Roei Tov | Searcharxiv

arXiv2018

Approximating Cycles in Directed Graphs: Fast Algorithms for Girth and Roundtrip Spanners

The girth of a graph, i.e. the length of its shortest cycle, is a fundamental graph parameter. Unfortunately all known algorithms for computing, even approximately, the girth and girth-related structures in directed weighted $m$-edge and $n$-node graphs require $Ω(\min\{n^ω, mn\})$ time (for $2\leqω<2.373$). In this paper, we drastically improve these runtimes as follows: * Multiplicative Approximations in Nearly Linear Time: We give an algorithm that in $\widetilde{O}(m)$ time computes an $\widetilde{O}(1)$-multiplicative approximation of the girth as well as an $\widetilde{O}(1)$-multiplicative roundtrip spanner with $\widetilde{O}(n)$ edges with high probability (w.h.p). * Nearly Tight Additive Approximations: For unweighted graphs and any $α\in (0,1)$ we give an algorithm that in $\widetilde{O}(mn^{1 - α})$ time computes an $O(n^α)$-additive approximation of the girth w.h.p, and partially derandomize it. We show that the runtime of our algorithm cannot be significantly improved without a breakthrough in combinatorial Boolean matrix multiplication. Our main technical contribution to achieve these results is the first nearly linear time algorithm for computing roundtrip covers, a directed graph decomposition concept key to previous roundtrip spanner constructions. Previously it was not known how to compute these significantly faster than $Ω(\min\{n^ω, mn\})$ time. Given the traditional difficulty in efficiently processing directed graphs, we hope our techniques may find further applications.

cs.DS↗

arXiv2014

New routing techniques and their applications

Let $G=(V,E)$ be an undirected graph with $n$ vertices and $m$ edges. We obtain the following new routing schemes: - A routing scheme for unweighted graphs that uses $\tilde O(\frac{1}ε n^{2/3})$ space at each vertex and $\tilde O(1/ε)$-bit headers, to route a message between any pair of vertices $u,v\in V$ on a $(2 + ε,1)$-stretch path, i.e., a path of length at most $(2+ε)\cdot d(u,v)+1$. This should be compared to the $(2,1)$-stretch and $\tilde O(n^{5/3})$ space distance oracle of Patrascu and Roditty [FOCS'10 and SIAM J. Comput. 2014] and to the $(2,1)$-stretch routing scheme of Abraham and Gavoille [DISC'11] that uses $\tilde O( n^{3/4})$ space at each vertex. - A routing scheme for weighted graphs with normalized diameter $D$, that uses $\tilde O(\frac{1}ε n^{1/3}\log D)$ space at each vertex and $\tilde O(\frac{1}ε\log D)$-bit headers, to route a message between any pair of vertices on a $(5+ε)$-stretch path. This should be compared to the $5$-stretch and $\tilde O(n^{4/3})$ space distance oracle of Thorup and Zwick [STOC'01 and J. ACM. 2005] and to the $7$-stretch routing scheme of Thorup and Zwick [SPAA'01] that uses $\tilde O( n^{1/3})$ space at each vertex. Since a $5$-stretch routing scheme must use tables of $Ω( n^{1/3})$ space our result is almost tight. - For an integer $\ell>1$, a routing scheme for unweighted graphs that uses $\tilde O(\ell\frac{1}ε n^{\ell/(2\ell \pm 1)})$ space at each vertex and $\tilde O(\frac{1}ε)$-bit headers, to route a message between any pair of vertices on a $(3\pm2/\ell+ε,2)$-stretch path. - A routing scheme for weighted graphs, that uses $\tilde O(\frac{1}εn^{1/k}\log D)$ space at each vertex and $\tilde O(\frac{1}ε\log D)$-bit headers, to route a message between any pair of vertices on a $(4k-7+ε)$-stretch path.

cs.DS↗