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Register a new userApproximating Cycles in Directed Graphs: Fast Algorithms for Girth and Roundtrip Spanners
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Virginia Vassilevska Williams
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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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 $Omega(min{n^{omega}, mn})$ time (for $2leqomega<2.373$). 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 $alpha in (0,1)$ we give an algorithm that in $widetilde{O}(mn^{1 - alpha})$ time computes an $O(n^alpha)$-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 $Omega(min{n^omega, mn})$ time. Given the traditional difficulty in efficiently processing directed graphs, we hope our techniques may find further applications.
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In this paper we give fast distributed graph algorithms for detecting and listing small subgraphs, and for computing or approximating the girth. Our algorithms improve upon the state of the art by polynomial factors, and for girth, we obtain an constant-time algorithm for additive +1 approximation in the Congested Clique, and the first parametrized algorithm for exact computation in CONGEST. In the Congested Clique, we develop a technique for learning small neighborhoods, and apply it to obtain an $O(1)$-round algorithm that computes the girth with only an additive +1 error. Next, we introduce a new technique (the partition tree technique) allowing for efficiently and deterministically listing all copies of any subgraph, improving upon the state-of the-art for non-dense graphs. We give two applications of this technique: First we show that for constant $k$, $C_{2k}$-detection can be solved in $O(1)$ rounds in the Congested Clique, improving on prior work which used matrix multiplication and had polynomial round complexity. Second, we show that in triangle-free graphs, the girth can be exactly computed in time polynomially faster than the best known bounds for general graphs. In CONGEST, we describe a new approach for finding cycles, and apply it in two ways: first we show a fast parametrized algorithm for girth with round complexity $tilde{O}(min(gcdot n^{1-1/Theta(g)},n))$ for any girth $g$; and second, we show how to find small even-length cycles $C_{2k}$ for $k = 3,4,5$ in $O(n^{1-1/k})$ rounds, which is a polynomial improvement upon the previous running times. Finally, using our improved $C_6$-freeness algorithm and the barrier on proving lower bounds on triangle-freeness of Eden et al., we show that improving the current $tildeOmega(sqrt{n})$ lower bound for $C_6$-freeness of Korhonen et al. by any polynomial factor would imply strong circuit complexity lower bounds.
It was recently found that there are very close connections between the existence of additive spanners (subgraphs where all distances are preserved up to an additive stretch), distance preservers (subgraphs in which demand pairs have their distance preserved exactly), and pairwise spanners (subgraphs in which demand pairs have their distance preserved up to a multiplicative or additive stretch) [Abboud-Godwin SODA 16, Godwin-Williams SODA 16]. We study these problems from an optimization point of view, where rather than studying the existence of extremal instances we are given an instance and are asked to find the sparsest possible spanner/preserver. We give an $O(n^{3/5 + epsilon})$-approximation for distance preservers and pairwise spanners (for arbitrary constant $epsilon > 0$). This is the first nontrivial upper bound for either problem, both of which are known to be as hard to approximate as Label Cover. We also prove Label Cover hardness for approximating additive spanners, even for the cases of additive 1 stretch (where one might expect a polylogarithmic approximation, since the related multiplicative 2-spanner problem admits an $O(log n)$-approximation) and additive polylogarithmic stretch (where the related multiplicative spanner problem has an $O(1)$-approximation). Interestingly, the techniques we use in our approximation algorithm extend beyond distance-based problem to pure connectivity network design problems. In particular, our techniques allow us to give an $O(n^{3/5 + epsilon})$-approximation for the Directed Steiner Forest problem (for arbitrary constant $epsilon > 0$) when all edges have uniform costs, improving the previous best $O(n^{2/3 + epsilon})$-approximation due to Berman et al.~[ICALP 11] (which holds for general edge costs).
