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检索条件"主题词=randomized algorithms"

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Approximating the k-Level in Three-Dimensional Plane Arrangements 16

Approximating the k-Level in Three-Dimensional Plane Arrange...

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Annual ACM-Society for Industrial and Applied Mathmatics Symposium on Discrete algorithms

作者： Sariel Har-Peled Haim Kaplan Micha Sharir Department of Computer Science University of Illinois School of Computer Science Tel Aviv University

ISBN: (纸本)9781510819672

Let H be a set of n non-vertical planes in three dimensions, and let r < n be a parameter. We give a simple alternative proof of the existence of a O(1/r)-cutting of the first n/r levels of A(H), which consists of O(r) semi-unbounded vertical triangular prisms. The same construction yields an approximation of the (n/r)-level by a terrain consisting of O(r/ε~3) triangular faces, which lies entirely between the levels (1 ± ε)n/r. The proof does not use sampling, and exploits techniques based on planar separators and various structural properties of levels in three-dimensional arrangements and of planar maps. The proof is constructive, and leads to a simple randomized algorithm, that computes the terrain in O(n + r~2ε~(-6) log~3 r) expected time. An application of this technique allows us to mimic Matousek's construction of cuttings in the plane [36], to obtain a similar construction of "layered" (1/r)-cutting of the entire arrangement A(H), of optimal size O(r~3). Another application is a simplified optimal approximate range counting algorithm in three dimensions, competing with that of Afshani and Chan [1].

关键词： terrains randomized algorithms plane landscapes Planar Proof Three Dimensions constructivity Platanus hybrida Planing Structural properties Cuttings Cutting approximation

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Fast Algorithm for Finding True Number of Clusters. Applications to Control Systems

Fast Algorithm for Finding True Number of Clusters. Applicat...

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第24届中国控制与决策会议

作者： Mikhail Morozkov Oleg Granichin Zeev Volkovich Department of Mathematics and Mechanics Saint Petersburg State University

One of the most difficult problems in cluster analysis is the identification of the number of groups in a given data set. In this paper we offer the randomized approach in the rate distortion framework. A randomized algorithm has been suggested to allocate this position. The scenario approach is used to significantly reduce the computational complexity. With ability to determine the true number of clusters and perform clustering in real-time operational mode we outline several applications in control systems and decision-making problems that can benefit from algorithm in question essentially. We also provide simulation results to show considerable speed optimization with guaranteed level of probability.

关键词： Clustering randomized algorithms Adaptive Control Optimitzaion

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A subexponential lower bound for the Random Facet algorithm for Parity Games 11

A subexponential lower bound for the Random Facet algorithm ...

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Annual ACM-Society for Industrial and Applied Mathmatics Symposium on Discrete algorithms

作者： Oliver Friedmann Thomas Dueholm Hansen Uri Zwick Department of Computer Science University of Munich Department of Computer Science Aarhus University

ISBN: (纸本)9780898719932

Parity Games form an intriguing family of infinite duration games whose solution is equivalent to the solution of important problems in automatic verification and automata theory. They also form a very natural subclass of Deterministic Mean Payoff Games, which in turn is a very natural subclass of turn-based Stochastic Mean Payoff Games. It is a major open problem whether these game families can be solved in polynomial time. The currently theoretically fastest algorithms for the solution of all these games are adaptations of the randomized algorithms of Kalai and of Matousek, Sharir and Welzl for LP-type problems, an abstract generalization of linear programming. The expected running time of both algorithms is subexponential in the size of the game, i.e., 2~(O((n log n)~(1/2))), where n is the number of vertices in the game. We focus in this paper on the algorithm of Matousek, Sharir and Welzl and refer to it as the Random Facet algorithm. Matousek constructed a family of abstract optimization problems such that the expected running time of the Random Facet algorithm, when run on a random instance from this family, is close to the subexponential upper bound given above. This shows that in the abstract setting, the 2~(O((n log n)~(1/2))) upper bound on the complexity of the Random Facet algorithm is essentially tight. It is not known, however, whether the abstract optimization problems constructed by Matousek correspond to games of any of the families mentioned above. There was some hope, therefore, that the Random Facet algorithm, when applied to, say, parity games, may run in polynomial time. We show, that this, unfortunately, is not the case by constructing explicit parity games on which the expected running time of the Random Facet algorithm is close to the subexponential upper bound. The games we use mimic the behavior of a randomized counter. They are also the first explicit LP-type problems on which the Random Facet algorithm is not polynomial.

