A vertex k-labeling of graph G is distinguishing if the only automorphism that preserves the labels of G is the identity map. The distinguishing number of G, D(G), is the smallest integer k for which G has a distinguishing k-labeling. In this paper, we apply the principle of inclusion-exclusion and develop recursive formulas to count the number of inequivalent distinguishing k-labelings of a graph. Along the way, we prove that the distinguishing number of a planar graph can be computed in time polynomial in the size of the graph.}

math/9610221

J. Maurice Rojas

Extensions and Corrections for: ``A Convex Geometric Approach to Counting the Roots of a Polynomial System''

MSRI 1996-075

math.AG cs.DS

This brief note corrects some errors in the paper quoted in the title, highlights a combinatorial result which may have been overlooked, and points to further improvements in recent literature.

math/9702221

J. Maurice Rojas

Some New Applications of Toric Geometry

MSRI 1997-016

math.AG cs.DS

This paper reexamines univariate reduction from a toric geometric point of view. We begin by constructing a binomial variant of the $u$-resultant and then retailor the generalized characteristic polynomial to fully exploit sparsity in the monomial structure of any given polynomial system. We thus obtain a fast new algorithm for univariate reduction and a better understanding of the underlying projections. As a corollary, we show that a refinement of Hilbert's Tenth Problem is decidable within single-exponential time. We also show how certain multisymmetric functions of the roots of polynomial systems can be calculated with sparse resultants.

math/9702222

J. Maurice Rojas

Toric Generalized Characteristic Polynomials

MSRI 1997-017

math.AG cs.DS

We illustrate an efficient new method for handling polynomial systems with degenerate solution sets. In particular, a corollary of our techniques is a new algorithm to find an isolated point in every excess component of the zero set (over an algebraically closed field) of any $n$ by $n$ system of polynomial equations. Since we use the sparse resultant, we thus obtain complexity bounds (for converting any input polynomial system into a multilinear factorization problem) which are close to cubic in the degree of the underlying variety -- significantly better than previous bounds which were pseudo-polynomial in the classical B\'ezout bound. By carefully taking into account the underlying toric geometry, we are also able to improve the reliability of certain sparse resultant based algorithms for polynomial system solving.

math/9704218

Alexander Barvinok

Alexander Barvinok

A simple polynomial time algorithm to approximate the permanent within a simply exponential factor

MSRI 1997-031, formerly math.LA/9704218

math.RA cs.DS

We present a simple randomized polynomial time algorithm to approximate the mixed discriminant of $n$ positive semidefinite $n \times n$ matrices within a factor $2^{O(n)}$. Consequently, the algorithm allows us to approximate in randomized polynomial time the permanent of a given $n \times n$ non-negative matrix within a factor $2^{O(n)}$. When applied to approximating the permanent, the algorithm turns out to be a simple modification of the well-known Godsil-Gutman estimator.

physics/0302034

Marek W. Gutowski

Marek W. Gutowski

Power and beauty of interval methods

Short, yet highly informative introduction into interval methods with immediate application to experimental data analysis. To be presented on May 26-29, 2003, VI Domestic Conference on Evolutionary Algorithms and Global Optimization, Poland (invited talk). 8 pages, no figures, LaTex2e. Improved layout, simplified notation, keyword list extended

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

Interval calculus is a relatively new branch of mathematics. Initially understood as a set of tools to assess the quality of numerical calculations (rigorous control of rounding errors), it became a discipline in its own rights today. Interval methods are usefull whenever we have to deal with uncertainties, which can be rigorously bounded. Fuzzy sets, rough sets and probability calculus can perform similar tasks, yet only the interval methods are able to (dis)prove, with mathematical rigor, the (non)existence of desired solution(s). Known are several problems, not presented here, which cannot be effectively solved by any other means. This paper presents basic notions and main ideas of interval calculus and two examples of useful algorithms.