A roundtrip spanner of a directed graph $G$ is a subgraph of $G$ preserving roundtrip distances approximately for all pairs of vertices. Despite extensive research, there is still a small stretch gap between roundtrip spanners in directed graphs and undirected graphs. For a directed graph with real edge weights in $[1,W]$, we first propose a new deterministic algorithm that constructs a roundtrip spanner with $(2k-1)$ stretch and $O(k n^{1+1/k}log (nW))$ edges for every integer $k> 1$, then remove the dependence of size on $W$ to give a roundtrip spanner with $(2k-1)$ stretch and $O(k n^{1+1/k}log n)$ edges. While keeping the edge size small, our result improves the previous $2k+epsilon$ stretch roundtrip spanners in directed graphs [Roditty, Thorup, Zwick02; Zhu, Lam18], and almost matches the undirected $(2k-1)$-spanner with $O(n^{1+1/k})$ edges [Althofer et al. 93] when $k$ is a constant, which is optimal under Erdos conjecture.
Let $P subset mathbb{R}^2$ be a planar $n$-point set such that each point $p in P$ has an associated radius $r_p > 0$. The transmission graph $G$ for $P$ is the directed graph with vertex set $P$ such that for any $p, q in P$, there is an edge from $p$ to $q$ if and only if $d(p, q) leq r_p$. Let $t > 1$ be a constant. A $t$-spanner for $G$ is a subgraph $H subseteq G$ with vertex set $P$ so that for any two vertices $p,q in P$, we have $d_H(p, q) leq t d_G(p, q)$, where $d_H$ and $d_G$ denote the shortest path distance in $H$ and $G$, respectively (with Euclidean edge lengths). We show how to compute a $t$-spanner for $G$ with $O(n)$ edges in $O(n (log n + log Psi))$ time, where $Psi$ is the ratio of the largest and smallest radius of a point in $P$. Using more advanced data structures, we obtain a construction that runs in $O(n log^5 n)$ time, independent of $Psi$. We give two applications for our spanners. First, we show how to use our spanner to find a BFS tree in $G$ from any given start vertex in $O(n log n)$ time (in addition to the time it takes to build the spanner). Second, we show how to use our spanner to extend a reachability oracle to answer geometric reachability queries. In a geometric reachability query we ask whether a vertex $p$ in $G$ can reach a target $q$ which is an arbitrary point in the plane (rather than restricted to be another vertex $q$ of $G$ in a standard reachability query). Our spanner allows the reachability oracle to answer geometric reachability queries with an additive overhead of $O(log nlog Psi)$ to the query time and $O(n log Psi)$ to the space.
We present online algorithms for directed spanners and Steiner forests. These problems fall under the unifying framework of online covering linear programming formulations, developed by Buchbinder and Naor (MOR, 34, 2009), based on primal-dual techniques. Our results include the following: For the pairwise spanner problem, in which the pairs of vertices to be spanned arrive online, we present an efficient randomized $tilde{O}(n^{4/5})$-competitive algorithm for graphs with general lengths, where $n$ is the number of vertices. With uniform lengths, we give an efficient randomized $tilde{O}(n^{2/3+epsilon})$-competitive algorithm, and an efficient deterministic $tilde{O}(k^{1/2+epsilon})$-competitive algorithm, where $k$ is the number of terminal pairs. These are the first online algorithms for directed spanners. In the offline setting, the current best approximation ratio with uniform lengths is $tilde{O}(n^{3/5 + epsilon})$, due to Chlamtac, Dinitz, Kortsarz, and Laekhanukit (TALG 2020). For the directed Steiner forest problem with uniform costs, in which the pairs of vertices to be connected arrive online, we present an efficient randomized $tilde{O}(n^{2/3 + epsilon})$-competitive algorithm. The state-of-the-art online algorithm for general costs is due to Chakrabarty, Ene, Krishnaswamy, and Panigrahi (SICOMP 2018) and is $tilde{O}(k^{1/2 + epsilon})$-competitive. In the offline version, the current best approximation ratio with uniform costs is $tilde{O}(n^{26/45 + epsilon})$, due to Abboud and Bodwin (SODA 2018). A small modification of the online covering framework by Buchbinder and Naor implies a polynomial-time primal-dual approach with separation oracles, which a priori might perform exponentially many calls. We convert the online spanner problem and the online Steiner forest problem into online covering problems and round in a problem-specific fashion.
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