关键词： Game theory Runtime Articular Bone Surface algorithms family Game animals Upper bound Random Polynomial time AUTOMATA THEORY randomized algorithms

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Beating Brute Force for Systems of Polynomial Equations over Finite Fields 17

Beating Brute Force for Systems of Polynomial Equations over...

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Annual ACM-Society for Industrial and Applied Mathmatics Symposium on Discrete algorithms

作者： Daniel Lokshtanov Ramamohan Paturi Suguru Tamaki Ryan Williams Huacheng Yu University of Bergen University of California San Diego Kyoto University Stanford University

ISBN: (纸本)9781510836358

We consider the problem of solving systems of multivariate polynomial equations of degree k over a finite field. For every integer k ≥ 2 and finite field F_q where q = p~d for a prime p, we give, to the best of our knowledge, the first algorithms that achieve an exponential speedup over the brute force O(q~n) time algorithm in the worst case. We present two algorithms, a randomized algorithm with running time q~(n+o(n)) · q~(-n/O(k)) time if q ≤ 2~(4ekd), and q~(n+o(n))·(log q/dek)~(-dn) otherwise, where e = 2.718... is Napier's constant, and a deterministic algorithm for counting solutions with running time q~(n+o(n))·q~(-n/O(kq~(6/7d))). For the important special case of quadratic equations in F_2, our randomized algorithm has running time O(2~(0.8765n)). For systems over F_2 we also consider the case where the input polynomials do not have bounded degree, but instead can be efficiently represented as a ΣΠΣ circuit, i.e., a sum of products of sums of variables. For this case we present a deterministic algorithm running in time 2~(n-δn) for δ = 1/O(log(s/n)) for instances with s product gates in total and n variables. Our algorithms adapt several techniques recently developed via the polynomial method from circuit complexity. The algorithm for systems of ΣΠΣ polynomials also introduces a new degree reduction method that takes an instance of the problem and outputs a subexponential-sized set of instances, in such a way that feasibility is preserved and every polynomial among the output instances has degree O(log(s/n)).

关键词： Galois fields deterministic algorithms randomized algorithms Polynomial Runtime algorithms circuit complexity Beating

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Line Transversals of Convex Polyhedra in R~3 09

Line Transversals of Convex Polyhedra in R~3

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Annual ACM-Society for Industrial and Applied Mathmatics Symposium on Discrete algorithms

作者： Haim Kaplan Natan Rubin Micha Sharir School of Computer Science Tel Aviv University

We establish a bound of O(n~2k~(1+ε)), for any ε > 0, on the combinatorial complexity of the set T of line transversals of a collection P of k convex polyhedra in R~3 with a total of n facets, and present a randomized algorithm which computes the boundary of T in comparable expected time. Thus, when k＜＜ n, the new bounds on the complexity (and construction cost) of T improve upon the previously best known bounds, which are nearly cubic in n. To obtain the above result, we study the set T_(l_0) of line transversals which emanate from a fixed line l_0, establish an almost tight bound of O(nk~(1+ε)) on the complexity of T_(l_0), and provide a randomized algorithm which computes T_(l_0) in comparable expected time. Slightly improved combinatorial bounds for the complexity of T_(l_0), and comparable improvements in the cost of constructing this set, are established for two special cases, both assuming that the polyhedra of P are pairwise disjoint: the case where l_0 is disjoint from the polyhedra of P, and the case where the polyhedra of P are unbounded in a direction parallel to l_0.

