The event B method provides a general framework for modelling both data structures and algorithms. B models are validated by discharging proof obligations ensuring safety properties. We address the problem of developm...
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The event B method provides a general framework for modelling both data structures and algorithms. B models are validated by discharging proof obligations ensuring safety properties. We address the problem of development of greedy algorithms using the seminal work of S. Curtis;she has formalised greedy algorithms in a relational calculus and has provided a list of results ensuring optimality results. Our first contribution is a re-modelling of Curtis's results in the event B framework and a mechanical checking of theorems on greedy algorithms The second contribution is the reuse of the mathematical framework for developing greedy algorithms from event B models;since the resulting event B models are generic, we show how to instantiate generic event B models to derive specific greedy algorithms;generic event B developments help in managing proofs complexity. Consequently, we contribute to the design of a library of proof-based developed algorithms.
We survey some recent developments in the analysis of greedy algorithms for assignment and transportation problems. We focus on the linear programming model for matroids and linear assignment problems with Monge prope...
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We survey some recent developments in the analysis of greedy algorithms for assignment and transportation problems. We focus on the linear programming model for matroids and linear assignment problems with Monge property, on general linear programs, probabilistic analysis for linear assignment and makespan minimization, and on-line algorithms for linear and non-linear assignment problems.
Given a set ofn positive integers and another positive integerW, the Subset-Sum Problem is to find that subset whose sum is closest to, without exceeding,W. We present a polynomial approximation scheme for this proble...
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Given a set ofn positive integers and another positive integerW, the Subset-Sum Problem is to find that subset whose sum is closest to, without exceeding,W. We present a polynomial approximation scheme for this problem and prove that its worst-case performance dominates that of Johnson's well-known scheme.
This paper provides new results on computing simultaneous sparse approximations of multichannel signals over redundant dictionaries using two greedy algorithms. The first one, p-thresholding, selects the S atoms that ...
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This paper provides new results on computing simultaneous sparse approximations of multichannel signals over redundant dictionaries using two greedy algorithms. The first one, p-thresholding, selects the S atoms that have the largest p-correlation while the second one, p-simultaneous matching pursuit (p-SOMP), is a generalisation of an algorithm studied by Tropp in (Signal Process. 86: 572-588, 2006). We first provide exact recovery conditions as well as worst case analyses of all algorithms. The results, expressed using the standard cumulative coherence, are very reminiscent of the single channel case and, in particular, impose stringent restrictions on the dictionary. We unlock the situation by performing an average case analysis of both algorithms. First, we set up a general probabilistic signal model in which the coefficients of the atoms are drawn at random from the standard Gaussian distribution. Second, we show that under this model, and with mild conditions on the coherence, the probability that p-thresholding and p-SOMP fail to recover the correct components is overwhelmingly small and gets smaller as the number of channels increases. Furthermore, we analyse the influence of selecting the set of correct atoms at random. We show that, if the dictionary satisfies a uniform uncertainty principle (Candes and Tao, IEEE Trans. Inf. Theory, 52(12):5406-5425, 2006), the probability that simultaneous OMP fails to recover any sufficiently sparse set of atoms gets increasingly smaller as the number of channels increases.
We study the convergence of certain greedy algorithms in Banach spaces. We introduce the WN property for Banach spaces and prove that the algorithms converge in the weak topology for general dictionaries in uniformly ...
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We study the convergence of certain greedy algorithms in Banach spaces. We introduce the WN property for Banach spaces and prove that the algorithms converge in the weak topology for general dictionaries in uniformly smooth Banach spaces with the WN property. We show that reflexive spaces with the uniform Opial property have theWN property. We show that our results do not extend to algorithms which employ a 'dictionary dual' greedy step.
We consider the problem of designing a centralized telecommunication network comprised of multipoint lines given a set of terminal locations, traffic requirements, and a common central site. The optimal solution to th...
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We consider the problem of designing a centralized telecommunication network comprised of multipoint lines given a set of terminal locations, traffic requirements, and a common central site. The optimal solution to this problem is a capacitated minimal spanning tree. We develop a class of heuristic algorithms for the solution of this problem by imbedding existing heuristics, referred to as first-order greedy algorithms, inside a loop where small, carefully chosen sets of arcs are alternately forced in and out of the solution. The resultant procedure is shown to be superior to existing techniques, producing solutions typically 2 percent better, while requiring only a modest amount of additional computer time.
