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23
Applying parallel computation algorithms in the design of serial algorithms
 J. ACM
, 1983
"... Abstract. The goal of this paper is to point out that analyses of parallelism in computational problems have practical implications even when multiprocessor machines are not available. This is true because, in many cases, a good parallel algorithm for one problem may turn out to be useful for design ..."
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Cited by 238 (7 self)
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Abstract. The goal of this paper is to point out that analyses of parallelism in computational problems have practical implications even when multiprocessor machines are not available. This is true because, in many cases, a good parallel algorithm for one problem may turn out to be useful for designing an efficient serial algorithm for another problem. A d ~ eframework d for cases like this is presented. Particular cases, which are discussed in this paper, provide motivation for examining parallelism in sorting, selection, minimumspanningtree, shortest route, maxflow, and matrix multiplication problems, as well as in scheduling and locational problems.
On the exponent of the all pairs shortest path problem
"... The upper bound on the exponent, ω, of matrix multiplication over a ring that was three in 1968 has decreased several times and since 1986 it has been 2.376. On the other hand, the exponent of the algorithms known for the all pairs shortest path problem has stayed at three all these years even for t ..."
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Cited by 73 (2 self)
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The upper bound on the exponent, ω, of matrix multiplication over a ring that was three in 1968 has decreased several times and since 1986 it has been 2.376. On the other hand, the exponent of the algorithms known for the all pairs shortest path problem has stayed at three all these years even for the very special case of directed graphs with uniform edge lengths. In this paper we give an algorithm of time O � n ν log 3 n � , ν = (3 + ω)/2, for the case of edge lengths in {−1, 0, 1}. Thus, for the current known bound on ω, we get a bound on the exponent, ν < 2.688. In case of integer edge lengths with absolute value bounded above by M, the time bound is O � (Mn) ν log 3 n � and the exponent is less than 3 for M = O(n α), for α < 0.116 and the current bound on ω.
All Pairs Shortest Paths using Bridging Sets and Rectangular Matrix Multiplication
 Journal of the ACM
, 2000
"... 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... ..."
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Cited by 68 (7 self)
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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...
Fast Approximation of Centrality
 Journal of Graph Algorithms and Applications
, 2001
"... Social studies researchers use graphs to model group activities in social networks. An important property in this context is the centrality of a vertex: the inverse of the average distance to each other vertex. We describe a randomized approximation algorithm for centrality in weighted graphs. For g ..."
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Cited by 38 (0 self)
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Social studies researchers use graphs to model group activities in social networks. An important property in this context is the centrality of a vertex: the inverse of the average distance to each other vertex. We describe a randomized approximation algorithm for centrality in weighted graphs. For graphs exhibiting the small world phenomenon, our method estimates the centrality of all vertices with high probability within a (1 + #) factor in nearlinear time. 1 Introduction In social network analysis, the vertices of a graph represent agents in a group and the edges represent relationships, such as communication or friendship. The idea of applying graph theory to analyze the connection between the structural centrality and group process was introduced by Bavelas [4]. Various measurement of centrality [7, 14, 15] have been proposed for analyzing communication activity, control, or independence within a social network. We are particularly interested in closeness centrality [5, 6, 24]...
All Pairs Shortest Paths in weighted directed graphs  exact and almost exact algorithms
, 1998
"... 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 abso ..."
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Cited by 37 (6 self)
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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(n 2+ ) time, where satisfies the equation !(1; ; 1) = 1 + 2 and !(1; ; 1) is the exponent of the multiplication of an n \Theta n matrix by an n \Theta n matrix. The currently best available bounds on !(1; ; 1), obtained by Coppersmith and Winograd, and by Huang and Pan, imply that ! 0:575. The running time of our algorithm is therefore O(n 2:575 ). Our algorithm improves on the ~ O(n (3+!)=2 ) time algorithm, where ! = !(1; 1; 1) ! 2:376 is the usual exponent of matrix multiplication, obtained by Alon, Galil and Margalit, whose running time is only known to be O(n 2:688 ). The second
Finding a maximum weight triangle in n 3−δ time, with applications
 In Proc. of STOC
, 2006
"... We present the first truly subcubic algorithms for finding a maximum nodeweighted triangle in directed and undirected graphs with arbitrary real weights. The first is an O(B · n 3+ω 2) = O(B · n 2.688) deterministic algorithm, where n is the number of nodes, ω is the matrix multiplication exponen ..."
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Cited by 17 (10 self)
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We present the first truly subcubic algorithms for finding a maximum nodeweighted triangle in directed and undirected graphs with arbitrary real weights. The first is an O(B · n 3+ω 2) = O(B · n 2.688) deterministic algorithm, where n is the number of nodes, ω is the matrix multiplication exponent, and B is the number of bits of precision. The second is a strongly polynomial randomized algorithm that runs in O(n 3+ω 2 log n) expected worstcase time. To achieve this, we show how to efficiently sample a weighted triangle uniformly at random, out of just those triangles whose total weight falls in some prescribed interval (W1, W2) for arbitrary weights W1 and W2. Previous approaches to the problem resulted in time bounds with either an exponential dependence on B, or a runtime of the form Ω(n 3 /(log n) c). The algorithms are easily extended to finding a maximum nodeweighted induced subgraph on 3k nodes in Õ(n (3+ω)k 2) = Õ(n2.688k) time. We give applications to a variety of problems, including a stable matching problem between buyers and sellers in computational economics, and discuss the possibility of extending our approach to a truly subcubic algorithm for computing allpairs shortest paths on directed graphs with arbitrary weights. ∗ Both authors were supported by the NSF ALADDIN
Subcubic Equivalences Between Path, Matrix, and Triangle Problems
"... We say an algorithm on n × n matrices with entries in [−M,M] (or nnode graphs with edge weights from [−M,M]) is truly subcubic if it runs in O(n 3−δ · poly(log M)) time for some δ> 0. We define a notion of subcubic reducibility, and show that many important problems on graphs and matrices solvab ..."
