on the efficiency of pairing heaps and related data structures the pairing heap is well regarded as an efficient data structure for implementing priority queue operations it is included in the gnu c library strikingly simple in design the pairing heap data structure nonetheless seems difficult to analyze belonging to the genre of self adjusting data structures with its design originating as a self adjusting analogue of the fibonacci heap it has been previously conjectured that the pairing heap provides constrant amortized time decrease key operations and experimental studies have supported this conjecture this paper demonstrates contrary to conjecture that the pairing heap requires more than constant amortized time to perform decrease key operations moreover new experimental findings are presented that reveal detectable growth in the amortized cost of the decrease key operation second a unifying framework is developed that includes both pairing heaps and fibonacci heaps the parameter of interest in this framework is the storage capacity available in the nodes of the data structure for auxiliary balance information fields in this respect fibonacci heaps require log log n bits per node when n items are present this is shown to be asymptotically optimal for data structures that achieve the same asymptotic performance bounds as fibonacci heaps and fall within this framework

the design of a precompensator for multivariable adaptive control a network theoretic approach a network theoretic approach to the design of a dynamic precompensator c s for a multiinput multioutput plant t s is considered the design is based on the relative degree of each element of t s specifically an efficient algorithm is presented for determining whether a given plant t s has a diagonal precompensator c s such that for almost all cases t s c s has a diagonal interactor the algorithm also finds any optimal precompensator in the sense that the total relative degree is minimal the algorithm can be easily modified to work even when a t s represented by a nonsquare matrix is given

three dimensional stochastic analysis of macrodispersion in aquifers the dispersive mixing resulting from complex flow in three dimensionally heterogeneous porous media is analyzed using stochastic continuum theory stochastic solutions of the perturbed steady flow and solute transport equations are used to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of a three dimensional statistically anisotropic input covariance describing the hydraulic conductivity with a statistically isotropic input covariance the longitudinal macrodispersivity is convectively controlled but the transverse macrodispersivity is proportional to the local dispersivity and is several orders of magnitude smaller than the longitudinal term with an arbitrarily oriented anisotropic conductivity covariance all components of the macrodispersivity tensor are convectively controlled and the ratio of transverse to longitudinal dispersivity is of the order of 10 1 in this case the off diagonal components of the dispersivity tensor are significant being numerically larger than the diagonal transverse terms and the transverse dispersion process can be highly anisotropic dispersivities predicted by the stochastic theory are shown to be consistent with controlled field experiments and monte carlo simulations the theory which treats the asymptotic condition of large displacement indicates that a classical gradient transport fickian relationship is valid for large scale displacements

approximation algorithms for single source unsplittable flow in the single source unsplittable flow problem we are given a network g a source vertex s and k commodities with sinks ti and real valued demands rho i 1 leq i leq k we seek to route the demand rho i of each commodity i along a single s ti flow path so that the total flow routed across any edge e is bounded by the edge capacity ue the conceptual difficulty of this np hard problem arises from combining packing constraints due to the existence of capacities with path selection in a graph of arbitrary topology in this paper we give a generic framework which yields approximation algorithms that are simpler than those previously known and achieve significant improvements upon the approximation ratios our framework with appropriate subroutines applies to all optimization versions previously considered and unlike previous work treats in a unified manner directed and undirected graphs we provide extensions of our algorithms which yield the best possible approximation guarantees for restricted sets of demand values and an associated scheduling problem

testing the necklace condition for shortest tours and optimal factors in the plane a tour of a finite set p of points is a necklace tour if there are disks with the points in p as centers such that two disks intersect if and only if their centers are adjacent in it has been observed by sanders that a necklace tour is an optimal traveling salesman tour

finding the shortest paths by node combination by repeatedly combining the source node s nearest neighbor we propose a node combination nc method to implement the dijkstra s algorithm the nc algorithm finds the shortest paths with three simple iterative steps find the nearest neighbor of the source node combine that node with the source node and modify the weights on edges that connect to the nearest neighbor the nc algorithm is more comprehensible and convenient for programming as there is no need to maintain a set with the nodes distances experimental evaluations on various networks reveal that the nc algorithm is as efficient as dijkstra s algorithm as the whole process of the nc algorithm can be implemented with vectors we also show how to find the shortest paths on a weight matrix

