No abstract

A (β, ǫ)-hopset for a weighted undirected n-vertex graph G = (V, E) is a set of edges, whose addition to the graph guarantees that every pair of vertices has a path between them that contains at most β edges, whose length is within 1 + ǫ of the shortest path. In her seminal paper, Cohen [Coh00, JACM 2000] introduced the notion of hopsets in the context of parallel computation of approximate shortest paths, and since then it has found numerous applications in various other settings, such as dynamic graph algorithms, distributed computing, and the streaming model.Cohen [Coh00] devised efficient algorithms for constructing hopsets with polylogarithmic in n number of hops. Her constructions remain the state-of-the-art since the publication of her paper in STOC'94, i.e., for more than two decades.In this paper we exhibit the first construction of sparse hopsets with a constant number of hops. We also find efficient algorithms for hopsets in various computational settings, improving the best known constructions. Generally, our hopsets strictly outperform the hopsets of [Coh00], both in terms of their parameters, and in terms of the resources required to construct them.We demonstrate the applicability of our results for the fundamental problem of computing approximate shortest paths from s sources. Our results improve the running time for this problem in the parallel, distributed and streaming models, for a vast range of s.

We prove that any graph G with n points has a distribution T over spanning trees such that for any edge (u, v) the expected stretch ET ∼T [dT (u, v)/dG(u, v)] is bounded byÕ(log n). Our result is obtained via a new approach of building "highways" between portals and a new strong diameter probabilistic decomposition theorem.

Miller et al. [43] devised a distributed 1 algorithm in the CONGEST model, that given a parameter k = 1, 2, . . ., constructs an O(k)-spanner of an input unweighted nvertex graph with O(n 1+1/k ) expected edges in O(k) rounds of communication. In this paper we improve the result of [43], by showing a k-round distributed algorithm in the same model, that constructs a (2k − 1)-spanner with O(n 1+1/k / ) edges, with probability 1 − , for any > 0. Moreover, when k = ω(log n), our algorithm produces (still in k rounds) ultra-sparse spanners, i.e., spanners of size n(1 + o(1)), with probability 1 − o(1). To our knowledge, this is the first distributed algorithm in the CONGEST or in the PRAM models that constructs spanners or skeletons (i.e., connected spanning subgraphs) that sparse. Our algorithm can also be implemented in linear time in the standard centralized model, and for large k, it provides spanners that are sparser than any other spanner given by a known (near-)linear time algorithm.We also devise improved bounds (and algorithms realizing these bounds) for (1 + , β)-spanners and emulators. In particular, we show that for any unweighted n-vertex graph and any > 0, there exists a (1 + , ( log log n ) log log n )-emulator with O(n) edges. All previous constructions of (1 + , β)-spanners and emulators employ a superlinear number of edges, for all choices of parameters.Finally, we provide some applications of our results to approximate shortest paths' computation in unweighted graphs. * Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel. Email: elkinm@cs.bgu.ac.il. 1 They actually showed a PRAM algorithm. The distributed algorithm with these properties is implicit in [43].

A maximal matching can be maintained in fully dynamic (supporting both addition and deletion of edges) n-vertex graphs using a trivial deterministic algorithm with a worstcase update time of O(n). No deterministic algorithm that outperforms the naïve O(n) one was reported up to this date. The only progress in this direction is due to Ivković and Lloyd [14], who in 1993 devised a deterministic algorithm with an amortized update time of O((n + m) √ 2/2 ), where m is the number of edges.In this paper we show the first deterministic fully dynamic algorithm that outperforms the trivial one. Specifically, we provide a deterministic worst-case update time of O( √ m). Moreover, our algorithm maintains a matching which is in fact a 3/2-approximate maximum cardinality matching (MCM). We remark that no fully dynamic algorithm for maintaining (2 − )-approximate MCM improving upon the naïve O(n) was known prior to this work, even allowing amortized time bounds and randomization.For low arboricity graphs (e.g., planar graphs and graphs excluding fixed minors), we devise another simple deterministic algorithm with sub-logarithmic update time. Specifically, it maintains a fully dynamic maximal matching with amortized update time of O(log n/ log log n). This result addresses an open question of Onak and Rubinfeld [19].We also show a deterministic algorithm with optimal space usage of O(n + m), that for arbitrary graphs maintains a maximal matching with amortized update time of O( √ m).

We prove that any weighted graph G = (V, E, w) with n points and m edges has a spanning tree T such that \sum \{ u,v\} \in E d T (u,v) w(u,v) = O(m log n log log n). Moreover, such a tree can be found in time O(m log n log log n). Our result is obtained using our new petal-decomposition approach which guarantees that the radius of each cluster in the tree is at most four times the radius of the induced subgraph of the cluster in the original graph.

We consider the problem of embedding finite metrics with slack: we seek to produce embeddings with small dimension and distortion while allowing a (small)

Abstract-Given a set of n points in 1, how many dimensions are needed to represent all pairwise distances within a specific distortion? This dimension-distortion tradeoff question is well understood for the 2 norm, where O((log n)/ 2 ) dimensions suffice to achieve 1 + distortion. In sharp contrast, there is a significant gap between upper and lower bounds for dimension reduction in 1. A recent result shows that distortion 1 + can be achieved with n/ 2 dimensions. On the other hand, the only lower bounds known are that distortion δ requires n Ω(1/δ2 ) dimensions and that distortion 1+ requires n 1/2−O( log(1/ )) dimensions. In this work, we show the first near linear lower bounds for dimension reduction in 1. In particular, we show that 1 + distortion requires at least n 1−O(1/ log(1/ )) dimensions. Our proofs are combinatorial, but inspired by linear programming. In fact, our techniques lead to a simple combinatorial argument that is equivalent to the LP based proof of BrinkmanCharikar for lower bounds on dimension reduction in 1.