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/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.
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Blelloch, G.E., Vassilevska, V., Williams, R. (2008). A New Combinatorial Approach for Sparse Graph Problems. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds) Automata, Languages and Programming. ICALP 2008. Lecture Notes in Computer Science, vol 5125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70575-8_10
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DOI: https://doi.org/10.1007/978-3-540-70575-8_10
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