43,521 research outputs found

    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

    Transfer matrix for spanning trees, webs and colored forests

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    We use the transfer matrix formalism for dimers proposed by Lieb, and generalize it to address the corresponding problem for arrow configurations (or trees) associated to dimer configurations through Temperley's correspondence. On a cylinder, the arrow configurations can be partitioned into sectors according to the number of non-contractible loops they contain. We show how Lieb's transfer matrix can be adapted in order to disentangle the various sectors and to compute the corresponding partition functions. In order to address the issue of Jordan cells, we introduce a new, extended transfer matrix, which not only keeps track of the positions of the dimers, but also propagates colors along the branches of the associated trees. We argue that this new matrix contains Jordan cells.Comment: 29 pages, 7 figure

    Parameterized Complexity of Edge Interdiction Problems

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    We study the parameterized complexity of interdiction problems in graphs. For an optimization problem on graphs, one can formulate an interdiction problem as a game consisting of two players, namely, an interdictor and an evader, who compete on an objective with opposing interests. In edge interdiction problems, every edge of the input graph has an interdiction cost associated with it and the interdictor interdicts the graph by modifying the edges in the graph, and the number of such modifications is constrained by the interdictor's budget. The evader then solves the given optimization problem on the modified graph. The action of the interdictor must impede the evader as much as possible. We focus on edge interdiction problems related to minimum spanning tree, maximum matching and shortest paths. These problems arise in different real world scenarios. We derive several fixed-parameter tractability and W[1]-hardness results for these interdiction problems with respect to various parameters. Next, we show close relation between interdiction problems and partial cover problems on bipartite graphs where the goal is not to cover all elements but to minimize/maximize the number of covered elements with specific number of sets. Hereby, we investigate the parameterized complexity of several partial cover problems on bipartite graphs

    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} )

    Changing Bases: Multistage Optimization for Matroids and Matchings

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    This paper is motivated by the fact that many systems need to be maintained continually while the underlying costs change over time. The challenge is to continually maintain near-optimal solutions to the underlying optimization problems, without creating too much churn in the solution itself. We model this as a multistage combinatorial optimization problem where the input is a sequence of cost functions (one for each time step); while we can change the solution from step to step, we incur an additional cost for every such change. We study the multistage matroid maintenance problem, where we need to maintain a base of a matroid in each time step under the changing cost functions and acquisition costs for adding new elements. The online version of this problem generalizes online paging. E.g., given a graph, we need to maintain a spanning tree TtT_t at each step: we pay ct(Tt)c_t(T_t) for the cost of the tree at time tt, and also TtTt1| T_t\setminus T_{t-1} | for the number of edges changed at this step. Our main result is an O(logmlogr)O(\log m \log r)-approximation, where mm is the number of elements/edges and rr is the rank of the matroid. We also give an O(logm)O(\log m) approximation for the offline version of the problem. These bounds hold when the acquisition costs are non-uniform, in which caseboth these results are the best possible unless P=NP. We also study the perfect matching version of the problem, where we must maintain a perfect matching at each step under changing cost functions and costs for adding new elements. Surprisingly, the hardness drastically increases: for any constant ϵ>0\epsilon>0, there is no O(n1ϵ)O(n^{1-\epsilon})-approximation to the multistage matching maintenance problem, even in the offline case

    Invariant Peano curves of expanding Thurston maps

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    We consider Thurston maps, i.e., branched covering maps f ⁣:S2S2f\colon S^2\to S^2 that are postcritically finite. In addition, we assume that ff is expanding in a suitable sense. It is shown that each sufficiently high iterate F=fnF=f^n of ff is semi-conjugate to zd ⁣:S1S1z^d\colon S^1\to S^1, where dd is equal to the degree of FF. More precisely, for such an FF we construct a Peano curve γ ⁣:S1S2\gamma\colon S^1\to S^2 (onto), such that Fγ(z)=γ(zd)F\circ \gamma(z) = \gamma(z^d) (for all zS1z\in S^1).Comment: 63 pages, 12 figure
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