a faster combinatorial algorithm for the generalized circulation problem this paper presents a modified version of algorithm mcf proposed by goldberg plotkin and tardos for the generalized circulation problem this new combinatorial algorithm has a worst case complexity that is better than the complexities of the mcf and fat path combinatorial algorithms of goldberg plotkin and tardos goldberg a v s a plotkin e tardos 1991 combinatorial algorithms for the generalized circulation problem math oper res 16 351 379

independence of l ell adic representations of geometric galois groups letk be an algebraically closed field of arbitrary characteristic letk k be a finitely generated field extension and letx be a separated scheme of finite type overk for each prime the absolute galois group of k acts on the adic etale cohomology modules ofx we prove that this family of representations varying over is almost independent in the sense of serre i e that the fixed fields inside an algebraic closure of k of the kernels of the representations for all become linearly disjoint over a finite extension of k in doing this we also prove a number of interesting facts on the images and on the ramification of this family of representations

a polynomial dual simplex algorithm for the generalized circulation problem this paper presents a polynomial time dual simplex algorithm for the generalized circulation problem an efficient implementation of this algorithm is given that has a worst case running time of o m2 m nlogn logb where n is the number of nodes m is the number of arcs and b is the largest integer used to represent the rational gain factors and integral capacities in the network this running time is as fast as the running time of any combinatorial algorithm that has been proposed thus far for solving the generalized circulation problem

relaxed most negative cycle and most positive cut canceling algorithms for minimum cost flow this paper presents two new scaling algorithms for the minimum cost network flow problem one a primal cycle canceling algorithm the other a dual cut canceling algorithm both algorithms scale a relaxed optimality parameter and create a second inner relaxation the primal algorithm uses the inner relaxation to cancel a most negative node disjoint family of cycles w r t the scaled parameter the dual algorithm uses it to cancel most positive cuts w r t the scaled parameter we show that in a network with n nodes and m arcs both algorithms need to cancel only omn objects per scaling phase furthermore we show how to efficiently implement both algorithms to yield weakly polynomial running times that are as fast as any other cycle or cut canceling algorithms our algorithms have potential practical advantages compared to some other canceling algorithms as well along the way we give a comprehensive survey of cycle and cut canceling algorithms for min cost flow we also clarify the formal duality between cycles and cuts

the soft heap an approximate priority queue with optimal error rate a simple variant of a priority queue called a soft heap is introduced the data structure supports the usual operations insert delete meld and findmin its novelty is to beat the logarithmic bound on the complexity of a heap in a comparison based model to break this information theoretic barrier the entropy of the data structure is reduced by artifically raising the values of certain keys given any mixed sequence of n operations a soft heap with error rate e for any 0 en of its items have their keys raised the amortized complexity of each operation is constant except for insert which takes 0 log 1 e time the soft heap is optimal for any value of e in a comparison based model the data structure is purely pointer based no arrays are move items across the data structure not individually as is customary but in groups in a data structuring equivalent of car pooling keys must be raised as a result in order to preserve the heap ordering of the data structure the soft heap can be used to compute exact or approximate medians and percentiles optimally it is also useful for approximate sorting and for computing minimum spanning trees of general graphs

improving space efficiency with path length prediction for finding k shortest simple paths finding mbi k shortest simple paths in a directed graph is a fundamental problem in many engineering applications most existing algorithms such as yen s algorithm and its variants have polynomial worst case time complexity but their average case running time is very high the heuristic algorithm mps can run significantly faster in practice however it requires an excessive amount of memory space in this paper we provide a new heuristic algorithm that achieves high space efficiency while maintaining similar average case running time we first propose a sidetrack representation of path with which a path can be stored in mbi o bf 1 space we then show how to categorize a candidate path as either partial or complete and restrict the number of paths added to the queue in addition we provide an empirical equation that can very accurately predict the mbi k th shortest path length provided that a much smaller number of shortest paths have been found extensive experiments prove that our algorithm can achieve an mbi o n speedup in practice over yen s algorithm in comparison with mps it runs up to three times faster and uses less space by an order of magnitude

