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43
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 60 (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...
Exact and Approximate Distances in Graphs  a survey
 In ESA
, 2001
"... We survey recent and not so recent results related to the computation of exact and approximate distances, and corresponding shortest, or almost shortest, paths in graphs. We consider many different settings and models and try to identify some remaining open problems. ..."
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Cited by 57 (0 self)
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We survey recent and not so recent results related to the computation of exact and approximate distances, and corresponding shortest, or almost shortest, paths in graphs. We consider many different settings and models and try to identify some remaining open problems.
On The Complexity Of Computing Determinants
 COMPUTATIONAL COMPLEXITY
, 2001
"... We present new baby steps/giant steps algorithms of asymptotically fast running time for dense matrix problems. Our algorithms compute the determinant, characteristic polynomial, Frobenius normal form and Smith normal form of a dense n n matrix A with integer entries in (n and (n bi ..."
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Cited by 47 (17 self)
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We present new baby steps/giant steps algorithms of asymptotically fast running time for dense matrix problems. Our algorithms compute the determinant, characteristic polynomial, Frobenius normal form and Smith normal form of a dense n n matrix A with integer entries in (n and (n bit operations; here denotes the largest entry in absolute value and the exponent adjustment by "+o(1)" captures additional factors for positive real constants C 1 , C 2 , C 3 . The bit complexity (n results from using the classical cubic matrix multiplication algorithm. Our algorithms are randomized, and we can certify that the output is the determinant of A in a Las Vegas fashion. The second category of problems deals with the setting where the matrix A has elements from an abstract commutative ring, that is, when no divisions in the domain of entries are possible. We present algorithms that deterministically compute the determinant, characteristic polynomial and adjoint of A with n and O(n ) ring additions, subtractions and multiplications.
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
Fast Sparse Matrix Multiplication
, 2004
"... Let A and B two n n matrices over a ring R (e.g., the reals or the integers) each containing at most m nonzero elements. We present a new algorithm that multiplies A and B using O(m ) algebraic operations (i.e., multiplications, additions and subtractions) over R. The naive matrix multi ..."
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Cited by 36 (2 self)
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Let A and B two n n matrices over a ring R (e.g., the reals or the integers) each containing at most m nonzero elements. We present a new algorithm that multiplies A and B using O(m ) algebraic operations (i.e., multiplications, additions and subtractions) over R. The naive matrix multiplication algorithm, on the other hand, may need to perform #(mn) operations to accomplish the same task. For , the new algorithm performs an almost optimal number of only n operations. For m the new algorithm is also faster than the best known matrix multiplication algorithm for dense matrices which uses O(n ) algebraic operations. The new algorithm is obtained using a surprisingly straightforward combination of a simple combinatorial idea and existing fast rectangular matrix multiplication algorithms. We also obtain improved algorithms for the multiplication of more than two sparse matrices.
Detecting Short Directed Cycles Using Rectangular Matrix Multiplication and Dynamic Programming
"... We present several new algorithms for detecting short fixed length cycles in digraphs. The new algorithms utilize fast rectangular matrix multiplication algorithms together with a dynamic programming approach similar to the one used in the solution of the classical chain matrix product problem. T ..."
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Cited by 18 (3 self)
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We present several new algorithms for detecting short fixed length cycles in digraphs. The new algorithms utilize fast rectangular matrix multiplication algorithms together with a dynamic programming approach similar to the one used in the solution of the classical chain matrix product problem. The new algorithms are instantiations of a generic algorithm that we present for finding a directed C k , i.e., a directed cycle of length k, in a digraph, for any fixed k 3. This algorithm partitions the prospective C k 's in the input digraph G = (V, E) into O(log V ) classes, according to the degrees of their vertices. For each cycle class we determine, in O(E log V ) time, whether G contains a C k from that class, where c k = c k (#) is a constant that depends only on #, the exponent of square matrix multiplication. The search for cycles from a given class is guided by the solution of a small dynamic programming problem. The total running time of the obtained deterministic algorithm is therefore O(E k+1 V ).
Computing the sign or the value of the determinant of an integer matrix, a complexity survey
 JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
, 2004
"... Computation of the sign of the determ9vSof amWDz[ and the determBvSitself is a challenge for both numhvB5q and exact mactv;W We survey the comWzqDvS of existingmistin to solve these problem when the input is an nnm;9q; A with integer entries. We study the bitcomvSW5[5Wv of the algorithm asymithmW5[ ..."