physics/0509039

Bernardo Huberman

Jure Leskovec, Lada A. Adamic and Bernardo A. Huberman

The Dynamics of Viral Marketing

Leskovec, J., Adamic, L. A., and Huberman, B. A. 2007. The dynamics of viral marketing. ACM Transactions on the Web, 1, 1 (May 2007)

10.1145/1232722.1232727

physics.soc-ph cond-mat.stat-mech cs.DB cs.DS

We present an analysis of a person-to-person recommendation network, consisting of 4 million people who made 16 million recommendations on half a million products. We observe the propagation of recommendations and the cascade sizes, which we explain by a simple stochastic model. We analyze how user behavior varies within user communities defined by a recommendation network. Product purchases follow a 'long tail' where a significant share of purchases belongs to rarely sold items. We establish how the recommendation network grows over time and how effective it is from the viewpoint of the sender and receiver of the recommendations. While on average recommendations are not very effective at inducing purchases and do not spread very far, we present a model that successfully identifies communities, product and pricing categories for which viral marketing seems to be very effective.

quant-ph/0101133

Paul Vitanyi

Harry Buhrman (CWI and Univ. Amsterdam), J. Tromp (CWI and BioInformatics Solutions), Paul Vitanyi (CWI and Univ. Amsterdam)

Time and Space Bounds for Reversible Simulation

11 pages LaTeX, Proc ICALP 2001, Lecture Notes in Computer Science, Vol xxx Springer-Verlag, Berlin, 2001

Journal of Physics A: Mathematical and General, 34(2001), 6821--6830.

10.1088/0305-4470/34/35/308

quant-ph cs.CC cs.DS

We prove a general upper bound on the tradeoff between time and space that suffices for the reversible simulation of irreversible computation. Previously, only simulations using exponential time or quadratic space were known. The tradeoff shows for the first time that we can simultaneously achieve subexponential time and subquadratic space. The boundary values are the exponential time with hardly any extra space required by the Lange-McKenzie-Tapp method and the ($\log 3$)th power time with square space required by the Bennett method. We also give the first general lower bound on the extra storage space required by general reversible simulation. This lower bound is optimal in that it is achieved by some reversible simulations.

quant-ph/0210064

Julia Kempe

Neil Shenvi, Julia Kempe, and K. Birgitta Whaley

A Quantum Random Walk Search Algorithm

13 pages, 3 figures

Phys. Rev. A, Vol. 67 (5), 052307 (2003)

10.1103/PhysRevA.67.052307

quant-ph cs.DS

Quantum random walks on graphs have been shown to display many interesting properties, including exponentially fast hitting times when compared with their classical counterparts. However, it is still unclear how to use these novel properties to gain an algorithmic speed-up over classical algorithms. In this paper, we present a quantum search algorithm based on the quantum random walk architecture that provides such a speed-up. It will be shown that this algorithm performs an oracle search on a database of $N$ items with $O(\sqrt{N})$ calls to the oracle, yielding a speed-up similar to other quantum search algorithms. It appears that the quantum random walk formulation has considerable flexibility, presenting interesting opportunities for development of other, possibly novel quantum algorithms.

quant-ph/0303081

Julia Kempe

Julia Kempe

Quantum random walks - an introductory overview

20 pages, 13 figures, to appear in Contemporary Physics

Contemporary Physics, Vol. 44 (4), p.307-327, 2003

10.1080/00107151031000110776

quant-ph cs.DS

This article aims to provide an introductory survey on quantum random walks. Starting from a physical effect to illustrate the main ideas we will introduce quantum random walks, review some of their properties and outline their striking differences to classical walks. We will touch upon both physical effects and computer science applications, introducing some of the main concepts and language of present day quantum information science in this context. We will mention recent developments in this new area and outline some open questions.