single source shortest paths in h minor free graphs we present an algorithm for the single source shortest paths sssp problem in directedh minor free graphs for every fixed h if g is a graph with n vertices having integer edge lengths and s is a designated source vertex of g the algorithm runs in o n 1 1 5 2logl o n 1 3 9 2logl time where l is the absolute value of the smallest edge length the algorithm computes the shortest paths and the distances from s to all vertices of the graph or else provides a certificate that g is not h minor free our result improves an earlier o n 1 5logl time algorithm for this problem which follows from a general sssp algorithm of goldberg

serre s modularity conjecture ii we provide proofs of theorems 4 1 and 5 1 of khare and wintenberger invent math doi 10 1007 s00222 009 0205 7 2009

oracles for distances avoiding a failed node or link we consider the problem of preprocessing an edge weighted directed graph g to answer queries that ask for the length and first hop of a shortest path from any given vertex x to any given vertex y avoiding any given vertex or edge as a natural application this problem models routing in networks subject to node or link failures we describe a deterministic oracle with constant query time for this problem that uses o n 2 log n space where n is the number of vertices in g the construction time for our oracle is o mn 2 n 3 log n however if one is willing to settle for theta n 2 5 space we can improve the preprocessing time to o mn 1 5 n 2 5 log n while maintaining the constant query time our algorithms can find the shortest path avoiding a failed node or link in time proportional to the length of the path

linear time approximation for maximum weight matching the maximum cardinality and maximum weight matching problems can be solved in o m n time a bound that has resisted improvement despite decades of research here m and n are the number of edges and vertices in this article we demonstrate that this m n barrier can be bypassed by approximation for any e 0 we give an algorithm that computes a 1 e approximate maximum weight matching in o me 1 log e 1 time that is optimal linear time for any fixed e our algorithm is dramatically simpler than the best exact maximum weight matching algorithms on general graphs and should be appealing in all applications that can tolerate a negligible relative error

a shortest augmenting path algorithm for the semi assignment problem the objective of this study is to develop a shortest augmenting path algorithm for solving the semi assignment problem and conduct an extensive computational comparison with the best alternative approaches the algorithm maintains dual feasibility and complementary slackness and works toward satisfying primal feasibility effective heuristics arc used to achieve an excellent advanced start and convergence is assured via the use of the shortest augmenting path procedure using reduced costs for arc lengths unlike other algorithms such as the primal simplex or the auction algorithm each iteration during the final phase of the procedure also known as the end game achieves one additional assignment the software implementations of our algorithm are fastest for the semi assignment problems that we tested our dense code is also faster than the best software for assignment problems in addition the algorithm has the best polynomial worst case running time bound that we have seen and the memory requirements are relatively small

all pairs shortest paths using bridging sets and rectangular matrix multiplication we present two new algorithms for solving the all pairs shortest paths apsp problem for weighted directed graphs both algorithms use fast matrix multiplication algorithms the first algorithm solves the apsp problem for weighted directed graphs in which the edge weights are integers of small absolute value in o n2 time where satisfies the equation 1 1 1 2 and 1 1 is the exponent of the multiplication of an n n matrix by an n n matrix currently the best available bounds on 1 1 obtained by coppersmith imply that 0 is an error parameter and w is the largest edge weight in the graph after the edge weights are scaled so that the smallest non zero edge weight in the graph is 1 it returns estimates of all the distances in the graph with a stretch of at most 1 e corresponding paths can also be found efficiently

on icosahedral artin representations ii we prove that some new infinite families of odd two dimensional icosahedral representations of the absolute galois group of inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i are modular and hence satsify the artin conjecture we also give an account of work of ramakrishna on lifting mod l galois representations to characteristic zero

modularity of 2 adic barsotti tate representations we prove a modularity lifting theorem for two dimensional 2 adic potentially barsotti tate representations this proves hypothesis h of khare wintenberger and completes the proof of serre s conjecture the main new ingredient is a classification of connected finite flat group schemes over rings of integers of finite extensions of 2

stochastic flow networks via multiple paths under time threshold and budget constraint this paper extends the quickest path problem to a stochastic flow network in which the capacity of each arc is variable we mainly evaluate the system reliability that d units of data can be sent from the source to the sink under both time threshold t and budget b in particular the data are transmitted through multiple disjoint minimal paths simultaneously in order to reduce the transmission time a simple algorithm is proposed to generate all lower boundary points for d t b and the system reliability can then be computed in terms of such points by utilizing a union of subsets computational complexity in both worst case and average cases show that the proposed algorithm can be executed efficiently

