15,490 research outputs found

    Approximate Minimum Diameter

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    We study the minimum diameter problem for a set of inexact points. By inexact, we mean that the precise location of the points is not known. Instead, the location of each point is restricted to a contineus region (\impre model) or a finite set of points (\indec model). Given a set of inexact points in one of \impre or \indec models, we wish to provide a lower-bound on the diameter of the real points. In the first part of the paper, we focus on \indec model. We present an O(21ϵdϵ2dn3)O(2^{\frac{1}{\epsilon^d}} \cdot \epsilon^{-2d} \cdot n^3 ) time approximation algorithm of factor (1+ϵ)(1+\epsilon) for finding minimum diameter of a set of points in dd dimensions. This improves the previously proposed algorithms for this problem substantially. Next, we consider the problem in \impre model. In dd-dimensional space, we propose a polynomial time d\sqrt{d}-approximation algorithm. In addition, for d=2d=2, we define the notion of α\alpha-separability and use our algorithm for \indec model to obtain (1+ϵ)(1+\epsilon)-approximation algorithm for a set of α\alpha-separable regions in time O(21ϵ2.n3ϵ10.sin(α/2)3)O(2^{\frac{1}{\epsilon^2}}\allowbreak . \frac{n^3}{\epsilon^{10} .\sin(\alpha/2)^3} )

    Distributed Edge Connectivity in Sublinear Time

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    We present the first sublinear-time algorithm for a distributed message-passing network sto compute its edge connectivity λ\lambda exactly in the CONGEST model, as long as there are no parallel edges. Our algorithm takes O~(n11/353D1/353+n11/706)\tilde O(n^{1-1/353}D^{1/353}+n^{1-1/706}) time to compute λ\lambda and a cut of cardinality λ\lambda with high probability, where nn and DD are the number of nodes and the diameter of the network, respectively, and O~\tilde O hides polylogarithmic factors. This running time is sublinear in nn (i.e. O~(n1ϵ)\tilde O(n^{1-\epsilon})) whenever DD is. Previous sublinear-time distributed algorithms can solve this problem either (i) exactly only when λ=O(n1/8ϵ)\lambda=O(n^{1/8-\epsilon}) [Thurimella PODC'95; Pritchard, Thurimella, ACM Trans. Algorithms'11; Nanongkai, Su, DISC'14] or (ii) approximately [Ghaffari, Kuhn, DISC'13; Nanongkai, Su, DISC'14]. To achieve this we develop and combine several new techniques. First, we design the first distributed algorithm that can compute a kk-edge connectivity certificate for any k=O(n1ϵ)k=O(n^{1-\epsilon}) in time O~(nk+D)\tilde O(\sqrt{nk}+D). Second, we show that by combining the recent distributed expander decomposition technique of [Chang, Pettie, Zhang, SODA'19] with techniques from the sequential deterministic edge connectivity algorithm of [Kawarabayashi, Thorup, STOC'15], we can decompose the network into a sublinear number of clusters with small average diameter and without any mincut separating a cluster (except the `trivial' ones). Finally, by extending the tree packing technique from [Karger STOC'96], we can find the minimum cut in time proportional to the number of components. As a byproduct of this technique, we obtain an O~(n)\tilde O(n)-time algorithm for computing exact minimum cut for weighted graphs.Comment: Accepted at 51st ACM Symposium on Theory of Computing (STOC 2019

