The string matching with mismatches problem requires finding the Hamming distance between a pattern P of length m and every length m substring of text T with length n. Fischer and Paterson's FFT-based algorithm so...
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The string matching with mismatches problem requires finding the Hamming distance between a pattern P of length m and every length m substring of text T with length n. Fischer and Paterson's FFT-based algorithm solves the problem without error in O (or n log m), where a is the size of the alphabet Sigma [SIAM-AMS Proc. 7 (1973) 113-125]. However, this in the worst case reduces to O (nm log m). Atallah, Chyzak and Dumas used the idea of randomly mapping the letters of the alphabet to complex roots of unity to estimate the score vector in time O (n log m) [Algorithmica 29 (2001) 468-486]. We show that the algorithm's score variance can be substantially lowered by using a bijective mapping, and specifically to zero in the case of binary and ternary alphabets. This result is extended via alphabet remappings to deterministically solve the string matching with mismatches problem with a constant factor of 2 improvement over Fischer-Paterson's method. (c) 2005 Elsevier Inc. All rights reserved.
uncertainty. In this setting, we study the problem of minimizing the expected value with respect to the uncertainty of the LS residual. For general nonlinear dependence of the data on the uncertain parameters, determi...
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uncertainty. In this setting, we study the problem of minimizing the expected value with respect to the uncertainty of the LS residual. For general nonlinear dependence of the data on the uncertain parameters, determining an exact solution to this problem is known to be computationally prohibitive. Here, we follow a probabilistic approach, and determine a probable near optimal solution by minimizing the empirical mean of the residual. Finite sample convergence of the proposed method is assessed using statistical learning methods. In particular, we prove that if one constructs the empirical approximation of the mean using a finite number N of samples, then the minimizer of this empirical approximation is, with high probability, an epsilon-suboptimal solution for the original problem. Moreover, this approximate solution can be efficiently determined numerically by a standard recursive algorithm. Comparisons with gradient algorithms for stochastic optimization are also discussed in the paper and several numerical examples illustrate the proposed methodology. (c) 2005 Elsevier B.V. All rights reserved.
The traditional zero-one principle for sorting networks states that "if a network with n input lines sorts all 2(n) binary sequences into nondecreasing order, then it will sort any arbitrary sequence of n numbers...
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The traditional zero-one principle for sorting networks states that "if a network with n input lines sorts all 2(n) binary sequences into nondecreasing order, then it will sort any arbitrary sequence of n numbers into nondecreasing order". We generalize this to the situation when a network sorts almost all binary sequences and relate it to the behavior of the sorting network on arbitrary inputs. We also present an application to mesh sorting. (c) 2004 Elsevier B.V. All rights reserved.
We propose and analyze a simple new randomized algorithm, called ResolveSat, for finding satisfying assignments of Boolean formulas in conjunctive normal form. The algorithm consists of two stages: a preprocessing sta...
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We propose and analyze a simple new randomized algorithm, called ResolveSat, for finding satisfying assignments of Boolean formulas in conjunctive normal form. The algorithm consists of two stages: a preprocessing stage in which resolution is applied to enlarge the set of clauses of the formula, followed by a search stage that uses a simple randomized greedy procedure to look for a satisfying assignment. Currently, this is the fastest known probabilistic algorithm for k-CNF satisfiability for k >= 4 (with a running time of O (2(0.5625n)) for 4-CNF). In addition, it is the fastest known probabilistic algorithm for k-CNF, k >= 3, that have at most one satisfying assignment (unique k-SAT) (with a running time O(2((2 ln 2-1)n+o(n))) = O(2(0.386...n)) in the case of 3-CNF). The analysis of the algorithm also gives an upper bound on the number of the codewords of a code defined by a k-CNF. This is applied to prove a lower bounds on depth 3 circuits accepting codes with nonconstant distance. In particular we prove a lower bound Omega(2(1.282...root n)) for an explicitly given Boolean function of n variables. This is the first such lower bound that is asymptotically bigger than 2(root n+o(root n)).
We study the problem of sampling uniformly at random from the set of k-colorings of a graph with maximum degree.. We focus attention on the Markov chain Monte Carlo method, particularly on a popular Markov chain for t...
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We study the problem of sampling uniformly at random from the set of k-colorings of a graph with maximum degree.. We focus attention on the Markov chain Monte Carlo method, particularly on a popular Markov chain for this problem, the Wang-Swendsen- Kotecky (WSK) algorithm. The second author recently proved that the WSK algorithm quickly converges to the desired distribution when k >= 11 triangle/6. We study how far these positive results can be extended in general. In this note we prove the first non-trivial results on when the WSK algorithm takes exponentially long to reach the stationary distribution and is thus called torpidly mixing. In particular, we show that the WSK algorithm is torpidly mixing on a family of bipartite graphs when 3 <= k < triangle/( 20 log.), and on a family of planar graphs for any number of colors. We also give a family of graphs for which, despite their small chromatic number, the WSK algorithm is not ergodic when k <= triangle/2, provided k is larger than some absolute constant k(0). (C) 2004 Elsevier B.V. All rights reserved.