solving shortest paths efficiently on nearly acyclic directed graphs shortest path problems can be solved very efficiently when a directed graph is nearly acyclic earlier results defined a graph decomposition now called the 1 dominator set which consists of a unique collection of acyclic structures with each single acyclic structure dominated by a single associated trigger vertex in this framework a specialised shortest path algorithm only spends delete min operations on trigger vertices thereby making the computation of shortest paths through non trigger vertices easier a previously presented algorithm computed the 1 dominator set in o mn worst case time which allowed it to be integrated as part of an o mn nrlogr time all pairs algorithm here m and n respectively denote the number of edges and vertices in the graph while r denotes the number of trigger vertices a new algorithm presented in this paper computes the 1 dominator set in just o m time this can be integrated as part of the o m rlogr time spent solving single source improving on the value of r obtained by the earlier tree decomposition single source algorithm in addition a new bidirectional form of 1 dominator set is presented which further improves the value of r by defining acyclic structures in both directions over edges in the graph the bidirectional 1 dominator set can similarly be computed in o m time and included as part of the o m rlogr time spent computing single source this paper also presents a new all pairs algorithm under the more general framework where r is defined as the size of any predetermined feedback vertex set of the graph improving the previous all pairs time complexity from o mn nr 2 to o mn r 3

faster scaling algorithms for network problems this paper presents algorithms for the assignment problem the transportation problem and the minimum cost flow problem of operations research the algorithms find a minimum cost solution yet run in time close to the best known bounds for the corresponding problems without costs for example the assignment problem equivalently minimum cost matching in a bipartite graph can be solved in o sqrt nm log nn time where n m and n denote the number of vertices number of edges and largest magnitude of a cost costs are assumed to be integral the algorithms work by scaling as in the work of goldberg and tarjan in each scaled problem an approximate optimum solution is found rather than an exact optimum

sharing information for the all pairs shortest path problem we improve the time complexity of the all pairs shortest path apsp problem for special classes of directed graphs one is a nearly acyclic directed graph and the other is a directed graph with limited edge costs the common idea for speed up is information sharing by n single source shortest path sssp problems that are solved for apsp we measure the degree of acyclicity by the size r of a given set of feedback vertices if r is small the given graph can be considered to be nearly acyclic we consider this parameter r in addition to the traditional parameters of the number of vertices n and that of edges m in the first part we improve the existing time complexity of o mn r 3 for the all pairs shortest path problem to o mn rnlogn this complexity is equal to or better than the previous one for all values of r under a reasonable assumption of m n in the second part we deal with a directed graph with non negative integer edge costs bounded by c we show the all pairs shortest path apsp problem can be solved in o mn n 2log c n time with the data structure of cascading bucket system we use the traditional computational model such that comparison addition operations on distance data and random access with o logn bits can be done in o 1 time also the graph is not separated meaning m n 1

personalized pagerank clustering a graph clustering algorithm based on random walks abstract graph clustering has been an essential part in many methods and thus its accuracy has a significant effect on many applications in addition exponential growth of real world graphs such as social networks biological networks and electrical circuits demands clustering algorithms with nearly linear time and space complexity in this paper we propose personalized pagerank clustering ppc that employs the inherent cluster exploratory property of random walks to reveal the clusters of a given graph we combine random walks and modularity to precisely and efficiently reveal the clusters of a graph ppc is a top down algorithm so it can reveal inherent clusters of a graph more accurately than other nearly linear approaches that are mainly bottom up it also gives a hierarchy of clusters that is useful in many applications ppc has a linear time and space complexity and has been superior to most of the available clustering algorithms on many datasets furthermore its top down approach makes it a flexible solution for clustering problems with different requirements