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Cited by 15 (6 self)
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We say an algorithm on n × n matrices with entries in [−M,M] (or nnode graphs with edge weights from [−M,M]) is truly subcubic if it runs in O(n 3−δ · poly(log M)) time for some δ> 0. We define a notion of subcubic reducibility, and show that many important problems on graphs and matrices solvable in O(n 3) time are equivalent under subcubic reductions. Namely, the following weighted problems either all have truly subcubic algorithms, or none of them do: • The allpairs shortest paths problem on weighted digraphs (APSP). • Detecting if a weighted graph has a triangle of negative total edge weight. • Listing up to n 2.99 negative triangles in an edgeweighted graph. • Finding a minimum weight cycle in a graph of nonnegative edge weights. • The replacement paths problem on weighted digraphs. • Finding the second shortest simple path between two nodes in a weighted digraph. • Checking whether a given matrix defines a metric. • Verifying the correctness of a matrix product over the (min,+)semiring. Therefore, if APSP cannot be solved in n 3−ε time for any ε> 0, then many other problems also
Finding, minimizing, and counting weighted subgraphs
 In Proceedings of the FourtyFirst Annual ACM Symposium on the Theory of Computing
, 2009
"... For a pattern graph H on k nodes, we consider the problems of finding and counting the number of (not necessarily induced) copies of H in a given large graph G on n nodes, as well as finding minimum weight copies in both nodeweighted and edgeweighted graphs. Our results include: • The number of cop ..."
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Cited by 15 (2 self)
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For a pattern graph H on k nodes, we consider the problems of finding and counting the number of (not necessarily induced) copies of H in a given large graph G on n nodes, as well as finding minimum weight copies in both nodeweighted and edgeweighted graphs. Our results include: • The number of copies of an H with an independent set of size s can be computed exactly in O ∗ (2 s n k−s+3) time. A minimum weight copy of such an H (with arbitrary real weights on nodes and edges) can be found in O(4 s+o(s) n k−s+3) time. (The O ∗ notation omits poly(k) factors.) These algorithms rely on fast algorithms for computing the permanent of a k × n matrix, over rings and semirings. • The number of copies of any H having minimum (or maximum) nodeweight (with arbitrary real weights on nodes) can be found in O(n ωk/3 + n 2k/3+o(1) ) time, where ω < 2.4 is the matrix multiplication exponent and k is divisible by 3. Similar results hold for other values of k. Also, the number of copies having exactly a prescribed weight can be found within this time. These algorithms extend the technique of Czumaj and Lingas (SODA 2007) and give a new (algorithmic) application of multiparty communication complexity. • Finding an edgeweighted triangle of weight exactly 0 in general graphs requires Ω(n 2.5−ε) time for all ε> 0, unless the 3SUM problem on N numbers can be solved in O(N 2−ε) time. This suggests that the edgeweighted problem is much harder than its nodeweighted version. 1
Determinant sums for undirected hamiltonicity
 in Prof. of FOCS’10, 2010
"... We present a Monte Carlo algorithm for Hamiltonicity detection in an nvertex undirected graph running in O ∗ (1.657 n) time. To the best of our knowledge, this is the first superpolynomial improvement on the worst case runtime for the problem since the O ∗ (2 n) bound established for TSP almost fif ..."
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Cited by 13 (0 self)
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We present a Monte Carlo algorithm for Hamiltonicity detection in an nvertex undirected graph running in O ∗ (1.657 n) time. To the best of our knowledge, this is the first superpolynomial improvement on the worst case runtime for the problem since the O ∗ (2 n) bound established for TSP almost fifty years ago (Bellman 1962, Held and Karp 1962). It answers in part the first open problem in Woeginger’s 2003 survey on exact algorithms for NPhard problems. For bipartite graphs, we improve the bound to O ∗ (1.414 n) time. Both the bipartite and the general algorithm can be implemented to use space polynomial in n. We combine several recently resurrected ideas to get the results. Our main technical contribution is a new reduction inspired by the algebraic sieving method for kPath (Koutis ICALP 2008, Williams IPL 2009). We introduce the Labeled Cycle Cover Sum in which weareset tocount weightedarclabeled cycle coversoverafinite field ofcharacteristic two. We reduce Hamiltonicity to Labeled Cycle Cover Sum and apply the determinant summation technique for Exact Set Covers (Björklund STACS 2010) to evaluate it. 1
Dynamic programming and fast matrix multiplication
 of LNCS
, 2006
"... Abstract. We give a novel general approach for solving NPhard optimization problems that combines dynamic programming and fast matrix multiplication. The technique is based on reducing much of the computation involved to matrix multiplication. We exemplify our approach on problems like Vertex Cover ..."
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Cited by 12 (4 self)
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Abstract. We give a novel general approach for solving NPhard optimization problems that combines dynamic programming and fast matrix multiplication. The technique is based on reducing much of the computation involved to matrix multiplication. We exemplify our approach on problems like Vertex Cover, Dominating Set and Longest Path. Our approach works faster than the usual dynamic programming solution for any vertex subset problem on graphs of bounded branchwidth. In particular, we obtain the currently fastest algorithms for Planar Vertex Cover of runtime O(2 2.52 √ n), for Planar Dominating Set of runtime exact O(2 3.99 √ n) and parameterized O(2 11.98 √ k) · n O(1) , and for Planar Longest Path of runtime O(2 5.58 √ n). The exponent of the running time is depending heavily on the running time of the fastest matrix multiplication algorithm that is currently o(n 2.376). 1