170 research outputs found

    Faster Clustering via Preprocessing

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    We examine the efficiency of clustering a set of points, when the encompassing metric space may be preprocessed in advance. In computational problems of this genre, there is a first stage of preprocessing, whose input is a collection of points MM; the next stage receives as input a query set QβŠ‚MQ\subset M, and should report a clustering of QQ according to some objective, such as 1-median, in which case the answer is a point a∈Ma\in M minimizing βˆ‘q∈QdM(a,q)\sum_{q\in Q} d_M(a,q). We design fast algorithms that approximately solve such problems under standard clustering objectives like pp-center and pp-median, when the metric MM has low doubling dimension. By leveraging the preprocessing stage, our algorithms achieve query time that is near-linear in the query size n=∣Q∣n=|Q|, and is (almost) independent of the total number of points m=∣M∣m=|M|.Comment: 24 page

    Sketching Cuts in Graphs and Hypergraphs

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    Sketching and streaming algorithms are in the forefront of current research directions for cut problems in graphs. In the streaming model, we show that (1βˆ’Ο΅)(1-\epsilon)-approximation for Max-Cut must use n1βˆ’O(Ο΅)n^{1-O(\epsilon)} space; moreover, beating 4/54/5-approximation requires polynomial space. For the sketching model, we show that rr-uniform hypergraphs admit a (1+Ο΅)(1+\epsilon)-cut-sparsifier (i.e., a weighted subhypergraph that approximately preserves all the cuts) with O(Ο΅βˆ’2n(r+log⁑n))O(\epsilon^{-2} n (r+\log n)) edges. We also make first steps towards sketching general CSPs (Constraint Satisfaction Problems)

    Fault-Tolerant Spanners: Better and Simpler

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    A natural requirement of many distributed structures is fault-tolerance: after some failures, whatever remains from the structure should still be effective for whatever remains from the network. In this paper we examine spanners of general graphs that are tolerant to vertex failures, and significantly improve their dependence on the number of faults rr, for all stretch bounds. For stretch kβ‰₯3k \geq 3 we design a simple transformation that converts every kk-spanner construction with at most f(n)f(n) edges into an rr-fault-tolerant kk-spanner construction with at most O(r3log⁑n)β‹…f(2n/r)O(r^3 \log n) \cdot f(2n/r) edges. Applying this to standard greedy spanner constructions gives rr-fault tolerant kk-spanners with O~(r2n1+2k+1)\tilde O(r^{2} n^{1+\frac{2}{k+1}}) edges. The previous construction by Chechik, Langberg, Peleg, and Roddity [STOC 2009] depends similarly on nn but exponentially on rr (approximately like krk^r). For the case k=2k=2 and unit-length edges, an O(rlog⁑n)O(r \log n)-approximation algorithm is known from recent work of Dinitz and Krauthgamer [arXiv 2010], where several spanner results are obtained using a common approach of rounding a natural flow-based linear programming relaxation. Here we use a different (stronger) LP relaxation and improve the approximation ratio to O(log⁑n)O(\log n), which is, notably, independent of the number of faults rr. We further strengthen this bound in terms of the maximum degree by using the \Lovasz Local Lemma. Finally, we show that most of our constructions are inherently local by designing equivalent distributed algorithms in the LOCAL model of distributed computation.Comment: 17 page

    Almost-Smooth Histograms and Sliding-Window Graph Algorithms

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    We study algorithms for the sliding-window model, an important variant of the data-stream model, in which the goal is to compute some function of a fixed-length suffix of the stream. We extend the smooth-histogram framework of Braverman and Ostrovsky (FOCS 2007) to almost-smooth functions, which includes all subadditive functions. Specifically, we show that if a subadditive function can be (1+Ο΅)(1+\epsilon)-approximated in the insertion-only streaming model, then it can be (2+Ο΅)(2+\epsilon)-approximated also in the sliding-window model with space complexity larger by factor O(Ο΅βˆ’1log⁑w)O(\epsilon^{-1}\log w), where ww is the window size. We demonstrate how our framework yields new approximation algorithms with relatively little effort for a variety of problems that do not admit the smooth-histogram technique. For example, in the frequency-vector model, a symmetric norm is subadditive and thus we obtain a sliding-window (2+Ο΅)(2+\epsilon)-approximation algorithm for it. Another example is for streaming matrices, where we derive a new sliding-window (2+Ο΅)(\sqrt{2}+\epsilon)-approximation algorithm for Schatten 44-norm. We then consider graph streams and show that many graph problems are subadditive, including maximum submodular matching, minimum vertex-cover, and maximum kk-cover, thereby deriving sliding-window O(1)O(1)-approximation algorithms for them almost for free (using known insertion-only algorithms). Finally, we design for every d∈(1,2]d\in (1,2] an artificial function, based on the maximum-matching size, whose almost-smoothness parameter is exactly dd

    Tight Bounds for Gomory-Hu-like Cut Counting

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    By a classical result of Gomory and Hu (1961), in every edge-weighted graph G=(V,E,w)G=(V,E,w), the minimum stst-cut values, when ranging over all s,t∈Vs,t\in V, take at most ∣Vβˆ£βˆ’1|V|-1 distinct values. That is, these (∣V∣2)\binom{|V|}{2} instances exhibit redundancy factor Ξ©(∣V∣)\Omega(|V|). They further showed how to construct from GG a tree (V,Eβ€²,wβ€²)(V,E',w') that stores all minimum stst-cut values. Motivated by this result, we obtain tight bounds for the redundancy factor of several generalizations of the minimum stst-cut problem. 1. Group-Cut: Consider the minimum (A,B)(A,B)-cut, ranging over all subsets A,BβŠ†VA,B\subseteq V of given sizes ∣A∣=Ξ±|A|=\alpha and ∣B∣=Ξ²|B|=\beta. The redundancy factor is Ωα,Ξ²(∣V∣)\Omega_{\alpha,\beta}(|V|). 2. Multiway-Cut: Consider the minimum cut separating every two vertices of SβŠ†VS\subseteq V, ranging over all subsets of a given size ∣S∣=k|S|=k. The redundancy factor is Ξ©k(∣V∣)\Omega_{k}(|V|). 3. Multicut: Consider the minimum cut separating every demand-pair in DβŠ†VΓ—VD\subseteq V\times V, ranging over collections of ∣D∣=k|D|=k demand pairs. The redundancy factor is Ξ©k(∣V∣k)\Omega_{k}(|V|^k). This result is a bit surprising, as the redundancy factor is much larger than in the first two problems. A natural application of these bounds is to construct small data structures that stores all relevant cut values, like the Gomory-Hu tree. We initiate this direction by giving some upper and lower bounds.Comment: This version contains additional references to previous work (which have some overlap with our results), see Bibliographic Update 1.
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