a weighted kt t free t factor algorithm for bipartite graphs for a simple bipartite graph and an integer t 2 we consider the problem of finding a minimum weight kt t free t factor which is a t factor containing no complete bipartite graph kt t as a subgraph when t 2 this problem amounts to the square free 2 factor problem in a bipartite graph for the unweighted square free 2 factor problem a combinatorial algorithm is given by hartvigsen and the weighted version of the problem is np hard for general t pap designed a combinatorial algorithm for the unweighted version and makai gave a dual integral description of kt t free t matchings for a certain case where the weight vector is vertex induced on any subgraph isomorphic to kt t for this class of weight vectors we propose a strongly polynomial algorithm to find a minimum weight kt t free t factor the algorithm adapts the unweighted algorithms of hartvigsen and pap and a primal dual approach to the minimum cost flow problem the algorithm is fully combinatorial and thus provides a dual integrality theorem which is tantamount to makai s one

an optimal algorithm for euclidean shortest paths in the plane we propose an optimal time algorithm for a classical problem in plane computational geometry computing a shortest path between two points in the presence of polygonal obstacles our algorithm runs in worst case time o n log n and requires o n log n space where n is the total number of vertices in the obstacle polygons the algorithm is based on an efficient implementation of wavefront propagation among polygonal obstacles and it actually computes a planar map encoding shortest paths from a fixed source point to all other points of the plane the map can be used to answer single source shortest path queries in o log n time the time complexity of our algorithm is a significant improvement over all previously published results on the shortest path problem finally we also discuss extensions to more general shortest path problems involving nonpoint and multiple sources

simulation relations for pattern matching in directed graphs we consider the problem of finding the occurrences of a pattern tree t in a directed graph g and propose two algorithms one for preprocessing and one for searching for t in g it is assumed that

efficient algorithms for updating betweenness centrality in fully dynamic graphs devise an algorithm for updating betweenness centrality in fully dynamic graphs recalculate centrality without computing all pairs shortest paths in the entire graph adapt a highest centrality edge finding algorithm based on the proposed algorithm adapt a community detection algorithm using the proposed algorithms experimental results show the proposed algorithm outperforms existing algorithms betweenness centrality of a vertex edge in a graph is a measure for the relative participation of the vertex edge in the shortest paths in the graph betweenness centrality is widely used in various areas such as biology transportation and social networks in this paper we study the update problem of betweenness centrality in fully dynamic graphs the proposed update algorithm substantially reduces the number of shortest paths which should be re computed when a graph is changed in addition we adapt a community detection algorithm using the proposed algorithm to show how much benefit can be obtained from the proposed algorithm in a practical application experimental results on real graphs show that the proposed algorithm efficiently update betweenness centrality and detect communities in a graph

fast approximate graph partitioning algorithms we study graph partitioning problems on graphs with edge capacities and vertex weights the problems of b balanced cuts and k balanced partitions are unified into a new problem called minimum capacity rho separators a rho separator is a subset of edges whose removal partitions the vertex set into connected components such that the sum of the vertex weights in each component is at most rho times the weight of the graph we present a new and simple o log n approximation algorithm for minimum capacity rho separators which is based on spreading metrics yielding an o log n approximation algorithm both for b balanced cuts and k balanced partitions in particular this result improves the previous best known approximation factor for k balanced partitions in undirected graphs by a factor of o log k we enhance these results by presenting a version of the algorithm that obtains an o log opt approximation factor the algorithm is based on a technique called spreading metrics that enables us to formulate directly the minimum capacity rho separator problem as an integer program we also introduce a generalization called the simultaneous separator problem where the goal is to find a minimum capacity subset of edges that separates a given collection of subsets simultaneously we extend our results to directed graphs for values of rho geq 1 2 we conclude with an efficient algorithm for computing an optimal spreading metric for rho separators this yields more efficient algorithms for computing b balanced cuts than were previously known

fibonacci heaps and their uses in improved network optimization algorithms in this paper we develop a new data structure for implementing heaps priority queues our structure fibonacci heaps abbreviated f heaps extends the binomial queues proposed by vuillemin and studied further by brown f heaps support arbitrary deletion from an n item heap in o log n amortized time and all other standard heap operations in o 1 amortized time using f heaps we are able to obtain improved running times for several network optimization algorithms in particular we obtain the following worst case bounds where n is the number of vertices and m the number of edges in the problem graph o n log n m for the single source shortest path problem with nonnegative edge lengths improved from o m log m n 2 n o n 2 log n nm for the all pairs shortest path problem improved from o nm log m n 2 n o n 2 log n nm for the assignment problem weighted bipartite matching improved from o nm log m n 2 n o m m n for the minimum spanning tree problem improved from o m log log m n 2 n where m n min i u log i n m n note that m n log n if m n of these results the improved bound for minimum spanning trees is the most striking although all the results give asymptotic improvements for graphs of appropriate densities