关键词： polyhedrons randomized algorithms Convex polyhedron Bound Phosphorus Complexity Construction costs Convexity Line mutual exclusion

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An Improved Distributed Algorithm for Maximal Independent Set 16

An Improved Distributed Algorithm for Maximal Independent Se...

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Annual ACM-Society for Industrial and Applied Mathmatics Symposium on Discrete algorithms

作者： Mohsen Ghaffari MIT

ISBN: (纸本)9781510819672

The Maximal Independent Set (MIS) problem is one of the basics in the study of locality in distributed graph algorithms. This paper presents a very simple randomized algorithm for this problem providing a near-optimal local complexity, which incidentally, when combined with some known techniques, also leads to a near-optimal global complexity. Classical MIS algorithms of Luby [STOC'85] and Alon, Babai and Itai [JALG'86] provide the global complexity guarantee that, with high probability, all nodes terminate after O(log n) rounds. In contrast, our initial focus is on the local complexity, and our main contribution is to provide a very simple algorithm guaranteeing that each particular node v terminates after O(log deg(v) + log 1/ε) rounds, with probability at least 1 - ε. The degree-dependency in this bound is optimal, due to a lower bound of Kuhn, Moscibroda, and Wattenhofer [PODC'04]. Interestingly, this local complexity smoothly transitions to a global complexity: by adding techniques of Barenboim, Elkin, Pettie, and Schneider [FOCS'12; arXiv: 1202.1983v3], we get an MIS algorithm with a high probability global complexity of O(logΔ) + 2~(O({the square root of}(log log n))), where Δ denotes the maximum degree. This improves over the O(log~2 Δ) + 2_(O({the square root of}(log log n))) result of Barenboim et al., and gets close to the Ω(min{log Δ, {the square root of}(log n)}) lower bound of Kuhn et al. Corollaries include improved algorithms for MIS in graphs of upper-bounded arboricity, or lower-bounded girth, for Ruling Sets, for MIS in the Local Computation algorithms (LCA) model, and a faster distributed algorithm for the Lovasz Local Lemma.

关键词： Management Information Systems distributed algorithms randomized algorithms Graph algorithms Square root algorithms Probability Complexity

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A Cubic Algorithm for Computing Gaussian Volume 14

A Cubic Algorithm for Computing Gaussian Volume

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Annual ACM-Society for Industrial and Applied Mathmatics Symposium on Discrete algorithms

作者： Ben Cousins Santosh Vempala Georgia Tech.

ISBN: (纸本)9781611973389

We present randomized algorithms for sampling the standard Gaussian distribution restricted to a convex set and for estimating the Gaussian measure of a convex set, in the general membership oracle model. The complexity of the integration algorithm is O*(n~3) while the complexity of the sampling algorithm is O*(n~3) for the first sample and O*(n~2) for every subsequent sample. These bounds improve on the corresponding state-of-the-art by a factor of n. Our improvement comes from several aspects: better isoperimetry, smoother annealing, avoiding trans-formation to isotropic position and the use of the "speedy walk" in the analysis.

关键词： integral algorithm randomized algorithms Convexity Gaussian distribution Gauss Cubic Corresponding states algorithms Information retrieval systems-Teletext and videotex

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Multi-agent Systems Reaching Optimal Consensus Based on Simple Bernoulli Decisions

Multi-agent Systems Reaching Optimal Consensus Based on Simp...

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Chinese Control Conference

作者： Guodong Shi Karl Henrik Johansson ACCESS Linnaeus Centre School of Electrical Engineering Royal Institute of Technology

ISBN: (纸本)9781467325813

In this paper, we formulate and solve a randomized optimal consensus problem for multi-agent systems with stochastically time-varying interconnection topology. The considered multi-agent system with a simple randomized iterating rule achieves an almost sure consensus meanwhile solving the optimization problem min_(z∈R~d) Σ_(i=1)~n f_i(z), in which the optimal solution set of objective function f_i corresponding to agent i can only be observed by agent i itself. At each time step, each agent independently and randomly chooses either taking an average among its neighbor set, or projecting onto the optimal solution set of its own optimization component. Both directed and bidirectional communication graphs are studied. Connectivity conditions are proposed to guarantee an optimal consensus almost surely with proper convexity and intersection assumptions. The convergence analysis is carried out using convex analysis. The results illustrate that a group of autonomous agents can reach an optimal opinion with probability one by each node simply making a randomized trade-off between following its neighbors or sticking to its own opinion at each time step.