We propose new greedy algorithms for learning the structure of a graphical model of a probability distribution, given samples drawn from the distribution. While structure learning of graphical models is a widely studi...
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We propose new greedy algorithms for learning the structure of a graphical model of a probability distribution, given samples drawn from the distribution. While structure learning of graphical models is a widely studied problem with several existing methods, greedy approaches remain attractive due to their low computational cost. The most natural greedy algorithm would be one which, essentially, adds neighbors to a node in sequence until stopping;it would do this for each node. While it is fast, simple and parallel, this naive greedy algorithm has the tendency to add non-neighbors that show high correlations with the given node. Our new algorithms overcome this problem in three different ways. The recursive greedy algorithm iteratively recovers the neighbors by running the greedy algorithm in an inner loop, but each time only adding the last added node to the neighborhood set. The second forward-backward greedy algorithm includes a node deletion step in each iteration that allows non-neighbors to be removed from the neighborhood set which may have been added in the previous steps. Finally, the greedy algorithm with pruning runs the greedy algorithm until completion and then removes all the incorrect neighbors. We provide both analytical guarantees and empirical performance for our algorithms. We show that in graphical models with strong non-neighbor interactions, our greedy algorithms can correctly recover the graph, whereas the previous greedy and convex optimization-based algorithms do not succeed.
Cloud computing allows users to access resources on demand. The size of data centers increase with the increasing demand for resources by users. Increase in the size of data centers is directly proportional to energy ...
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ISBN:
(纸本)9781479953646
Cloud computing allows users to access resources on demand. The size of data centers increase with the increasing demand for resources by users. Increase in the size of data centers is directly proportional to energy consumption. The total energy requirement has to be minimized by distributing virtual machine requests over data centers optimally, with the consideration of prices of distribution of virtual machines. These two parameters are taken into account to frame the objective function for the Virtual Machine Distribution across Data Centers. Here both servers and workloads are classified as IO bound and CPU bound. A greedy algorithm framework has been used to obtain sub-optimal solutions for virtual machine distribution problem. Simulation results obtained indicates in favor of best fit allocation.
Many problems in machine learning can be presented in the form of convex optimization problems with objective function as a loss function. The paper examines two weak relaxed greedy algorithms for finding the solution...
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ISBN:
(纸本)9783319694047;9783319694030
Many problems in machine learning can be presented in the form of convex optimization problems with objective function as a loss function. The paper examines two weak relaxed greedy algorithms for finding the solutions of convex optimization problems over convex hulls of atomic sets. Such problems arise as the natural convex relaxations of cardinality-type constrained problems, many of which are well-known to be NP-hard. Both algorithms utilize one atom from a dictionary per iteration, and therefore, guarantee designed sparsity of the approximate solutions. algorithms employ the so called 'gradient greedy step' that maximizes a linear functional which uses gradient information of the element obtained in the previous iteration. Both algorithms are 'weak' in the sense that they solve the linear subproblems at the gradient greedy step only approximately. Moreover, the second algorithm employs an approximate solution at the line-search step. Following ideas of [5] we put up the notion of the duality gap, the values of which are computed at the gradient greedy step of the algorithms on each iteration, and therefore, they are inherent upper bounds for primal errors, i.e. differences between values of objective function at current and optimal points on each step. We obtain dual convergence estimates for the weak relaxed greedy algorithms.
The greedy algorithms are efficient ways in reconstructing the sparse signal. Among all the greedy recovery algorithms for practical compressive sampling(CS), Subspace Pursuit(SP) can offer reliable recovery accuracy ...
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ISBN:
(纸本)9781612846835;9781612846828
The greedy algorithms are efficient ways in reconstructing the sparse signal. Among all the greedy recovery algorithms for practical compressive sampling(CS), Subspace Pursuit(SP) can offer reliable recovery accuracy with low costs. In this paper, the SP algorithm is optimized by reducing the complexity of the least square(LS) caculation in each iteration. The Optimized SP performs well in LTE channel estimation when compared with the other greedy algorithms. The utilization of Partial Fourier Matrix helps reduce the matrix storage in SP hardware implemention. The matrix inverse caculation is also simplified by taking advatage of the Hermite Toeplitz matrix generated from the Fourier Matrix.
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