Showing papers by "Liam Roditty published in 2014"

Better approximation algorithms for the graph diameter

Abstract: The diameter is a fundamental graph parameter and its computation is necessary in many applications. The fastest known way to compute the diameter exactly is to solve the All-Pairs Shortest Paths (APSP) problem.In the absence of fast algorithms, attempts were made to seek fast algorithms that approximate the diameter. In a seminal result Aingworth, Chekuri, Indyk and Motwani [SODA'96 and SICOMP'99] designed an algorithm that computes in O (n2 + m√n) time an estimate D for the diameter D in directed graphs with nonnegative edge weights, such that [EQUATION], where M is the maximum edge weight in the graph. In recent work, Roditty and Vassilevska W. [STOC 13] gave a Las Vegas algorithm that has the same approximation guarantee but improves the (expected) runtime to O (m√n). Roditty and Vassilevska W. also showed that unless the Strong Exponential Time Hypothesis fails, no O (n2-e) time algorithm for sparse unweighted undirected graphs can achieve an approximation ratio better than 3/2. Thus their algorithm is essentially tight for sparse unweighted graphs. For weighted graphs however, the approximation guarantee can be meaningless, as M can be arbitrarily large.In this paper we exhibit two algorithms that achieve a genuine 3/2-approximation for the diameter, one running in O (m3/2) time, and one running in O (mn2/3). time. Furthermore, our algorithms are deterministic, and thus we present the first deterministic (2 -- e)-approximation algorithm for the diameter that takes subquadratic time in sparse graphs.In addition, we address the question of obtaining an additive c-approximation for the diameter, i.e. an estimate D such that D -- c ≤ D ≤ D. An extremely simple O (mn1-e) time algorithm achieves an additive ne-approximation; no better results are known. We show that for any e > 0, getting an additive ne-approximation algorithm for the diameter running in O (n2-e) time for any δ > 2e would falsify the Strong Exponential Time Hypothesis. Thus the simple algorithm is probably essentially tight for sparse graphs, and moreover, obtaining a subquadratic time additive c-approximation for any constant c is unlikely.Finally, we consider the problem of computing the eccentricities of all vertices in an undirected graph, i.e. the largest distance from each vertex. Roditty and Vassilevska W. [STOC 13] show that in O (m√n) time, one can compute for each v e V in an undirected graph, an estimate e(v) for the eccentricity e (v) such that max{R, 2/3 · e(v)} ≤ e (v) ≤ min {D, 3/2 · e(v)} where R = minv e (v) is the radius of the graph. Here we improve the approximation guarantee by showing that a variant of the same algorithm can achieve estimates e' (v) with 3/5 · e (v) ≤ e' (v) ≤ e (v).

On Nash Equilibria for a Network Creation Game

Abstract: We study a basic network creation game proposed by Fabrikant et al. [2003]. In this game, each player (vertex) can create links (edges) to other players at a cost of α per edge. The goal of every player is to minimize the sum consisting of (a) the cost of the links he has created and (b) the sum of the distances to all other players.Fabrikant et al. conjectured that there exists a constant A such that, for any α > A, all nontransient Nash equilibria graphs are trees. They showed that if a Nash equilibrium is a tree, the price of anarchy is constant. In this article, we disprove the tree conjecture. More precisely, we show that for any positive integer n0, there exists a graph built by n ≥ n0 players which contains cycles and forms a nontransient Nash equilibrium, for any α with 1

Distance Oracles beyond the Thorup--Zwick Bound

Abstract: We give the first improvement to the space/approximation trade-off of distance oracles since the seminal result of Thorup and Zwick. For unweighted undirected graphs, our distance oracle has size $O(n^{5/3})$ and, when queried about vertices at distance $d$, returns a path of length at most 2d+1. For weighted undirected graphs with $m=n^2/\alpha$ edges, our distance oracle has size $O(n^2 / \sqrt[3]{\alpha})$ and returns a factor 2 approximation. Based on a plausible conjecture about the hardness of set intersection queries, we show that a 2-approximate distance oracle requires space $\widetilde{\Omega}(n^2 / \sqrt{\alpha})$. For unweighted graphs, this implies a $\widetilde{\Omega}(n^{1.5})$ space lower bound to achieve approximation 2d+1.

New routing techniques and their applications

Abstract: 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 $\tilde O(\frac{1}{\epsilon} n^{2/3})$ space at each vertex and $\tilde O(1/\epsilon)$-bit headers, to route a message between any pair of vertices $u,v\in V$ on a $(2 + \epsilon,1)$-stretch path, i.e., a path of length at most $(2+\epsilon)\cdot d(u,v)+1$. 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 $\tilde O(\frac{1}{\epsilon} n^{1/3}\log D)$ space at each vertex and $\tilde O(\frac{1}{\epsilon}\log D)$-bit headers, to route a message between any pair of vertices on a $(5+\epsilon)$-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 $\Omega( n^{1/3})$ space our result is almost tight. - For an integer $\ell>1$, a routing scheme for unweighted graphs that uses $\tilde O(\ell\frac{1}{\epsilon} n^{\ell/(2\ell \pm 1)})$ space at each vertex and $\tilde O(\frac{1}{\epsilon})$-bit headers, to route a message between any pair of vertices on a $(3\pm2/\ell+\epsilon,2)$-stretch path. - A routing scheme for weighted graphs, that uses $\tilde O(\frac{1}{\epsilon}n^{1/k}\log D)$ space at each vertex and $\tilde O(\frac{1}{\epsilon}\log D)$-bit headers, to route a message between any pair of vertices on a $(4k-7+\epsilon)$-stretch path.

On the Efficiency of the Hamming C-Centerstring Problems

Abstract: The Consensus String Problem is that of finding a string, such that the maximum Hamming distance from it to a given set of strings of the same length is minimized. However, a generalization is necessary for clustering. One needs to consider a partition into a number of sets, each with a distinct centerstring. In this paper we define two natural versions of the consensus problem for c centerstrings. We analyse the hardness and fixed parameter tractability of these problems and provide approximation algorithms.

Multiply Balanced k −Partitioning

Abstract: The problem of partitioning an edge-capacitated graph on n vertices into k balanced parts has been amply researched. Motivated by applications such as load balancing in distributed systems and market segmentation in social networks, we propose a new variant of the problem, called Multiply Balanced k Partitioning, where the vertex-partition must be balanced under d vertex-weight functions simultaneously.

Close to Linear Space Routing Schemes

Abstract: Let G = (V,E) be an unweighted undirected graph with n-vertices and m-edges, and let k > 2 be an integer. We present a routing scheme with a poly-logarithmic header size, that given a source s and a destination t at distance Δ from s, routes a message from s to t on a path whose length is O(kΔ + m 1/k ). The total space used by our routing scheme is \(\tilde{O}(mn^{O(1/\sqrt{\log n})})\), which is almost linear in the number of edges of the graph. We present also a routing scheme with \(\tilde{O}(n^{O(1/\sqrt{\log n})})\) header size, and the same stretch (up to constant factors). In this routing scheme, the routing table of every v ∈ V is at most \(\tilde{O}(kn^{O(1/\sqrt{\log n})}deg(v))\), where deg(v) is the degree of v in G. Our results are obtained by combining a general technique of Bernstein [6], that was presented in the context of dynamic graph algorithms, with several new ideas and observations.