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computing the Independence Number of Intersection Graphs 11

Computing the Independence Number of Intersection Graphs

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Annual acm-Society for Industrial and applied Mathmatics symposium on Discrete Algorithms

作者： Jacob Fox Janos Pach Department of Mathematics Princeton City College CUNY and Courant Institute

ISBN: (纸本)9780898719932

computing the maximum number of disjoint elements in a collection C of geometric objects is a classical problem in computational geometry with applications ranging from frequency assignment in cellular networks to map labeling in computational cartography. The problem is equivalent to finding the independence number, α(G_C), of the intersection graph G_C of C, obtained by connecting two elements of C with an edge if and only if their intersection is nonempty. This is known to be an NP-hard task even for systems of segments in the plane with at most two different slopes. The best known polynomial time approximation algorithm for systems of arbitrary segments is due to Agarwal and Mustafa, and returns in the worst case an n~(1/2+o(1))-approximation forα. Using extensions of the Lipton-Tarjan separator theorem, we improve this result and present, for every ε> 0, a polynomial time algorithm for computing α(G_C) with approximation ratio at most n~ε. In contrast, for general graphs, for anyε > 0 it is NP-hard to approximate the independence number within a factor of n~(1-ε). We also give a subexponential time exact algorithm for computing the independence number of intersection graphs of arcwise connected sets in the plane.

关键词： Intersection Crossing Nondeterministic polynomial-time hard plane Platanus hybrida frequency assignment Computational geometry Approximation algorithms Polynomial time Cellular networks

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Faster quantum algorithm for evaluating game trees 11

Faster quantum algorithm for evaluating game trees

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Annual acm-Society for Industrial and applied Mathmatics symposium on Discrete Algorithms

作者： Ben W. Reichardt School of Computer Science and Institute for Quantum Computing University of Waterloo

ISBN: (纸本)9780898719932

We give an O(n~(1/2) log n)-query quantum algorithm for evaluating size-n AND-OR formulas. Its running time is poly-logarithmically greater after efficient preprocessing. Unlike previous approaches, the algorithm is based on a quantum walk on a graph that is not a tree. Instead, the algorithm is based on a hybrid of direct-sum span program composition, which generates tree-like graphs, and a novel tensor-product span program composition method, which generates graphs with vertices corresponding to minimal zero-certificates.

关键词： game tree algorithms quanta Runtime formula

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computing Replacement Paths in Surface Embedded Graphs 11

Computing Replacement Paths in Surface Embedded Graphs

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Annual acm-Society for Industrial and applied Mathmatics symposium on Discrete Algorithms

作者： Jeff Erickson Amir Nayyeri Department of Computer Science University of Illinois at Urbana-Champaign

ISBN: (纸本)9780898719932

Let s and t be vertices in a directed graph G with non-negative edge weights. The replacement paths problem asks us to compute, for each edge e in G, the length of the shortest path from s to t that does not traverse e. We describe an algorithm that solves the replacement paths problem for directed graphs embedded on a surface of any genus g in O(gn log n) time, generalizing a recent O(n log n)-time algorithm of Wulff-Nilsen for planar graphs [SODA 2010].

关键词： directed graphs Paths Surgical Replantation shortest path Shortest path problem embedding Byways Napier's constant algorithms

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computing Shortest Paths amid Pseudodisks 11

Computing Shortest Paths amid Pseudodisks

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Annual acm-Society for Industrial and applied Mathmatics symposium on Discrete Algorithms

作者： Danny Z. Chen Haitao Wang Department of Computer Science and Engineering University of Notre Dame

ISBN: (纸本)9780898719932

Multiple objects in the plane are called pseudodisks if they are convex and the boundaries of any two of them intersect transversely at most twice. Given a set of n (possibly intersecting) pseudodisks of O(1) complexity each and two points s and t in the plane, we develop an O(n~2) time algorithm for computing a shortest s-to-t path avoiding the pseudodisks. In over two decades, the previously best algorithms for this problem take O(n~2 log n) time, even when all pseudodisks are pairwise-disjoint disks. Our technique is also applicable to a motion planning problem of finding a shortest path to translate a convex object in the plane from one location to another avoiding a given set of polygonal obstacles, improving the previously best known solution. Our algorithm actually solves a more general version of the motion planning problem. Further, as a by-product of our approach, we present an O(n~2) time algorithm for computing the visibility graph of a set of n (possibly intersecting) pseudodisks in the plane. The previously best known time bound of this visibility problem is O(n~2 log n). In addition, for n pairwise disjoint (non-polygonal) convex objects of O(1) complexity each in the plane, we compute a shortest s-to-t path avoiding all objects in O(n log n + k) time, where k is the size of the visibility graph of the objects.

