Edith Cohen

Size-Estimation Framework with Applications to Transitive Closure and Reachability

Computing the transitive closure in directed graphs is a fundamental graph problem. We consider the more restricted problem of computing the number of nodes reachable from every node and the size of the transitive closure. The fastest known transitive closure algorithms run inO(min{mn,n2.38}) time, wherenis the number of nodes andmthe number of edges in the graph. We present anO(m) time randomized (Monte Carlo) algorithm that estimates, with small relative error, the sizes of all reachability sets and the transitive closure. Another ramification of our estimation scheme is a O(m) time algorithm for estimating sizes of neighborhoods in directed graphs with nonnegative edge lengths. Our size-estimation algorithms are much faster than performing the respective explicit computations.

Strongly polynomial-time and NC algorithms for detecting cycles in periodic graphs

This paper is concerned with the problem of recognizing, in a graph with rational vector-weights associates with the edges, the existence of a cycle whose total weight is the zero vector. This problem is known to be equivalent to the problem of recognizing the existence of cycles in periodic (dynamic) graphs and to the validity of systems of recursive formulas. It was previously conjectured that combinatorial algorithms exist for the cases of two- and three-dimensional vector-weights. It is shown that strongly polynomial algorithms exist for any fixed dimension d. Moreover, these algorithms also establish membership in the class NC. On the other hand, it is shown that when the dimension of the weights is not fixed, the problem is equivalent to the general linear programming problem under strongly polynomial and logspace reductions

Fast algorithms for constructing t-spanners and paths with stretch t

The distance between two vertices in a weighted graph is the weight of a minimum-weight path between them. A path has stretch t if its weight is at most t times the distance between its end points. We consider a weighted undirected graph G=(V, E) and present algorithms that compute paths with stretch 2/spl les/t/spl les/log n. We present a O/spl tilde/((m+k)n/sup (2+/spl epsiv///t)) time randomized algorithm that finds paths between k specified pairs of vertices and a O/spl tilde/((m+ns)n/sup 2(1+log(n)/ /sup m+/spl epsiv/)/t/) deterministic algorithm that finds paths from s specified sources to all other vertices (for any fixed /spl epsiv/>0), where n=|V| and m=|E|. This improves significantly over the slower O/spl tilde/(min{k, n}m) exact shortest paths algorithms and a previous O/spl tilde/(mn/sup 64/t/+kn/sup 32/t/) time algorithm by Awerbuch et al. A t-spanner of a graph G is a set of weighted edges on the vertices of G such that distances in the spanner are not smaller and within a factor of t from the corresponding distances in G. Previous work was concerned with bounding the size and efficiently constructing t-spanners. We construct t-spanners of size O/spl tilde/(n/sup 1+(2+/spl epsiv///t)) in O/spl tilde/(mn/sup (2+/spl epsiv///t)) expected time (for any fixed /spl epsiv/>0), what constitutes a faster construction (by a factor of n/sup (3+2//t)) of sparser spanners than was previously attainable. We also provide efficient parallel constructions. Our algorithms are based on new structures called pairwise-covers and a novel approach to construct them efficiently. >

Strongly polynomial-time and NC algorithms for detecting cycles in dynamic graphs

This paper is concerned with the problem of recognizing, in a graph with rational vector-weights associated with the edges, the existence of a cycle whose total weight is the zero vector. This problem is known to be equivalent to the problem of recognizing the existence of cycles in dynamic graphs and to the validity of some systems of recursive formulas. It was previously conjectured that combinatorial algorithms exist for the cases of two- and three-dimensional vector-weights. The present paper gives strongly polynomial algorithms for any fixed dimension. Moreover, these algorithms also establish membership in the class NC. On the other hand, it is shown that when the dimension of the weights is not fixed, the problem is equivalent to the general linear programming problem under strongly polynomial and logspace reductions.