efficient implementation of graph algorithms using contraction the component merging problem is a new graph problem versions of this problem appear as bottlenecks in various graph algorithms a new data structure solves this problem efficiently and two special cases of the problem have even more efficient solutions based on other data structures the performance of the data structures is sped up by introducing a new algorithmic tool called packets the algorithms that use these solutions to the component merging problem also exploit new properties of two existing data structures specifically trees can be used simultaneously as a priority queue and a concatenable queue similarly f heaps support some kinds of split operations with no loss of efficiency an immediate application of the solution to the simplest version of the merging problem is an o t m n algorithm for finding minimum spanning trees in undirected graphs without using f heaps where t m n m log 2 log 2 log d n the graph has n vertices and m edges and d max m n 2 packets also improve the f heap minimum spanning tree algorithm giving the fastest algorithm currently known for this problem the efficient solutions to the merging problem and the new observation about f heaps lead to an o n t m n n log n algorithm for finding a maximum weighted matching in general graphs this settles an open problem posed by tarjan 15 p 123 where the weaker bound of o nm log n 2 m was conjectured

time version of the shortest path problem in a stochastic flow network many studies on hardware framework and routing policy are devoted to reducing the transmission time for a flow network a time version of the shortest path problem thus arises to find a quickest path which sends a given amount of data from the unique source to the unique sink with minimum transmission time more specifically the capacity of each arc in the flow network is assumed to be deterministic however in many real life networks such as computer systems telecommunication systems etc the capacity of each arc should be stochastic due to failure maintenance etc such a network is named a stochastic flow network hence the minimum transmission time is not a fixed number we extend the quickest path problem to evaluating the probability that d units of data can be sent under the time constraint t such a probability is named the system reliability in particular the data are transmitted through two minimal paths simultaneously in order to reduce the transmission time a simple algorithm is proposed to generate all d t mps and the system reliability can then be computed in terms of d t mps moreover the optimal pair of minimal paths with highest system reliability could be obtained

effective hydraulic conductivity of three dimensional heterogeneous formations of lognormal permeability distribution the impact of connectivity uniform mean flow takes place in a 3 d heterogeneous formation of normal hydraulic logconductivity y ln k the aim of the study is to derive the dependence of the horizontal kefh and vertical kefv effective conductivities on the structural parameters of hydraulic conductivity and investigate the impact of departure from multi gaussianity on kef by numerical simulations of flow in formations that share the same pdf and covariance of y but differ in the connectivity of classes of y the main result is that for the extreme models of connected and disconnected high y zones the ratio between the effective conductivities in isotropic media is much smaller than in 2 d the dependence of kefh and kefv upon the logconductivity variance and the anisotropy ratio is compared with existing approximations first order landau matheron conjecture self consistent approximation besides the theoretical interest the results offer a basis for empirical relationships to be used in applications

worst case analysis of set union algorithms this paper analyzes the asymptotic worst case running time of a number of variants of the well known method of path compression for maintaining a collection of disjoint sets under union we show that two one pass methods proposed by van leeuwen and van der weide are asymptotically optimal whereas several other methods including one proposed by rein and advocated by dijkstra are slower than the best methods

a 2 adic automorphy lifting theorem for unitary groups over cm fields we prove a minimal type automorphy lifting theorem for 2 adic galois representations of unitary type over imaginary cm fields we use this to improve an automorphy lifting theorem of kisin for mathrm gl 2

