46,729 research outputs found

    Submodular Stochastic Probing on Matroids

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    In a stochastic probing problem we are given a universe EE, where each element eEe \in E is active independently with probability pep_e, and only a probe of e can tell us whether it is active or not. On this universe we execute a process that one by one probes elements --- if a probed element is active, then we have to include it in the solution, which we gradually construct. Throughout the process we need to obey inner constraints on the set of elements taken into the solution, and outer constraints on the set of all probed elements. This abstract model was presented by Gupta and Nagarajan (IPCO '13), and provides a unified view of a number of problems. Thus far, all the results falling under this general framework pertain mainly to the case in which we are maximizing a linear objective function of the successfully probed elements. In this paper we generalize the stochastic probing problem by considering a monotone submodular objective function. We give a (11/e)/(kin+kout+1)(1 - 1/e)/(k_{in} + k_{out}+1)-approximation algorithm for the case in which we are given kink_{in} matroids as inner constraints and koutk_{out} matroids as outer constraints. Additionally, we obtain an improved 1/(kin+kout)1/(k_{in} + k_{out})-approximation algorithm for linear objective functions

    Locating a robber with multiple probes

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    We consider a game in which a cop searches for a moving robber on a connected graph using distance probes, which is a slight variation on one introduced by Seager. Carragher, Choi, Delcourt, Erickson and West showed that for any nn-vertex graph GG there is a winning strategy for the cop on the graph G1/mG^{1/m} obtained by replacing each edge of GG by a path of length mm, if mnm\geq n. The present authors showed that, for all but a few small values of nn, this bound may be improved to mn/2m\geq n/2, which is best possible. In this paper we consider the natural extension in which the cop probes a set of kk vertices, rather than a single vertex, at each turn. We consider the relationship between the value of kk required to ensure victory on the original graph and the length of subdivisions required to ensure victory with k=1k=1. We give an asymptotically best-possible linear bound in one direction, but show that in the other direction no subexponential bound holds. We also give a bound on the value of kk for which the cop has a winning strategy on any (possibly infinite) connected graph of maximum degree Δ\Delta, which is best possible up to a factor of (1o(1))(1-o(1)).Comment: 16 pages, 2 figures. Updated to show that Theorem 2 also applies to infinite graphs. Accepted for publication in Discrete Mathematic

    Fast and Powerful Hashing using Tabulation

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    Randomized algorithms are often enjoyed for their simplicity, but the hash functions employed to yield the desired probabilistic guarantees are often too complicated to be practical. Here we survey recent results on how simple hashing schemes based on tabulation provide unexpectedly strong guarantees. Simple tabulation hashing dates back to Zobrist [1970]. Keys are viewed as consisting of cc characters and we have precomputed character tables h1,...,hch_1,...,h_c mapping characters to random hash values. A key x=(x1,...,xc)x=(x_1,...,x_c) is hashed to h1[x1]h2[x2].....hc[xc]h_1[x_1] \oplus h_2[x_2].....\oplus h_c[x_c]. This schemes is very fast with character tables in cache. While simple tabulation is not even 4-independent, it does provide many of the guarantees that are normally obtained via higher independence, e.g., linear probing and Cuckoo hashing. Next we consider twisted tabulation where one input character is "twisted" in a simple way. The resulting hash function has powerful distributional properties: Chernoff-Hoeffding type tail bounds and a very small bias for min-wise hashing. This also yields an extremely fast pseudo-random number generator that is provably good for many classic randomized algorithms and data-structures. Finally, we consider double tabulation where we compose two simple tabulation functions, applying one to the output of the other, and show that this yields very high independence in the classic framework of Carter and Wegman [1977]. In fact, w.h.p., for a given set of size proportional to that of the space consumed, double tabulation gives fully-random hashing. We also mention some more elaborate tabulation schemes getting near-optimal independence for given time and space. While these tabulation schemes are all easy to implement and use, their analysis is not

    Equivalence Theorem and Probing the Electroweak Symmetry Breaking Sector

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    We examine the Lorentz non-invariance ambiguity in longitudinal weak-boson scatterings and the precise conditions for the validity of the Equivalence Theorem (ET). {\it Safe} Lorentz frames for applying the ET are defined, and the intrinsic connection between the longitudinal weak-boson scatterings and probing the symmetry breaking sector is analyzed. A universal precise formulation of the ET is presented for both the Standard Model and the Chiral Lagrangian formulated Electro-Weak Theories. It is shown that in electroweak theories with strongly interacting symmetry breaking sector, the longitudinal weak-boson scattering amplitude {\it under proper conditions} can be replaced by the corresponding Goldstone-boson scattering amplitude in which all the internal weak-boson lines and fermion loops are ignored.Comment: 20 pages, in LaTeX, to appear in Phys. Rev. D (1995). A few minor corrections were made to clarify our viewpoint of the Equivalence Theorem and compare our conclusion with those in the literatur
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