quant-ph/0311001

Andris Ambainis

Andris Ambainis

Quantum walk algorithm for element distinctness

33 pages, 1 figure, v9 typos with signs corrected on pages 11-12

SIAM Journal on Computing, 37(1):210-239, 2007

quant-ph cs.DS

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

We use quantum walks to construct a new quantum algorithm for element distinctness and its generalization. For element distinctness (the problem of finding two equal items among N given items), we get an O(N^{2/3}) query quantum algorithm. This improves the previous O(N^{3/4}) query quantum algorithm of Buhrman et.al. (quant-ph/0007016) and matches the lower bound by Shi (quant-ph/0112086). The algorithm also solves the generalization of element distinctness in which we have to find k equal items among N items. For this problem, we get an O(N^{k/(k+1)}) query quantum algorithm.

quant-ph/0402107

Julia Kempe

Andris Ambainis, Julia Kempe and Alexander Rivosh

Coins Make Quantum Walks Faster

25 pages, no figures

Proc. 16th ACM-SIAM SODA, p. 1099-1108 (2005)

quant-ph cs.DS

We show how to search N items arranged on a $\sqrt{N}\times\sqrt{N}$ grid in time $O(\sqrt N \log N)$, using a discrete time quantum walk. This result for the first time exhibits a significant difference between discrete time and continuous time walks without coin degrees of freedom, since it has been shown recently that such a continuous time walk needs time $\Omega(N)$ to perform the same task. Our result furthermore improves on a previous bound for quantum local search by Aaronson and Ambainis. We generalize our result to 3 and more dimensions where the walk yields the optimal performance of $O(\sqrt{N})$ and give several extensions of quantum walk search algorithms for general graphs. The coin-flip operation needs to be chosen judiciously: we show that another ``natural'' choice of coin gives a walk that takes $\Omega(N)$ steps. We also show that in 2 dimensions it is sufficient to have a two-dimensional coin-space to achieve the time $O(\sqrt{N} \log N)$.

quant-ph/0403120

Andris Ambainis

Andris Ambainis

Quantum walks and their algorithmic applications

11 pages, 3 figures, short survey on applications of quantum walks, v2: added a reference

International Journal of Quantum Information, 1:507-518, 2003.

quant-ph cs.DS

Quantum walks are quantum counterparts of Markov chains. In this article, we give a brief overview of quantum walks, with emphasis on their algorithmic applications.

quant-ph/0404060

Chris Lomont

Chris Lomont

A quantum Fourier transform algorithm

18 pages. Minor corrections were made, and some new material was added. Particularly, simulation results were added to show output of the algorithm, and to suggest possible improvements

quant-ph cs.DS

Algorithms to compute the quantum Fourier transform over a cyclic group are fundamental to many quantum algorithms. This paper describes such an algorithm and gives a proof of its correctness, tightening some claimed performance bounds given earlier. Exact bounds are given for the number of qubits needed to achieve a desired tolerance, allowing simulation of the algorithm.

quant-ph/0412033

Francois Le Gall

Yoshifumi Inui and Francois Le Gall

Efficient Quantum Algorithms for the Hidden Subgroup Problem over a Class of Semi-direct Product Groups

10 pages, final version. Algorithms modified to work with black-box groups too

Quantum Information and Computation, Vol. 7, No. 5&6 (2007), 559-570

10.26421/QIC7.5-6-9

quant-ph cs.DS

In this paper, we consider the hidden subgroup problem (HSP) over the class of semi-direct product groups $\mathbb{Z}_{p^r}\rtimes\mathbb{Z}_q$, for p and q prime. We first present a classification of these groups in five classes. Then, we describe a polynomial-time quantum algorithm solving the HSP over all the groups of one of these classes: the groups of the form $\mathbb{Z}_{p^r}\rtimes\mathbb{Z}_p$, where p is an odd prime. Our algorithm works even in the most general case where the group is presented as a black-box group with not necessarily unique encoding. Finally, we extend this result and present an efficient algorithm solving the HSP over the groups $\mathbb{Z}^m_{p^r}\rtimes\mathbb{Z}_p$.