prediction of steady state flow in nonuniform geologic media by conditional moments exact nonlocal formalism effective conductivities and weak approximation we consider the effect of measuring randomly varying local hydraulic conductivities k x on one s ability to predict steady state flow within a bounded domain driven by random source and boundary functions more precisely we consider the prediction of local hydraulic head h x and darcy flux q x by means of their unbiased ensemble moments h x and q x conditioned on measurements of k x these predictors satisfy a deterministic flow equation in which q x x h x r x where x is a relatively smooth unbiased estimate of k x and r x is a residual flux we derive a compact integral expression for r x which is rigorously valid for a broad class of k x fields including fractals it demonstrates that q x is nonlocal and non darcian so that an effective hydraulic conductivity does not generally exist we show analytically that under uniform mean flow the effective conductivity may be a scalar a symmetric or a nonsymmetric tensor or a set of directional scalars which do not form a tensor we demonstrate numerically that in two dimensional mean radial flow it may increase from the harmonic mean of k x near interior and boundary sources to the geometric mean far from such sources for cases where r x can neither be expressed nor approximated by a local expression we propose a weak integral approximation closure which appears to work well in media with pronounced heterogeneity and improves as the quantity and quality of k x measurements increase the nonlocal deterministic flow equation can be solved numerically by standard methods our theory shows clearly how the scale of grid discretization should relate to the scale quantity and quality of available data after providing explicit approximations for the second moments of head and flux prediction errors we conclude by discussing practical methods to compute x from noisy measurements of k x and to calculate required second moments of the associated estimation errors when k x is lognormal

all pairs shortest paths in o n 2 time with high probability we present an all pairs shortest path algorithm whose running time on a complete directed graph on n vertices whose edge weights are chosen independently and uniformly at random from 0 1 is o n2 in expectation and with high probability this resolves a long standing open problem the algorithm is a variant of the dynamic all pairs shortest paths algorithm of demetrescu and italiano 2006 the analysis relies on a proof that the number of locally shortest paths in such randomly weighted graphs is o n2 in expectation and with high probability we also present a dynamic version of the algorithm that recomputes all shortest paths after a random edge update in o log2n expected time

efficient approximation algorithms for bandwidth consecutive multicolorings of graphs let g be a graph in which each vertex v has a positive integer weight b v and each edge v w has a nonnegative integer weight b v w a bandwidth consecutive multicoloring simply called a b coloring of g assigns each vertex v a specified number b v of consecutive positive integers as colors of v so that for each edge v w all integers assigned to vertex v differ from all integers assigned to vertex w by more than b v w the maximum integer assigned to vertices is called the span of the coloring the b coloring problem asks to find a b coloring of a given graph g with the minimum span in the paper we present four efficient approximation algorithms for the problem which have theoretical performance guarantees for the computation time the span of a found b coloring and the approximation ratio we also obtain several upper bounds on the minimum span expressed in terms of the maximum b degrees one of which is an extension of brooks theorem on an ordinary coloring

the scaling network simplex algorithm in this paper we present a new primal simplex pivot rule and analyze the worst case complexity of the resulting simplex algorithm for the minimum cost flow the assignment and the shortest path problems we consider networks with n nodes m arcs integral arc capacities bounded by an integer number u and integral arc costs whose magnitudes are bounded by an integer number c our pivot rule may be regarded as a scaling version of dantzig s pivot rule our pivot rule defines a threshold value which is initially at most 2c and the rule permits any arc with a violation of at least 2 to be the editing variable we select the leaving arc so that strong feasibility of the basis is maintained when there is no arc satisfying this rule then we replace by 2 and repeat the process the algorithm terminates when 1 we show that the simplex algorithm using this rule performs o nmu log c pivots and can be implemented to run in o m2u log c time specializing these results for the assignment and short

a polynomial algorithm for the multicriteria cent dian location problem given a network with several weights per node and several lengths per edge we address the problem of locating a facility on the network such that the convex combinations of the center and median objective functions are minimized since we consider several weights and several lengths various objective functions should be minimized and hence we have to solve a multicriteria cent dian location problem a polynomial algorithm to characterize the efficient location point set on the network is developed furthermore this model can generalize other problems such as the multicriteria center problem and the multicriteria median problem computing time results on random planar networks considering different combinations of weights and lengths are reported which strengthen the polynomial complexity of the procedure

technical note a note on parametric analysis in linear assignment a classic application of the linear assignment problem is the assignment of people to jobs or jobs to people in this context it is interesting to measure competition for jobs and to generate a suitable list of jobs from which a person can choose the length of the list is a parameter a known list generation procedure is based on an interior point method followed by a parametric analysis we describe a more efficient procedure exploiting linear assignment theory and shortest path computations further we propose an alternative list generation procedure based on a special type of dual values for the linear assignment problem