Efficient Parallel Shortest-Paths in Digraphs with a Separator Decomposition

We consider111An extended abstract appeared in “Proceedings 5th Annual ACM Symposium on Parallel Algorithms and Architectures”, 1993.shortest-paths and reachability problems on directed graphs with real-valued edge weights. For sparser graphs, the knownNCalgorithms for these problems perform much more work than their sequential counterparts. In this paper we present efficient parallel algorithms for families of graphs, where a separator decomposition either is provided with the input or is easily obtainable. (A separator is a subset of the vertices that its removal splits the graph into connected components, such that the number of vertices in each component is at most a fixed fraction of the number of vertices in the graph. A separator decomposition is a recursive decomposition of the graph using separators.) LetG=(V,E), wheren=|V|, be a weighted directed graph with ak?-separator decomposition (where subgraphs withkvertices have separators of sizeO(k?)). We present anNCalgorithm that computes shortest-paths fromssources to all other vertices usingO(n3?+s(n+n2?)) work. A sequential version of our algorithm improves over previously known time bounds as well. Reachability fromssources can be computed usingO(M(n?)+s(n+n2?)) work, whereM(r)=o(r2.37) is the best known work bound forr×rmatrix multiplication. The algorithm is based on augmentingGwith a set ofO(n2?) edges such that in the augmented graph, all distances can be obtained by paths of sizeO(logn). The above bounds, with ?=0.5, are applicable to planar graphs, since ak0.5-separator decomposition can be computed within these bounds. We obtain further improvements for graphs with planar embeddings where all vertices lie on a small number of faces.

Multi-rate detection for the IS-95 CDMA forward traffic channels

The IS-95 direct sequence code division multiple access (DS-CDMA) system has become a US digital cellular standard. In the forward traffic channels (from the base station to the mobiles), the speech encoder rate can vary, depending on the speech activity in the transmitter. At the receiver, the decoder does not know which one of the speech encoder rate the transmitter used. This paper focuses on the IS-95 system that uses the 8 kbps speech encoder and describes an algorithm to determine the encoder rate used for the transmitted sequence, with high confidence, so that the appropriate Viterbi decoder can be applied only once and not four times. This can reduce the overall decoder complexity and increase the throughput by a factor of two. It can also reduce the power consumption in the decoder.

New algorithms for generalized network flows

AbstractThis paper, of which a preliminary version appeared in ISTCS92, is concerned with generalized network flow problems. In a generalized network, each edgee = (u, v) has a positive ‘flow multiplier’ae associated with it. The interpretation is that if a flow ofxe enters the edge at nodeu, then a flow ofaexe exits the edge atv.nThe uncapacitated generalized transshipment problem (UGT) is defined on a generalized network where demands and supplies (real numbers) are associated with the vertices and costs (real numbers) are associated with the edges. The goal is to find a flow such that the excess or deficit at each vertex equals the desired value of the supply or demand, and the sum over the edges of the product of the cost and the flow is minimized. Adler and Cosares [Operations Research 39 (1991) 955–960] reduced the restricted uncapacitated generalized transshipment problem, where only demand nodes are present, to a system of linear inequalities with two variables per inequality. The algorithms presented by the authors in [SIAM Journal on Computing, to appear result in a faster algorithm for restricted UGT.Generalized circulation is defined on a generalized network with demands at the nodes and capacity constraints on the edges (i.e., upper bounds on the amount of flow). The goal is to find a flow such that the excesses at the nodes are proportional to the demands and maximized. We present a new algorithm that solves the capacitated generalized flow problem by iteratively solving instances of UGT. The algorithm can be used to find an optimal flow or an approximation thereof. When used to find a constant factor approximation, the algorithm is not only more efficient than previous algorithms but also strongly polynomial. It is believed to be the first strongly polynomial approximation algorithm for generalized circulation. The existence of such an approximation algorithm is interesting since it is not known whether the exact problem has a strongly polynomial algorithm.