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Cited by 16 (3 self)
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Computation of the sign of the determ9vSof amWDz[ and the determBvSitself is a challenge for both numhvB5q and exact mactv;W We survey the comWzqDvS of existingmistin to solve these problem when the input is an nnm;9q; A with integer entries. We study the bitcomvSW5[5Wv of the algorithm asymithmW5[5 n and thenorm of A. Existing approaches rely onnumBDWzv approximW[ comoximW[z5 on exactcomvB[;5vSW or on both types of arithmvSW incom9zqvSWqD c 2003 Elsevier B.V. All rights reserved. Keywords: Determ9v9vm Bitcom9vmv Integer mteger Approxim59 comoxim599 Exact comv55DvS Random5D algorithm 1. I517251716 Com;vm9 the sign or the value of thedetermDvS; nmBqq A is a classicalproblem Numblem mmble are usually focused oncomBW9v the sign via an accurateapproxim;B99 of the determvS;;5 Amer the applications areimvWD;qW problem ofcom9qWvS;;5[ geom9qW that can be reduced to the determ5vS; question; the readerma refer to [11,12,9,10,46,45] and to the bibliography therein. InsymDW;B comW;BvS55 theproblem ofcomDzWv the exact value of the ThismisvqB; is based on work supported in part by the National Science Foundation under grants Nrs. DMS9977392, CCR9988177, and CCR0113121 (Kaltofen) and by the Centre National de la Recherche Scienti#que, Actions Incitatives No. 5929 et STIC LINBOX 2001 (Villard).
On the Complexity of Fixed Parameter Clique and Dominating Set
, 2004
"... We provide simple, faster algorithms for the detection of cliques and dominating sets of fixed order. Our algorithms are based on reductions to rectangular matrix multiplication. We also describe an improved algorithm for diamonds detection. ..."
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Cited by 16 (2 self)
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We provide simple, faster algorithms for the detection of cliques and dominating sets of fixed order. Our algorithms are based on reductions to rectangular matrix multiplication. We also describe an improved algorithm for diamonds detection.
Nonuniform ACC circuit lower bounds
, 2010
"... The class ACC consists of circuit families with constant depth over unbounded fanin AND, OR, NOT, and MODm gates, where m> 1 is an arbitrary constant. We prove: • NTIME[2 n] does not have nonuniform ACC circuits of polynomial size. The size lower bound can be slightly strengthened to quasipolynom ..."
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Cited by 16 (4 self)
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The class ACC consists of circuit families with constant depth over unbounded fanin AND, OR, NOT, and MODm gates, where m> 1 is an arbitrary constant. We prove: • NTIME[2 n] does not have nonuniform ACC circuits of polynomial size. The size lower bound can be slightly strengthened to quasipolynomials and other less natural functions. • ENP, the class of languages recognized in 2O(n) time with an NP oracle, doesn’t have nonuniform ACC circuits of 2no(1) size. The lower bound gives an exponential sizedepth tradeoff: for every d there is a δ> 0 such that ENP doesn’t have depthd ACC circuits of size 2nδ. Previously, it was not known whether EXP NP had depth3 polynomial size circuits made out of only MOD6 gates. The highlevel strategy is to design faster algorithms for the circuit satisfiability problem over ACC circuits, then prove that such algorithms entail the above lower bounds. The algorithm combines known properties of ACC with fast rectangular matrix multiplication and dynamic programming, while the second step requires a subtle strengthening of the author’s prior work [STOC’10]. Supported by the Josef Raviv Memorial Fellowship.
Finding the smallest Hsubgraph in real weighted graphs and related problems
 In Proc. of ICALP, SpringerVerlag LNCS 4051:262–273
, 2006
"... Abstract. Let G be a graph with real weights assigned to the vertices (edges). The weight of a subgraph of G is the sum of the weights of its vertices (edges). The MIN HSUBGRAPH problem is to find a minimum weight subgraph isomorphic to H, if one exists. Our main results are new algorithms for the ..."
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Cited by 11 (10 self)
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Abstract. Let G be a graph with real weights assigned to the vertices (edges). The weight of a subgraph of G is the sum of the weights of its vertices (edges). The MIN HSUBGRAPH problem is to find a minimum weight subgraph isomorphic to H, if one exists. Our main results are new algorithms for the MIN HSUBGRAPH problem. The only operations we allow on real numbers are additions and comparisons. Our algorithms are based, in part, on fast matrix multiplication. For vertexweighted graphs with n vertices we obtain the following results. We present an O(n t(ω,h) ) time algorithm for MIN HSUBGRAPH in case H is a fixed graph with h vertices and ω < 2.376 is the exponent of matrix multiplication. The value of t(ω, h) is determined by solving a small integer program. In particular, the smallest triangle can be found in O(n 2+1/(4−ω) ) ≤ o(n 2.616) time, the smallest K4 in O(n ω+1) time, the smallest K7 in O(n 4+3/(4−ω) ) time. As h grows, t(ω, h) converges to 3h/(6 − ω) < 0.828h. Interestingly, only for h = 4, 5, 8 the running time of our algorithm essentially matches that of the (unweighted) Hsubgraph detection problem. Already for triangles, our results improve upon the main result of [VW06]. Using rectangular matrix multiplication, the value of t(ω, h) can be improved; for example, the runtime for triangles becomes O(n 2.575). We also present an algorithm whose running time is a function of m, the number of edges. In particular, the smallest triangle can be found in O(m (18−4ω)/(13−3ω) ) ≤ o(m 1.45) time. For edgeweighted graphs we present an O(m 2−1/k log n) time algorithm that finds the smallest cycle of length 2k or 2k − 1. This running time is identical, up to a logarithmic factor, to the running time of the algorithm of Alon et al. for the unweighted case. Using the color coding method and a recent algorithm of Chan for distance products, we obtain an O(n 3 / log n) time randomized algorithm for finding the smallest cycle of any fixed length. 1