optimal paths in weighted timed automata we consider the optimal reachability problem for a timed automaton with respect to a linear cost function which results in a weighted timed automaton our solution to this optimization problem consists of reducing it to computing parametric shortest paths in a finite weighted directed graph we call this graph a parametric sub region graph it refines the region graph a standard tool for the analysis of timed automata by adding the information which is relevant to solving the optimal teachability problem we present an algorithm to solve the optimal reachability problem for weighted timed automata that takes time exponential in o n a w max where n is the number of clocks a is the size of the clock constraints and w max is the size of the largest weight we show that this algorithm can be improved if we restrict to weighted timed automata with a single clock in case we consider a single starting state for the optimal reachability problem our approach yields an algorithm that takes exponential time only in the length of clock constraints

self adjusting heaps in this paper we explore two themes in data structure design amortized computational complexity and self adjustment we are motivated by the following observations in most applications of data structures we wish to perform not just a single operation but a sequence of operations possibly having correlated behavior by averaging the running time per operation over a worst case sequence of operations we can sometimes obtain an overall time bound much smaller than the worst case time per operation multiplied by the number of operations we call this kind of averaging amortization standard kinds of data structures such as the many varieties of balanced trees are specifically designed so that the worst case time per operation is small such efficiency is achieved by imposing an explicit structural constraint that must be maintained during updates at a cost of both running time and storage space however if amortized running time is the complexity measure of interest we can guarantee efficiency withou

an analogue of hoffman s circulation conditions for max balanced flows letd v a be a directed graph a real valued vectorx defined on the arc seta is amax balanced flow ford if for every cutw the maximum weight over arcs leavingw equals the maximum weight over arcs enteringw for vectorsl u defined ona we describe an analogue of hoffman s circulation conditions for the existence of a max balanced flowx satisfyingl x u we describe an algorithm for computing such a vector but show that minimizing a linear function over the set of max balanced flows satisfyingl x u is np hard we show that the set of all max balanced flows satisfyingl x u has a greatest element under the usual coordinate partial order and we describe an algorithm for computing this element this allows us to solve several related approximation problems we also investigate the set of minimal elements under the coordinate partial order we describe an algorithm for finding a minimal element and show that counting the number of minimal elements is p hard many of our algorithms exploit the relationship between max balanced flows and bottleneck paths

system reliability for quickest path problems under time threshold and budget many studies on hardware framework and routing policy are devoted to reducing the transmission time for a computer network the quickest path problem thus arises to find a path which sends a given amount of data from the source to the sink such that the transmission time is minimized more specifically the capacity of each arc in the network is assumed to be deterministic however in many real life networks such as computer systems telecommunication systems etc the capacity of each arc is stochastic due to failure maintenance etc such a network is named stochastic flow network hence the minimum transmission time is not a fixed number we extend the quickest path problem to evaluating the probability that d units of data can be sent from the source to the sink under both time threshold t and budget b such a probability is named system reliability a simple algorithm is proposed to generate all lower boundary points for d t b and the system reliability can then be computed in terms of such points

approximate distance oracles let g v e be an undirected weighted graph with vvv n and vev m let k 1 be an integer we show that g v e can be preprocessed in o kmn1 k expected time constructing a data structure of size o kn1p1 k such that any subsequent distance query can be answered approximately in o k time the approximate distance returned is of stretch at most 2k 1 that is the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k 1 a 1963 girth conjecture of erdos implies that n1p1 k space is needed in the worst case for any real stretch strictly smaller than 2kp1 the space requirement of our algorithm is therefore essentially optimal the most impressive feature of our data structure is its constant query time hence the name oracle previously data structures that used only o n1p1 k space had a query time of n1 k our algorithms are extremely simple and easy to implement efficiently they also provide faster constructions of sparse spanners of weighted graphs and improved tree covers and distance labelings of weighted or unweighted graphs

the partial sum criterion for steiner trees in graphs and shortest paths many optimization problems are defined in a weighted graph generally the criterion is min sum i e the total weight of all edges in the solution here we consider a different criterion