We investigate the distribution of the depth of a node containing a specific key or, equivalently, the number of steps needed to retrieve an item stored in a randomly grown binary search tree. Using a representation i...
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We investigate the distribution of the depth of a node containing a specific key or, equivalently, the number of steps needed to retrieve an item stored in a randomly grown binary search tree. Using a representation in terms of mixed and compounded standard distributions, we derive approximations by Poisson and mixed Poisson distributions;these lead to asymptotic normality results. We are particularly interested in the influence of the key value on the distribution of the node depth. Methodologically our message is that the explicit representation may provide additional insight if compared to the standard approach that is based on the recursive structure of the trees. Further, in order to exhibit the influence of the key on the distributional asymptotics, a suitable choice of distance of probability distributions is important. Our results are also applicable in connection with the number of recursions needed in Hoare's [Comm. ACM 4 (1961) 321-322] selection algorithm FIND.
In this paper a few "difficult" problems related to simultaneous stabilization of three plants (equivalent to a certain problem related to unit interpolation in H.) have been addressed through the framework ...
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In this paper a few "difficult" problems related to simultaneous stabilization of three plants (equivalent to a certain problem related to unit interpolation in H.) have been addressed through the framework of randomized algorithms. These problems which were proposed by Blondel (Simultaneous Stabilization of Linear Systems, Springer, Berlin, 1994) and Blondel and Gevers (Math. Control Signals Systems 6 (1994) 135) concern the existence of a controller. (C) 2002 Elsevier Science Ltd. All rights reserved.
We present an optimal parallel randomized algorithm for the Voronoi diagram of a set of n nonintersecting (except possibly at endpoints) line segments in the plane. Our algorithm runs in O(log n) time with high probab...
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We present an optimal parallel randomized algorithm for the Voronoi diagram of a set of n nonintersecting (except possibly at endpoints) line segments in the plane. Our algorithm runs in O(log n) time with high probability using O (n) processors on a CRCW PRAM. This algorithm is optimal in terms of work done since the sequential time bound for this problem is O(n log n). Our algorithm improves by an O(log n) factor the previously best known deterministic parallel algorithm, given by Goodrich, O'Dunlaing, and Yap, which runs in O(log(2) n) time using O(n) processors. We obtain this result by using a new "two-stage" random sampling technique. By choosing large samples in the first stage of the algorithm, we avoid the hurdle of problem-size "blow-up" that is typical in recursive parallel geometric algorithms. We combine the two-stage sampling technique with efficient search and merge procedures to obtain an optimal algorithm. This technique gives an alternative optimal algorithm for the Voronoi diagram of points as well (all other optimal parallel algorithms for this problem use the transformation to three-dimensional half-space intersection).
Tuning or fine tuning of a tracker system turns out to be a hard job in practice. The main reason for this is that in a practical (surveillance) tracker system there are a lot of design parameters and a lot of competi...
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Tuning or fine tuning of a tracker system turns out to be a hard job in practice. The main reason for this is that in a practical (surveillance) tracker system there are a lot of design parameters and a lot of competing requirements to be met. An algorithm to tune a tracker system automatically and at the same time obtain quantitative results in terms of the optimality of the solution is provided here. The theory of randomized algorithms is used to obtain probabilistic statements on the quality of the output of the tuning process. A simplified example illustrates how the developed theory is to be used.
Reducing the error of quantum algorithms is often achieved by applying a primitive called amplitude amplification. Its use leads in many instances to quantum algorithms that are quadratically faster than any classical...
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ISBN:
(纸本)3540309357
Reducing the error of quantum algorithms is often achieved by applying a primitive called amplitude amplification. Its use leads in many instances to quantum algorithms that are quadratically faster than any classical algorithm. Amplitude amplification is controlled by choosing two complex numbers phi(s) and phi(t) of unit norm, called phase factors. If the phases are well-chosen, amplitude amplification reduces the error of quantum algorithms, if not, it may increase the error. We give an analysis of amplitude amplification with a emphasis on the influence of the phase factors on the error of quantum algorithms. We introduce a so-called phase matrix and use it to give a straightforward and novel analysis of amplitude amplification processes. We show that we may always pick identical phase factors phi(s) = phi(t) with argument in the range pi/3 <= arg(phi(s)) <= pi We also show that identical phase factors phi(s) = phi(t) with -pi/2 < arg(phi(s)) < pi/2 never leads to an increase in the error, generalizing a recent result of Lov Grover who shows that amplitude amplification becomes a quantum analogue of classical repetition if we pick phase factors phi(s) = phi(t) with arg(phi(s)) = pi/3.
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