A simpler algorithm for the all pairs shortest path problem with o(n2 log n) expected time
COCOA'10 Proceedings of the 4th international conference on Combinatorial optimization and applications - Volume Part II
On the $k$ Shortest Simple Paths Problem in Weighted Directed Graphs
SIAM Journal on Computing
Networks cannot compute their diameter in sublinear time
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Optimal distributed all pairs shortest paths and applications
PODC '12 Proceedings of the 2012 ACM symposium on Principles of distributed computing
Fast approximation algorithms for the diameter and radius of sparse graphs
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Max flows in O(nm) time, or better
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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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/log2 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.