关键词： plane Platanus hybrida shortest path Planing motion planning Convexity solidification visibility logs logarithm (log) algorithms

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Fast Information Spreading in Graphs with Large Weak Conductance 11

Fast Information Spreading in Graphs with Large Weak Conduct...

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Annual acm-Society for Industrial and applied Mathmatics symposium on Discrete Algorithms

作者： Keren Censor-Hillel Hadas Shachnai Department of Computer Science Technion

ISBN: (纸本)9780898719932

Gathering data from nodes in a network is at the heart of many distributed applications, most notably, while performing a global task. We consider information spreading among n nodes of a network, where each node v has a message m(v) which must be received by all other nodes. The time required for information spreading has been previously upper-bounded with an inverse relationship to the conductance of the underlying communication graph. This implies high running times for graphs with small conductance. The main contribution of this paper is an information spreading algorithm which overcomes communication bottlenecks and thus achieves fast information spreading for a wide class of graphs, despite their small conductance. As a key tool in our study we use the recently defined concept of weak conductance, a generalization of classic graph conductance which measures how well-connected the components of a graph are. Our hybrid algorithm, which alternates between random and deterministic communication phases, exploits the connectivity within components by first applying partial information spreading, after which messages are sent across bottlenecks, thus spreading further throughout the network. This yields substantial improvements over the best known running times of algorithms for information spreading on any graph that has a large weak conductance, from polynomial to polylogarithmic number of rounds. We demonstrate the power of fast information spreading in accomplishing global tasks on the leader election problem, which lies at the core of distributed computing. Our results yield an algorithm for leader election that has a scalable running time on graphs with large weak conductance, improving significantly upon previous results.

关键词： Distributed computing Randomized algorithms Weak conductance Information spreading Leader election Distributed computing Conductance randomized algorithms algorithms

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Randomized Diffusion for Indivisible Loads 11

Randomized Diffusion for Indivisible Loads

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Annual acm-Society for Industrial and applied Mathmatics symposium on Discrete Algorithms

作者： Petra Berenbrink Colin Cooper Tom Friedetzky Tobias Friedrich Thomas Sauerwald School of Computing Science Simon Fraser University Department of Computer Science King's College London School of Engineering and Computing Sciences Durham University Max-Planck-Institut fur Informatik

ISBN: (纸本)9780898719932

We present a new randomized diffusion-based algorithm for balancing indivisible tasks (tokens) on a network. Our aim is to minimize the discrepancy between the maximum and minimum load. The algorithm works as follows. Every vertex distributes its tokens as evenly as possible among its neighbors and itself. If this is not possible without splitting some tokens, the vertex redistributes its excess tokens among all its neighbors randomly (without replacement). In this paper we prove several upper bounds on the load discrepancy for general networks. These bounds depend on some expansion properties of the network, that is, the second largest eigenvalue, and a novel measure which we refer to as refined local divergence. We then apply these general bounds to obtain results for some specific networks. For constant-degree expanders and torus graphs, these yield exponential improvements on the discrepancy bounds compared to the algorithm of Rabani, Sinclair, and Wanka [14]. For hypercubes we obtain a polynomial improvement. In contrast to previous papers, our algorithm is vertex-based and not edge-based. This means excess tokens are assigned to vertices instead to edges, and the vertex reallocates all of its excess tokens by itself. This approach avoids nodes having "negative loads" (like in [8, 10]), but causes additional dependencies for the analysis.

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New Approximation Algorithms for Minimum Enclosing Convex Shapes 11

New Approximation Algorithms for Minimum Enclosing Convex Sh...

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Annual acm-Society for Industrial and applied Mathmatics symposium on Discrete Algorithms

作者： Ankan Saha S. V. N. Vishwanathan Xinhua Zhang Department of Computer Science University of Chicago Departments of Statistics and Computer Science Purdue University Department of of Computing Science University of Alberta

ISBN: (纸本)9780898719932

Given n points in a d dimensional Euclidean space, the Minimum Enclosing Ball (MEB) problem is to find the ball with the smallest radius which contains all n points. We give two approximation algorithms for producing an enclosing ball whose radius is at most ε away from the optimum. The first requires O(ndL/ε~(1/2)) effort, where L is a constant that depends on the scaling of the data. The second is a O~*(ndQ/ε~(1/2)) approximation algorithm, where Q is an upper bound on the norm of the points. This is in contrast with coresets based algorithms which yield a O(nd/ε) greedy algorithm. Finding the Minimum Enclosing Convex Polytope (MECP) is a related problem wherein a convex polytope of a fixed shape is given and the aim is to find the smallest magnification of the polytope which encloses the given points. For this problem we present O(mndL/ε) and O~*(mndQ/ε) approximation algorithms, where m is the number of faces of the polytope. Our algorithms borrow heavily from convex duality and recently developed techniques in non-smooth optimization, and are in contrast with existing methods which rely on geometric arguments. In particular, we specialize the excessive gap framework of Nesterov [19] to obtain our results.

