All-pairs bottleneck paths for general graphs in truly sub-cubic time

Vassilevska, Virginia; Williams, Ryan; Yuster, Raphael

doi:10.1145/1250790.1250876

Cited by 52 publications

(75 citation statements)

References 19 publications

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“…Pollack [27] introduced the problem and showed how to solve it in O(n 3 ) time. Vassilevska, Williams, and Yuster [31] gave an O(n 2+ω/3 )-time algorithm, where ω is the exponent of matrix multiplication over a ring. This was recently improved by Duan and Pettie [14] to O(n (3+ω)/2 ).…”

Section: Relation To Previous Workmentioning

confidence: 99%

On Cartesian Trees and Range Minimum Queries

2012

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Abstract.We present new results on Cartesian trees with applications in range minimum queries and bottleneck edge queries. We introduce a cacheoblivious Cartesian tree for solving the range minimum query problem, a Cartesian tree of a tree for the bottleneck edge query problem on trees and undirected graphs, and a proof that no Cartesian tree exists for the two-dimensional version of the range minimum query problem.

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Section: Relation To Previous Workmentioning

confidence: 99%

On Cartesian Trees and Range Minimum Queries

2012

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“…Matrix multiplication over semirings has a multitude of applications in computer science, and in particular in the area of graph algorithms (e.g., [6,19,21,22,23,25]). One example is Boolean matrix multiplication, related for instance to the computation of the transitive closure of a graph, where the product of two n × n Boolean matrices A and B is defined as the n × n Boolean matrix C = A · B such that C[i, j] = 1 if and only if there exists a k ∈ {1, .…”

Section: Introductionmentioning

confidence: 99%

“…The (max, min)-product has mainly been studied in the field in fuzzy logic [7] under the name composition of relations and in the context of computing the all-pairs bottleneck paths of a graph (i.e., computing, for all pairs (s,t) of vertices in a graph, the maximum flow that can be routed between s and t). More precisely, it is well known (see, e.g., [6,19,23]) that if the (max, min)-product of two n × n matrices can be computed in time T (n), then the all-pairs bottleneck paths of a graph with n vertices can be computed in timeÕ(T (n)). As an application of Theorem 1.1, we thus obtain a O(n 2.473 )-time quantum algorithm computing the all-pairs bottleneck paths of a graph of n vertices, while classically the best upper bound for this task is O(n 2.687 ), again from [6].…”

Section: Introductionmentioning

confidence: 99%

“…We use this technique in various ways to improve the combinatorial steps in several classical approaches [1,6,23,26] that are based on a combination of algebraic steps (computing some matrix products over a field) and combinatorial steps. Moreover, the speedup obtained by quantum enumeration enables us to depart from these original approaches and optimize the combinatorial and algebraic steps in different ways, for instance relying on rectangular matrix multiplication instead of square matrix multiplication.…”

Section: Introductionmentioning

confidence: 99%

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Gall¹,

Nishimura²

2017

Chicago J. of Theoretical Comp. Sci.

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In this paper we construct quantum algorithms for matrix products over several algebraic structures called semirings, including the (max, min)-matrix product, the distance matrix product and the Boolean matrix product. In particular, we obtain the following results.• We construct a quantum algorithm computing the product of two n × n matrices over the (max, min) semiring with time complexity O(n 2.473 ). In comparison, the best known classical algorithm for the same problem, by Duan and Pettie (SODA'09), has complexity O(n 2.687 ).• We construct a quantum algorithm computing the most significant bits of each entry of the distance product of two n × n matrices in time O(2 0.64 n 2.46 ). In comparison, the best known classical algorithm for the same problem, by Vassilevska and Williams (STOC'06) and Yuster (SODA'09), has complexity O(2 n 2.69 ).The above two algorithms are the first quantum algorithms that perform better than thẽ O(n 5/2 )-time straightforward quantum algorithm based on quantum search for matrix multiplication over these semirings. We also consider the Boolean semiring, and construct a quantum algorithm computing the product of two n × n Boolean matrices that outperforms the best known classical algorithms for sparse matrices.

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“…, n. More informatively, we wish to compute D(u, v), the number of coordinates i for which u i ≤ v i . Matousek's algorithm solves this problem in O(n (ω+3)/2 ) time, if |V | = n. Permutation dominance is an important ingredient in more recent results; on all-pairs shortest paths [4], on all-pairs bottleneck paths [14], and on finding heaviest vertexweighted subgraphs [13]. In Section 2 we show how to compute the dominance matrix D P slightly faster than was previously known.…”

Section: Introductionmentioning

confidence: 99%

Efficient algorithms on sets of permutations, dominance, and real-weighted APSP

Yuster

2009

Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms

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Sets of permutations play an important role in the design of some efficient algorithms. In this paper we design two algorithms that manipulate sets of permutations. Both algorithms, each solving a different problem, use fast matrix multiplication techniques to achieve a significant improvement in the running time over the naive solutions.For a set of permutations P ⊂ S n we say that i k-dominates j if the number of permutations π ∈ P for which π(i) < π(j) is k. The dominance matrix of P is the n × n matrix D P where D P (i, j) = k if and only if i k-dominates j. We give an efficient algorithm for computing D P using fast rectangular matrix multiplication. In particular, when |P | = n our algorithm runs in O(n 2.684 ) time. Computing the dominance matrix of permutations is computationally equivalent to the dominance problem in computational geometry. Thus, our algorithm slightly improves upon a well-known O(n 2.688 ) time algorithm of Matousek for the dominance problem. Permutation dominance is used, together with several other ingredients, to obtain a truly sub-cubic algorithm for the All Pairs Shortest Paths (APSP) problem in real-weighted directed graphs, where the number of distinct weights emanating from each vertex is O(n 0.338 ). A special case of this algorithm implies an O(n 2.842 ) time algorithm for real vertexweighted APSP, which slightly improves a recent result of Chan .A set of permutations P ⊂ S n is fully expanding if the product of any two elements of P yields a distinct permutation. Stated otherwise, |P 2 | = |P | 2 where P 2 ⊂ S n is the set of products of two elements of P . We present a randomized algorithm that computes |P 2 | and hence decides if P is fully expanding. The algorithm also produces a table that, for any σ 1 , σ 2 , σ 3 , σ 4 ∈ P , answers the query σ 1 σ 2 = σ 3 σ 4 inÕ(1) time. The algorithm uses, among other ingredients, a combination of fast matrix multiplication and polynomial identity testing. In particular, for |P | = n our algorithm runs in O(n ω ) time where ω < 2.376 is the matrix multiplication *

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All-pairs bottleneck paths for general graphs in truly sub-cubic time

Cited by 52 publications

References 19 publications

On Cartesian Trees and Range Minimum Queries

On Cartesian Trees and Range Minimum Queries

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Efficient algorithms on sets of permutations, dominance, and real-weighted APSP

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