LCA Queries in Directed Acyclic Graphs

Kowaluk, Mirosław; Lingas, Andrzej

doi:10.1007/11523468_20

Cited by 22 publications

(24 citation statements)

References 6 publications

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“…The first application, considered in [3], is All-Pairs Lowest Common Ancestors in directed acyclic graphs. The fastest algorithm for this problem, due to Kowaluk and Lingas [13,14], runs in O(n 2+µ ) time. We show that this problem can be easily reduced to a special case of closed-APBP.…”

Section: The New Resultsmentioning

confidence: 99%

All-Pairs Bottleneck Paths in Vertex Weighted Graphs

2009

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Let G = (V, E, w) be a directed graph, where w : V → R is an arbitrary weight function defined on its vertices. The bottleneck weight, or the capacity, of a path is the smallest weight of a vertex on the path. For two vertices u, v the bottleneck weight, or the capacity, from u to v, denoted c (u, v), is the maximum bottleneck weight of a path from u to v. In the All-Pairs Bottleneck Paths (APBP) problem we have to find the bottleneck weights for all ordered pairs of vertices. Our main result is an O(n 2.575 ) time algorithm for the APBP problem. The exponent is derived from the exponent of fast matrix multiplication. Our algorithm is the first sub-cubic algorithm for this problem. Unlike the sub-cubic algorithm for the all-pairs shortest paths (APSP) problem, that only applies to bounded (or relatively small) integer edge or vertex weights, the algorithm presented for APBP problem works for arbitrary large vertex weights.The APBP problem has numerous applications, and several interesting problems that have recently attracted attention can be reduced to it, with no asymptotic loss in the running times of the known algorithms for these problems. Some examples are a result of Vassilevska and Williams [STOC 2006] on finding a triangle of maximum weight, a result of Bender et al. [SODA 2001] on computing least common ancestors in DAGs and a result of Kowaluk and Lingas [ICALP 2005] on finding maximum witnesses for boolean matrix multiplication. Thus, the APBP problem provides a uniform framework for these applications. For some of these problems, we can in fact show that their complexity is equivalent to that of the APBP problem.A slight modification of our algorithm enables us to compute shortest paths of maximum bottleneck weight. Let d(u, v) denote the (unweighted) distance from u to v, and let sc (u, v) denote the maximum bottleneck weight of a path from u to v having length d (u, v). The AllPairs Bottleneck Shortest Paths (APBSP) problem is to compute sc (u, v) for all ordered pairs of vertices. We present an algorithm for the APBSP problem whose running time is O(n 2.86 ).

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Section: The New Resultsmentioning

confidence: 99%

All-Pairs Bottleneck Paths in Vertex Weighted Graphs

2009

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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%

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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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Finding the Smallest H-Subgraph in Real Weighted Graphs and Related Problems

Vassilevska

Williams

Yuster

2006

Lecture Notes in Computer Science

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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 H-SUBGRAPH problem is to find a minimum weight subgraph isomorphic to H, if one exists. Our main results are new algorithms for the MIN H-SUBGRAPH problem. The only operations we allow on real numbers are additions and comparisons. Our algorithms are based, in part, on fast matrix multiplication. For vertex-weighted graphs with n vertices we obtain the following results. We present an O(n t(ω,h) ) time algorithm for MIN H-SUBGRAPH 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 edge-weighted 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.

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LCA Queries in Directed Acyclic Graphs

Cited by 22 publications

References 6 publications

All-Pairs Bottleneck Paths in Vertex Weighted Graphs

All-Pairs Bottleneck Paths in Vertex Weighted Graphs

A New Combinatorial Approach for Sparse Graph Problems

Finding the Smallest H-Subgraph in Real Weighted Graphs and Related Problems

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