Faster algorithms for finding lowest common ancestors in directed acyclic graphs

Czumaj, Artur; Kowaluk, Mirosaw; Lingas, Andrzej

doi:10.1016/j.tcs.2007.02.053

Cited by 52 publications

(56 citation statements)

References 12 publications

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“…For each edge (i, j), the maximum witness k of C [i, j] (if any) forms the heaviest triangle including this edge. Hence, using the O(n 2.616 ) time-bound for the maximum witness problem established in [13] and improved to O(n 2.575 ) by rectangular matrix multiplication [7], Vassilevska, Williams, and Yuster [18] obtained the same upper time-bounds for finding a maximum-weight triangle in a vertex-weighted graph.…”

Section: Introductionmentioning

confidence: 79%

Finding a Heaviest Vertex-Weighted Triangle Is not Harder than Matrix Multiplication

Czumaj¹,

Lingas²

2009

SIAM J. Comput.

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Abstract. We show that a maximum-weight triangle in an undirected graph with n vertices and real weights assigned to vertices can be found in time O(n ω + n 2+o(1) ), where ω is the exponent of the fastest matrix multiplication algorithm. By the currently best bound on ω, the running time of our algorithm is O(n 2.376 ). Our algorithm substantially improves the previous time-bounds for this problem, and its asymptotic time complexity matches that of the fastest known algorithm for finding any triangle (not necessarily a maximum-weight one) in a graph. We can extend our algorithm to improve the upper bounds on finding a maximum-weight triangle in a sparse graph and on finding a maximum-weight subgraph isomorphic to a fixed graph. We can find a maximum-weight triangle in a vertex-weighted graph with m edges in asymptotic time required by the fastest algorithm for finding any triangle in a graph with m edges, i.e., in time O(m 1.41 ). Our algorithms for a maximum-weight fixed subgraph (in particular any clique of constant size) are asymptotically as fast as the fastest known algorithms for a fixed subgraph.

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Section: Introductionmentioning

confidence: 79%

Finding a Heaviest Vertex-Weighted Triangle Is not Harder than Matrix Multiplication

Czumaj¹,

Lingas²

2009

SIAM J. Comput.

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“…For all pairs of nodes s and t we want to compute the highest node in topological order that still has a path to both s and t. Such a node is called a least common ancestor (LCA) of s and t. The all pairs LCA problem is to determine an LCA for every pair of vertices in a DAG. In terms of n, the best algebraic algorithm for finding all pairs LCAs uses the minimum witness product and runs in O(n 2.575 ) [12,7]. Czumaj, Kowaluk, and Lingas [12,7] gave an algorithm for finding all pairs LCAs in a sparse DAG in O(mn) time.…”

Section: All Pairs Shortest Paths (Apsp)mentioning

confidence: 99%

“…In terms of n, the best algebraic algorithm for finding all pairs LCAs uses the minimum witness product and runs in O(n 2.575 ) [12,7]. Czumaj, Kowaluk, and Lingas [12,7] gave an algorithm for finding all pairs LCAs in a sparse DAG in O(mn) time. We improve this runtime to O(mn log( n 2 m )/ log n).…”

Section: All Pairs Shortest Paths (Apsp)mentioning

confidence: 99%

“…It has been used to compute all pairs least common ancestors in DAGs [12,7], to find minimum weight triangles in node-weighted graphs [17], and to compute all pairs bottleneck paths in a node weighted graph [16]. The best known algorithm for the minimum witness product is by Czumaj, Kowaluk and Lingas [12,7] and runs in O(n 2.575 ) time. However, when one of the matrices has at most m = o(n 1.575 ) nonzeros, nothing better than O(mn) was known (in terms of m) for combinatorially computing the minimum witness product.…”

Section: Timementioning

confidence: 99%

“…To our knowledge, the best algorithm in terms of m and n for finding all pairs least common ancestors (LCAs) in a DAG, is a dynamic programming algorithm by Czumaj, Kowaluk and Lingas [12,7] which runs in O(mn) time. We improve the runtime of this algorithm to O(mn log(n 2 /m)/ log n) using the following generalization of Theorem 1, the proof of which is omitted.…”

Section: All Pairs Least Common Ancestors In a Dagmentioning

confidence: 99%

See 2 more Smart Citations

A New Combinatorial Approach for Sparse Graph Problems

Blelloch

Vassilevska

Williams

2008

Automata, Languages and Programming

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Abstract. We give a new combinatorial data structure for representing arbitrary Boolean matrices. After a short preprocessing phase, the data structure can perform fast vector multiplications with a given matrix, where the runtime depends on the sparsity of the input vector. The data structure can also return minimum witnesses for the matrix-vector product. Our approach is simple and implementable: the data structure works by precomputing small problems and recombining them in a novel way. It can be easily plugged into existing algorithms, achieving an asymptotic speedup over previous results. As a consequence, we achieve new running time bounds for computing the transitive closure of a graph, all pairs shortest paths on unweighted undirected graphs, and finding a maximum node-weighted triangle. Furthermore, any asymptotic improvement on our algorithms would imply a o(n 3 / log 2 n) combinatorial algorithm for Boolean matrix multiplication, a longstanding open problem in the area. We also use the data structure to give the first asymptotic improvement over O(mn) for all pairs least common ancestors on directed acyclic graphs.

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Quantum and Approximation Algorithms for Maximum Witnesses of Boolean Matrix Products

Kowaluk

Lingas

2021

Algorithms and Discrete Applied Mathematics

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Faster algorithms for finding lowest common ancestors in directed acyclic graphs

Cited by 52 publications

References 12 publications

Finding a Heaviest Vertex-Weighted Triangle Is not Harder than Matrix Multiplication

Finding a Heaviest Vertex-Weighted Triangle Is not Harder than Matrix Multiplication

A New Combinatorial Approach for Sparse Graph Problems

Quantum and Approximation Algorithms for Maximum Witnesses of Boolean Matrix Products

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