quant-ph/0503238

Vladimir Korepin

Vladimir Korepin

Optimization of Partial Search

5 pages

Journal of Physics A: Math. Gen. vol 38, pages L731-L738, 2005

10.1088/0305-4470/38/44/L02

YITP-SB-05-08

quant-ph cs.DS

Quantum Grover search algorithm can find a target item in a database faster than any classical algorithm. One can trade accuracy for speed and find a part of the database (a block) containing the target item even faster, this is partial search. A partial search algorithm was recently suggested by Grover and Radhakrishnan. Here we optimize it. Efficiency of the search algorithm is measured by number of queries to the oracle. The author suggests new version of Grover-Radhakrishnan algorithm which uses minimal number of queries to the oracle. The algorithm can run on the same hardware which is used for the usual Grover algorithm.

quant-ph/0504012

Andris Ambainis

Andris Ambainis

Quantum search algorithms

12 pages, short survey on selected topics for SIGACT News Complexity Column, published in June 2004

SIGACT News, 35 (2):22-35, 2004.

quant-ph cs.CC cs.DS

We review some of quantum algorithms for search problems: Grover's search algorithm, its generalization to amplitude amplification, the applications of amplitude amplification to various problems and the recent quantum algorithms based on quantum walks.

quant-ph/0506265

Ashwin Nayak

Frederic Magniez (CNRS-LRI) and Ashwin Nayak (U. Waterloo and Perimeter Inst.)

Quantum Complexity of Testing Group Commutativity

10 pages, requires fullpage,amsthm,amsfonts,amsmath; To appear in Algorithmica; earlier version appeared in ICALP 2005; corrects minor typos, results are unchanged

quant-ph cs.DS

We consider the problem of testing the commutativity of a black-box group specified by its k generators. The complexity (in terms of k) of this problem was first considered by Pak, who gave a randomized algorithm involving O(k) group operations. We construct a quite optimal quantum algorithm for this problem whose complexity is in O (k^{2/3}). The algorithm uses and highlights the power of the quantization method of Szegedy. For the lower bound of Omega(k^{2/3}), we give a reduction from a special case of Element Distinctness to our problem. Along the way, we prove the optimality of the algorithm of Pak for the randomized model.

quant-ph/0608026

J\'er\'emie Roland

Fr\'ed\'eric Magniez, Ashwin Nayak, J\'er\'emie Roland and Miklos Santha

Search via Quantum Walk

21 pages. Various modifications and improvements, especially in Section 4

SIAM Journal on Computing, 40(1):142-164, 2011

10.1137/090745854

quant-ph cs.CC cs.DS

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

We propose a new method for designing quantum search algorithms for finding a "marked" element in the state space of a classical Markov chain. The algorithm is based on a quantum walk \'a la Szegedy (2004) that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimation to the quantum walk in order to implement an approximate reflection operator. This operator is then used in an amplitude amplification scheme. As a result we considerably expand the scope of the previous approaches of Ambainis (2004) and Szegedy (2004). Our algorithm combines the benefits of these approaches in terms of being able to find marked elements, incurring the smaller cost of the two, and being applicable to a larger class of Markov chains. In addition, it is conceptually simple and avoids some technical difficulties in the previous analyses of several algorithms based on quantum walk.

quant-ph/0609205

Vladimir Korepin

Vladimir E. Korepin and Brenno C. Vallilo

Group Theoretical Formulation of Quantum Partial Search Algorithm

12 pages

Prog. Theor. Phys. Vol. 116, No. 5 (2006), p. 783

10.1143/PTP.116.783

YIPT-SB-06-41

quant-ph cs.DS math.GR

Searching and sorting used as a subroutine in many important algorithms. Quantum algorithm can find a target item in a database faster than any classical algorithm. One can trade accuracy for speed and find a part of the database (a block) containing the target item even faster, this is partial search. An example is the following: exact address of the target item is given by a sequence of many bits, but we need to know only some of them. More generally partial search considers the following problem: a database is separated into several blocks. We want to find a block with the target item, not the target item itself. In this paper we reformulate quantum partial search algorithm in terms of group theory.