关键词： Multi-agent systems Optimal consensus Set convergence Distributed optimization randomized algorithms Multi-agent systems randomized algorithms optimal solution 3-chloro-4,4-dimethyl-2-oxazolidinone Intelligent agents Bernoulli

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On the Sample Complexity of Uncertain Linear and Bilinear Matrix Inequalities

On the Sample Complexity of Uncertain Linear and Bilinear Ma...

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IEEE Annual Conference on Decision and Control

作者： Mohammadreza Chamanbaz Fabrizio Dabbene Roberto Tempo Venkatakrishnan Venkataramanan Qing-Guo Wang Data Storage Institute Singapore IEIIT-CNR Torino Italy Department of Electrical and Computer Engineering National University of Singapore

ISBN: (纸本)9781467357159

In this paper, we consider uncertain linear and bilinear matrix inequalities which depend in a possibly nonlinear way on a vector of uncertain parameters. Motivated by recent results in statistical learning, we show that probabilistic guaranteed solutions can be obtained by means of randomized algorithms. In particular, we show that the Vapnik-Chevonenkis dimension (VC-dimension) of the two problems is finite, and we compute upper bounds on it. In turn, these bounds allow us to derive explicitly the sample complexity of the problems. Using these bounds, in the second part of the paper, we derive a sequential scheme, based on a sequence of optimization and validation steps. The algorithm is on the same lines of recent schemes proposed for similar problems, but improves both in terms of complexity and generality.

关键词： sample complexity Plant uncertain bilinearity Statistical learning randomized algorithms inequalities Upper bound Uncertainty Finite

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Hardness of Continuous Local Search: Query Complexity and Cryptographic Lower Bounds 17

Hardness of Continuous Local Search: Query Complexity and Cr...

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Annual ACM-Society for Industrial and Applied Mathmatics Symposium on Discrete algorithms

作者： Pavel Hubacek Eylon Yogev Weizmann Institute of Science

ISBN: (纸本)9781510836358

Local search proved to be an extremely useful tool when facing hard optimization problems (e.g., via the simplex algorithm, simulated annealing, or genetic algorithms). Although powerful, it has its limitations: there are functions for which exponentially many queries are needed to find a local optimum. In many contexts the optimization problem is defined by a continuous function, which might offer an advantage when performing the local search. This leads us to study the following natural question: How hard is continuous local search? The computational complexity of such search problems is captured by the complexity class CLS (Daskalakis and Papadimitriou SODA'11) which is contained in the intersection of PLS and PPAD, two important subclasses of TFNP (the class of NP search problems with a guaranteed solution). In this work, we show the first hardness results for CLS (the smallest non-trivial class among the currently defined subclasses of TFNP). Our hardness results are in terms of black-box (where only oracle access to the function is given) and white-box (where the function is represented succinctly by a circuit). In the black-box case, we show instances for which any (computationally unbounded) randomized algorithm must perform exponentially many queries in order to find a local optimum. In the white-box case, we show hardness for computationally bounded algorithms under cryptographic assumptions. Our results demonstrate a strong conceptual barrier precluding design of efficient algorithms for solving local search problems even over continuous domains. As our main technical contribution we introduce a new total search problem which we call END-OF-METERED-LINE. The special structure of END-OF-METERED-LINE enables us to: (1) show that it is contained in CLS, and (2) prove hardness for it both in the black-box and the white-box setting.

关键词： tabu search Soundness hardness local optimum Function efficient algorithm randomized algorithms BLACK-BOX Continuous Function Subclasses complexity classes

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