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19
More algorithms for allpairs shortest paths in weighted graphs
 In Proceedings of 39th Annual ACM Symposium on Theory of Computing
, 2007
"... In the first part of the paper, we reexamine the allpairs shortest paths (APSP) problem and present a new algorithm with running time O(n 3 log 3 log n / log 2 n), which improves all known algorithms for general realweighted dense graphs. In the second part of the paper, we use fast matrix multipl ..."
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Cited by 75 (3 self)
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In the first part of the paper, we reexamine the allpairs shortest paths (APSP) problem and present a new algorithm with running time O(n 3 log 3 log n / log 2 n), which improves all known algorithms for general realweighted dense graphs. In the second part of the paper, we use fast matrix multiplication to obtain truly subcubic APSP algorithms for a large class of “geometrically weighted ” graphs, where the weight of an edge is a function of the coordinates of its vertices. For example, for graphs embedded in Euclidean space of a constant dimension d, we obtain a time bound near O(n 3−(3−ω)/(2d+4)), where ω < 2.376; in two dimensions, this is O(n 2.922). Our framework greatly extends the previously considered case of smallintegerweighted graphs, and incidentally also yields the first truly subcubic result (near O(n 3−(3−ω)/4) = O(n 2.844) time) for APSP in realvertexweighted graphs, as well as an improved result (near O(n (3+ω)/2) = O(n 2.688) time) for the allpairs lightest shortest path problem for smallintegerweighted graphs. 1
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 42 (11 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
LinearSpace Data Structures for Range Mode Query in Arrays
"... A mode of a multiset S is an element a ∈ S of maximum multiplicity; that is, a occurs at least as frequently as any other element in S. Given an array A[1: n] of n elements, we consider a basic problem: constructing a static data structure that efficiently answers range mode queries on A. Each query ..."
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Cited by 18 (8 self)
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A mode of a multiset S is an element a ∈ S of maximum multiplicity; that is, a occurs at least as frequently as any other element in S. Given an array A[1: n] of n elements, we consider a basic problem: constructing a static data structure that efficiently answers range mode queries on A. Each query consists of an input pair of indices (i, j) for which a mode of A[i: j] must be returned. The best previous data structure with linear space, by Krizanc, Morin, and Smid (ISAAC 2003), requires O ( √ n log log n) query time. We improve their result and present an O(n)space data structure that supports range mode queries in O ( p n / log n) worstcase time. Furthermore, we present strong evidence that a query time significantly below √ n cannot be achieved by purely combinatorial techniques; we show that boolean matrix multiplication of two √ n × √ n matrices reduces to n range mode queries in an array of size O(n). Additionally, we give linearspace data structures for orthogonal range mode in higher dimensions (queries in near O(n 1−1/2d) time) and for halfspace range mode in higher dimensions (queries in O(n 1−1/d2) time).
Popular conjectures imply strong lower bounds for dynamic problems
 CoRR
"... Abstract—We consider several wellstudied problems in dynamic algorithms and prove that sufficient progress on any of them would imply a breakthrough on one of five major open problems in the theory of algorithms: 1) Is the 3SUM problem on n numbers in O(n2−ε) time for some ε> 0? 2) Can one dete ..."
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Cited by 11 (3 self)
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Abstract—We consider several wellstudied problems in dynamic algorithms and prove that sufficient progress on any of them would imply a breakthrough on one of five major open problems in the theory of algorithms: 1) Is the 3SUM problem on n numbers in O(n2−ε) time for some ε> 0? 2) Can one determine the satisfiability of a CNF formula on n variables and poly n clauses in O((2 − ε)npoly n) time for some ε> 0? 3) Is the All Pairs Shortest Paths problem for graphs on n vertices in O(n3−ε) time for some ε> 0? 4) Is there a linear time algorithm that detects whether a given graph contains a triangle? 5) Is there an O(n3−ε) time combinatorial algorithm for n×n Boolean matrix multiplication? The problems we consider include dynamic versions of bipartite perfect matching, bipartite maximum weight matching, single source reachability, single source shortest paths, strong connectivity, subgraph connectivity, diameter approximation and some nongraph problems such as Pagh’s problem defined in a recent paper by Pǎtraşcu[STOC 2010]. Index Terms—dynamic algorithms; all pairs shortest paths; 3SUM; lower bounds; I.
Grothendiecktype inequalities in combinatorial optimization
 COMM. PURE APPL. MATH
, 2011
"... We survey connections of the Grothendieck inequality and its variants to combinatorial optimization and computational complexity. ..."
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Cited by 9 (3 self)
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We survey connections of the Grothendieck inequality and its variants to combinatorial optimization and computational complexity.
Spectral methods for matrices and tensors
 IN PROCEEDINGS OF THE 42ND ACM SYMPOSIUM ON THEORY OF COMPUTING
, 2010
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Improved OutputSensitive Quantum Algorithms for Boolean Matrix Multiplication
"... We present new quantum algorithms for Boolean Matrix Multiplication in both the time complexity and the query complexity settings. As far as time complexity is concerned, our results show that the product of two n × n Boolean matrices can be computed on a quantum computer in time Õ(n 3/2 + nℓ 3/4), ..."
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Cited by 6 (1 self)
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We present new quantum algorithms for Boolean Matrix Multiplication in both the time complexity and the query complexity settings. As far as time complexity is concerned, our results show that the product of two n × n Boolean matrices can be computed on a quantum computer in time Õ(n 3/2 + nℓ 3/4), where ℓ is the number of nonzero entries in the product, improving over the outputsensitive quantum algorithm by Buhrman and ˇ Spalek that runs in Õ(n3/2√ℓ) time. This is done by constructing a quantum version of a recent algorithm by Lingas, using quantum techniques such as quantum counting to exploit the sparsity of the output matrix. As far as query complexity is concerned, our results improve over the quantum algorithm by Vassilevska Williams and Williams based on a reduction to the triangle finding problem. One of the main contributions leading to this improvement is the construction of a triangle finding quantum algorithm tailored especially for the tripartite graphs appearing in the reduction. 1
Reducing the Worst Case Running Times of a Family of RNA and CFG Problems, Using Valiant’s Approach
"... Abstract. We study Valiant’s classical algorithm for Context Free Grammar recognition in subcubic time, and extract features that are common to problems on which Valiant’s approach can be applied. Based on this, we describe several problem templates, and formulate generic algorithms that use Valian ..."
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Cited by 4 (1 self)
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Abstract. We study Valiant’s classical algorithm for Context Free Grammar recognition in subcubic time, and extract features that are common to problems on which Valiant’s approach can be applied. Based on this, we describe several problem templates, and formulate generic algorithms that use Valiant’s technique and can be applied to all problems which abide by these templates. These algorithms obtain new worst case running time bounds for a large family of important problems within the world of RNA Secondary Structures and Context Free Grammars. 1