quant-ph/9607014

Christophe Durr

Christoph Durr and Peter Hoyer

A Quantum Algorithm for Finding the Minimum

2 pages

quant-ph cs.DS

We give a quantum algorithm to find the index y in a table T of size N such that in time O(c sqrt N), T[y] is minimum with probability at least 1-1/2^c.

quant-ph/9703009

Ming Li (University of Waterloo), John Tromp (CWI), Paul Vitanyi (CWI and University of Amsterdam)

Reversible Simulation of Irreversible Computation by Pebble Games

11 pages, Latex, Submitted to Physica D

Physica D120 (1998) 168-176

10.1016/S0167-2789(98)00052-9

CWI Tech Report 1996

quant-ph cs.CC cs.DS

Reversible simulation of irreversible algorithms is analyzed in the stylized form of a `reversible' pebble game. While such simulations incur little overhead in additional computation time, they use a large amount of additional memory space during the computation. The reacheable reversible simulation instantaneous descriptions (pebble configurations) are characterized completely. As a corollary we obtain the reversible simulation by Bennett and that among all simulations that can be modelled by the pebble game, Bennett's simulation is optimal in that it uses the least auxiliary space for the greatest number of simulated steps. One can reduce the auxiliary storage overhead incurred by the reversible simulation at the cost of allowing limited erasing leading to an irreversibility-space tradeoff. We show that in this resource-bounded setting the limited erasing needs to be performed at precise instants during the simulation. We show that the reversible simulation can be modified so that it is applicable also when the simulated computation time is unknown.

quant-ph/9703022

Ming Li (University of Waterloo), Paul Vitanyi (CWI and University of Amsterdam)

Reversibility and Adiabatic Computation: Trading Time and Space for Energy

30 pages, Latex. Lemma 2.3 should be replaced by the slightly better ``There is a winning strategy with $n+2$ pebbles and $m-1$ erasures for pebble games $G$ with $T_G= m2^n$, for all $m \geq 1$'' with appropriate further changes (as pointed out by Wim van Dam). This and further work on reversible simulations as in Section 2 appears in quant-ph/9703009

Proc. Royal Society of London, Series A, 452(1996), 769-789

10.1098/rspa.1996.0039

quant-ph cs.CC cs.CE cs.DS

Future miniaturization and mobilization of computing devices requires energy parsimonious `adiabatic' computation. This is contingent on logical reversibility of computation. An example is the idea of quantum computations which are reversible except for the irreversible observation steps. We propose to study quantitatively the exchange of computational resources like time and space for irreversibility in computations. Reversible simulations of irreversible computations are memory intensive. Such (polynomial time) simulations are analysed here in terms of `reversible' pebble games. We show that Bennett's pebbling strategy uses least additional space for the greatest number of simulated steps. We derive a trade-off for storage space versus irreversible erasure. Next we consider reversible computation itself. An alternative proof is provided for the precise expression of the ultimate irreversibility cost of an otherwise reversible computation without restrictions on time and space use. A time-irreversibility trade-off hierarchy in the exponential time region is exhibited. Finally, extreme time-irreversibility trade-offs for reversible computations in the thoroughly unrealistic range of computable versus noncomputable time-bounds are given.

quant-ph/9902053

Andris Ambainis

Andris Ambainis

A better lower bound for quantum algorithms searching an ordered list

10 pages, LaTeX

quant-ph cs.CC cs.DS

We show that any quantum algorithm searching an ordered list of n elements needs to examine at least 1/12 log n-O(1) of them. Classically, log n queries are both necessary and sufficient. This shows that quantum algorithms can achieve only a constant speedup for this problem. Our result improves lower bounds of Buhrman and de Wolf(quant-ph/9811046) and Farhi, Goldstone, Gutmann and Sipser (quant-ph/9812057).