    Distributed Symmetry Breaking in Hypergraphs

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    Fundamental local symmetry breaking problems such as Maximal Independent Set (MIS) and coloring have been recognized as important by the community, and studied extensively in (standard) graphs. In particular, fast (i.e., logarithmic run time) randomized algorithms are well-established for MIS and Δ+1\Delta +1-coloring in both the LOCAL and CONGEST distributed computing models. On the other hand, comparatively much less is known on the complexity of distributed symmetry breaking in {\em hypergraphs}. In particular, a key question is whether a fast (randomized) algorithm for MIS exists for hypergraphs. In this paper, we study the distributed complexity of symmetry breaking in hypergraphs by presenting distributed randomized algorithms for a variety of fundamental problems under a natural distributed computing model for hypergraphs. We first show that MIS in hypergraphs (of arbitrary dimension) can be solved in O(log2n)O(\log^2 n) rounds (nn is the number of nodes of the hypergraph) in the LOCAL model. We then present a key result of this paper --- an O(Δϵpolylog(n))O(\Delta^{\epsilon}\text{polylog}(n))-round hypergraph MIS algorithm in the CONGEST model where Δ\Delta is the maximum node degree of the hypergraph and ϵ>0\epsilon > 0 is any arbitrarily small constant. To demonstrate the usefulness of hypergraph MIS, we present applications of our hypergraph algorithm to solving problems in (standard) graphs. In particular, the hypergraph MIS yields fast distributed algorithms for the {\em balanced minimal dominating set} problem (left open in Harris et al. [ICALP 2013]) and the {\em minimal connected dominating set problem}. We also present distributed algorithms for coloring, maximal matching, and maximal clique in hypergraphs.Comment: Changes from the previous version: More references adde

    Parallel Algorithms for Geometric Graph Problems

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    We give algorithms for geometric graph problems in the modern parallel models inspired by MapReduce. For example, for the Minimum Spanning Tree (MST) problem over a set of points in the two-dimensional space, our algorithm computes a (1+ϵ)(1+\epsilon)-approximate MST. Our algorithms work in a constant number of rounds of communication, while using total space and communication proportional to the size of the data (linear space and near linear time algorithms). In contrast, for general graphs, achieving the same result for MST (or even connectivity) remains a challenging open problem, despite drawing significant attention in recent years. We develop a general algorithmic framework that, besides MST, also applies to Earth-Mover Distance (EMD) and the transportation cost problem. Our algorithmic framework has implications beyond the MapReduce model. For example it yields a new algorithm for computing EMD cost in the plane in near-linear time, n1+oϵ(1)n^{1+o_\epsilon(1)}. We note that while recently Sharathkumar and Agarwal developed a near-linear time algorithm for (1+ϵ)(1+\epsilon)-approximating EMD, our algorithm is fundamentally different, and, for example, also solves the transportation (cost) problem, raised as an open question in their work. Furthermore, our algorithm immediately gives a (1+ϵ)(1+\epsilon)-approximation algorithm with nδn^{\delta} space in the streaming-with-sorting model with 1/δO(1)1/\delta^{O(1)} passes. As such, it is tempting to conjecture that the parallel models may also constitute a concrete playground in the quest for efficient algorithms for EMD (and other similar problems) in the vanilla streaming model, a well-known open problem

    Theoretically Efficient Parallel Graph Algorithms Can Be Fast and Scalable

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    There has been significant recent interest in parallel graph processing due to the need to quickly analyze the large graphs available today. Many graph codes have been designed for distributed memory or external memory. However, today even the largest publicly-available real-world graph (the Hyperlink Web graph with over 3.5 billion vertices and 128 billion edges) can fit in the memory of a single commodity multicore server. Nevertheless, most experimental work in the literature report results on much smaller graphs, and the ones for the Hyperlink graph use distributed or external memory. Therefore, it is natural to ask whether we can efficiently solve a broad class of graph problems on this graph in memory. This paper shows that theoretically-efficient parallel graph algorithms can scale to the largest publicly-available graphs using a single machine with a terabyte of RAM, processing them in minutes. We give implementations of theoretically-efficient parallel algorithms for 20 important graph problems. We also present the optimizations and techniques that we used in our implementations, which were crucial in enabling us to process these large graphs quickly. We show that the running times of our implementations outperform existing state-of-the-art implementations on the largest real-world graphs. For many of the problems that we consider, this is the first time they have been solved on graphs at this scale. We have made the implementations developed in this work publicly-available as the Graph-Based Benchmark Suite (GBBS).Comment: This is the full version of the paper appearing in the ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), 201
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