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Distributed Selfish Load Balancing on Networks 11

Distributed Selfish Load Balancing on Networks

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Annual acm-Society for Industrial and applied Mathmatics symposium on Discrete Algorithms

作者： Petra Berenbrink Martin Hoefer Thomas Sauerwald School of Computing Science Simon Fraser University RWTH Aachen University Max-Planck-Institut fur Informatik

ISBN: (纸本)9780898719932

We study distributed load balancing in networks with selfish agents. In the simplest model considered here, there are n identical machines represented by vertices in a network and >>n selfish agents that unilaterally decide to move from one vetex to another if this improves their experienced load. We present several protocols for concurrent migration that satisfy desirable properties such as being based only on local information and computation and the absence of global coordination or cooperation of agents. Our main contribution is to show rapid convergence of the resulting migration process to states that satisfy different stability or balance criteria. In particular, the convergence time to a Nash equilibrium is only logarithmic in m and polynomial in n, where the polynomial depends on the graph structure. Using a slight modification with neutral moves, a perfectly balanced state can be reached after additional time polynomial in n. In addition, we show reduced convergence times to approximate Nash equilibria. Finally, we extend our results to networks of machines with different speeds or to agents that have different weights and show similar results for convergence to approximate and exact Nash equilibria.

关键词： Nash equilibrium Polynomial Network Load balancing Converge local information. Approximate machine Agents

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Reflections for quantum query algorithms 11

Reflections for quantum query algorithms

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Annual acm-Society for Industrial and applied Mathmatics symposium on Discrete Algorithms

作者： Ben W. Reichardt School of Computer Science and Institute for Quantum Computing University of Waterloo

ISBN: (纸本)9780898719932

We show that any boolean function can be evaluated optimally by a quantum query algorithm that alternates a certain fixed, input-independent reflection with a second reflection that coherently queries the input string. Originally introduced for solving the unstructured search problem, this two-reflections structure is therefore a universal feature of quantum algorithms. Our proof goes via the general adversary bound, a semi-definite program (SDP) that lower-bounds the quantum query complexity of a function. By a quantum algorithm for evaluating span programs, this lower bound is known to be tight up to a sub-logarithmic factor. The extra factor comes from converting a continuous-time query algorithm into a discrete-query algorithm. We give a direct and simplified quantum algorithm based on the dual SDP, with a boundederror query complexity that matches the general adversary bound. Therefore, the general adversary lower bound is tight; it is in fact an SDP for quantum query complexity. This implies that the quantum query complexity of the composition f o (g,..., g) of two boolean functions f and g matches the product of the query complexities of f and g, without a logarithmic factor for error reduction. It efficiently characterizes the quantum query complexity of a read-once formula over any finite gate set. It further shows that span programs are equivalent to quantum query algorithms.

关键词： quanta Boolean functions algorithms Reflection Service Discovery Protocol input string

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A fast approximation strategy for summarizing a set of streaming time series 10

A fast approximation strategy for summarizing a set of strea...

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25th Annual acm symposium on applied computing, SAC 2010

作者： Marascu, Alice Masseglia, Florent Lechevallier, Yves Inria Sophia Antipolis 2004 route des Lucioles 06902 Sophia Antipolis France Inria Paris - Rocquencourt 11 domaine de Voluceau 78153 Le Chesnay France

ISBN: (纸本)9781605586380

Summarizing a set of streaming time series is an important issue that reliably allows information to be monitored and stored in domains such as finance [12], networks [2, 1], etc. To date, most of existing algorithms have focused on this problem by summarizing the time series separately [12, 4]. Moreover, the same amount of memory has been allocated to each time series. Yet, memory management is an important subject in the data stream field, but a framework allocating equal amount of memory to each sequence is not appropriate. We introduce an effective and efficient method which succeeds to respond to both challenges: (1) a memory optimized framework along with (2) a fast novel sequence merging method. Experiments with real data show that this method is effective and efficient. © 2010 acm.

关键词： Time series

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