Using Selective Path-Doubling for Parallel Shortest-Path Computations

We11A preliminary version appeared in the “Proceedings of the 2nd Israeli Symposium on the Theory of Computing and Systems, 1993.”consider parallel shortest-paths computations in weighted undirected graphsG=(V,E), wheren=|V| andm=|E|. The standardO(n3) work path-doubling (Floyd-Warshall) algorithm consists ofO(logn) phases, where in each phase, for every triplet of vertices (u1,u2,u3)?V3, the distance betweenu1andu3is updated to be no more than the sum of the previous-phase distances between {u1,u2} and {u2,u3}. We introduce a new NC algorithm that for ?=o(n), considers onlyO(n?2) triplets. Our algorithm performsO(n?2) work and augmentsEwithO(n?) new weighted edges such that between every pair of vertices, there exists a minimum weight path of size (number of edges)O(n/?) (whereO(f)?O(fpolylogn)). To compute shortest-paths, we apply to the augmented graph algorithms that are efficient for small-size shortest paths. We obtain anO(t) timeO(|S|n2+n3/t2) work deterministic PRAM algorithm for computing shortest-paths from |S| sources to all other vertices, wheret?nis a parameter. When the ratio of the largest edge weight and the smallest edge weight isnO(polylogn), the algorithm computes shortest paths. When weights are arbitrary, it computes paths within a factor of 1+n??(polylogn)of shortest.

Estimating the size of the transitive closure in linear time

Computing transitive closure and reachability information in directed graphs is a fundamental graph problem with many applications. The fastest known algorithms run in O(sm) time for computing all nodes reachable from each of 1/spl les/s/spl les/n source nodes, or, using fast matrix multiplication, in O(n/sup 2.38/) time for computing the transitive closure, where n is the number of nodes and m the number of edges in the graph. In query optimization in database applications it is often the case that only estimates on the size of the transitive closure and on the number of nodes reachable from certain nodes are needed. We present an O(m) time randomized algorithm that estimates the number of nodes reachable from every node and the size of the transitive closure. We also obtain a O/spl tilde/(m) time algorithm for estimating sizes of neighborhoods in directed graphs with nonnegative weights, avoiding the O/spl tilde/(mn) time bound of explicitly computing these neighborhoods. Our size-estimation algorithms are much faster than performing the actual computations and improve significantly over previous estimation methods.<<ETX>>

Polylog-time and near-linear work approximation scheme for undirected shortest paths

Shortest paths computations constitute one of the most fundamental network problems. Nonetheless, known parallel shortest-paths algorithms are generally inefficient: they perform significantly more work (product of time and processors) than their sequential counterparts. This gap, known in the literature as the “transitive closure bottleneck,” poses along-standing open problem. Our main result is an O(mn’O + s(m + nl+’O )) work polylog-time randomized algorithm that computes paths within (1+0(1/ polylog n)) of shortest from s source nodes to alI other nodes in weighted undirected networks with n nodes and m edges (for any fixed co > O). This work bound nearly matches the O(sm) sequential time. In contrast, previous polylog-time algorithms required min { 6( n3 ), ~ (m2 ) } work (even when s = 1), and previous near-linear work algorithms required near-O(n) time. Another result is faster shortest-paths algorithms if accurate distances are required only between “distant” vertices: We obtain an O((m + sn)n’” ) time algorithm that computes paths of weight (1 + 0(1/ pdybg n))dist + o(w~.. polylog n), where dist is the corresponding distance and wmaX is the maximum edge weight. Our chief instrument, which is of independent int crest, are efficient constructions of sparse hop sets. A (d, c)-hop set of a network G = (V, E) is a set E* of new weighted edges such that minimum-weight d-edge pat hs in (V, E u E*) have weight within (1 + c) of the respective dist antes in G. We construct hop sets of size O(nl+’O ) where e = 0(1/ polylog n) and d = O(polylog n).