The random simplex algorithm for linear programming proceeds as follows: at each step, it moves from a vertex upsilon of the polytope to a randomly chosen neighbor of upsilon, the random choice being made from those n...
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The random simplex algorithm for linear programming proceeds as follows: at each step, it moves from a vertex upsilon of the polytope to a randomly chosen neighbor of upsilon, the random choice being made from those neighbors of upsilon that improve the objective function. We exhibit a polytope defined by n constraints in three dimensions with height O(log n), for which the expected running time of the random simplex algorithm is Omega(n/log n).
We show that the probability of the exceptional set decays exponentially for a broad class of randomized algorithms approximating solutions of ODEs, admitting a certain error decomposition. This class includes randomi...
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We show that the probability of the exceptional set decays exponentially for a broad class of randomized algorithms approximating solutions of ODEs, admitting a certain error decomposition. This class includes randomized explicit and implicit Euler schemes, and the randomized two-stage Runge-Kutta scheme (under inexact information). We design a confidence interval for the exact solution of an IVP and perform numerical experiments to illustrate the theoretical results.
An accelerated least-squares approach is introduced in this work by incorporating a greedy point selection method with randomized singular value decomposition (rSVD) to reduce the computational complexity of missing d...
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An accelerated least-squares approach is introduced in this work by incorporating a greedy point selection method with randomized singular value decomposition (rSVD) to reduce the computational complexity of missing data reconstruction. The rSVD is used to speed up the computation of a low-dimensional basis that is required for the least-squares projection by employing randomness to generate a small matrix instead of a large matrix from high-dimensional data. A greedy point selection algorithm, based on the discrete empirical interpolation method, is then used to speed up the reconstruction process in the least-squares approximation. The accuracy and computational time reduction of the proposed method are demonstrated through three numerical experiments. The first two experiments consider standard testing images with missing pixels uniformly distributed on them, and the last numerical experiment considers a sequence of many incomplete two-dimensional miscible flow images. The proposed method is shown to accelerate the reconstruction process while maintaining roughly the same order of accuracy when compared to the standard least-squares approach.
The singular value decomposition (SVD) and the principal component analysis are fundamental tools and probably the most popular methods for data dimension reduction. The rapid growth in the size of data matrices has l...
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The singular value decomposition (SVD) and the principal component analysis are fundamental tools and probably the most popular methods for data dimension reduction. The rapid growth in the size of data matrices has lead to a need for developing efficient large-scale SVD algorithms. randomized SVD was proposed, and its potential was demonstrated for computing a low-rank SVD (Rokhlin et al., 2009). In this article, we introduce a consistency notion for random projections used in randomized SVD and provide a consistency theorem for it. We also present a numerical example to show how the random projections to low dimension affect the consistency. (C) 2020 Elsevier B.V. All rights reserved.
Clique counting is considered to be a challenging problem in graph mining. The reason is combinatorial explosion;even moderate graphs with a few million edges could have clique counts in the order of many billions. In...
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ISBN:
(纸本)9781450391283
Clique counting is considered to be a challenging problem in graph mining. The reason is combinatorial explosion;even moderate graphs with a few million edges could have clique counts in the order of many billions. In this paper, we propose a fast and scalable algorithm for approximating 4-clique counts in a single-pass streaming model. By leveraging a combination of sampling approaches, we estimate the 4-clique count with high accuracy. Our algorithm performs well on massive graphs containing several billions of 4-cliques, and terminates within a reasonable amount of time.
This paper presents block-coordinate descent algorithms for the approximate solution of large structured convex programming problems. The constraints of such problems consist of K disjoint convex compact sets B-k call...
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This paper presents block-coordinate descent algorithms for the approximate solution of large structured convex programming problems. The constraints of such problems consist of K disjoint convex compact sets B-k called blocks, and M nonnegative-valued convex block-separable inequalities called coupling or resource constraints. The algorithms are based on an exponential po- tential function reduction technique. It is shown that feasibility as well as min-max resource-sharing problems for such constraints can be solved to a relative accuracy E in O(K ln M(epsilon(-2) + In K)) iterations, each of which solves K block problems to a comparable accuracy, either sequentially or in parallel. The same bound holds for the expected number of iterations of a randomized variant of the algorithm which uniformly selects a random block to process at each iteration. An extension to objective and constraint functions of arbitrary sign is also presented. The above results yield fast approximation schemes for a number of applications such as problems with additively separable functions, generalized concurrent hows with side constraints, linear and nonlinear supply-sharing transportation networks, and deterministic equivalents of certain two-stage stochastic programs. Another consequence of this analysis is that, for a fixed relative accuracy, the approximate solution of matrix games is in NC.
In the Equal-Subset-Sum problem, we are given a set S of n integers and the problem is to decide if there exist two disjoint nonempty subsets A, B subset of S, whose elements sum up to the same value. The problem is N...
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ISBN:
(纸本)9783959771245
In the Equal-Subset-Sum problem, we are given a set S of n integers and the problem is to decide if there exist two disjoint nonempty subsets A, B subset of S, whose elements sum up to the same value. The problem is NP-complete. The state-of-the-art algorithm runs in O*(3(n/2)) <= O*(1.7321(n)) time and is based on the meet-in-the-middle technique. In this paper, we improve upon this algorithm and give O*(1.7088(n)) worst case Monte Carlo algorithm. This answers a question suggested by Woeginger in his inspirational survey. Additionally, we analyse the polynomial space algorithm for Equal-Subset-Sum. A naive polynomial space algorithm for Equal-Subset-Sum runs in O*(3(n)) time. With read-only access to the exponentially many random bits, we show a randomized algorithm running in O*(2.6817(n)) time and polynomial space.
The subset-sum problem (SSP) is defined as follows: given a positive integer bound and a set of n positive integers find a subset whose sum is closest to, but not greater than, the bound. We present a randomized appro...
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The subset-sum problem (SSP) is defined as follows: given a positive integer bound and a set of n positive integers find a subset whose sum is closest to, but not greater than, the bound. We present a randomized approximation algorithm for this problem with linear space complexity and time complexity of O(n log n). Experiments with random uniformly-distributed instances of SSP show that our algorithm outperforms, both in running time and average error, Martello and Toth's (1984) quadratic greedy search, whose time complexity is O(n2). We propose conjectures on the expected error of our algorithm for uniformly-distributed instances of SSP and provide some analytical arguments justifying these conjectures. We present also results of numerous tests. International Federation of Operational Research Societies 2002.
Clustering analysis is one of the important problems in the fields of data mining and machine learning. There are many different clustering methods. Among them, k-means clustering is one of the most popular schemes ow...
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ISBN:
(纸本)9781424420957
Clustering analysis is one of the important problems in the fields of data mining and machine learning. There are many different clustering methods. Among them, k-means clustering is one of the most popular schemes owing to its simple and practicality. This paper investigates the approximate algorithm for the k-means clustering by means of selecting the k initial points from the input point set. An expected 2-approximation algorithm is presented in this paper. Meanwhile, an efficient algorithm for selecting the initial points is also proposed. At last some experimental results are given to test the valid of these algorithms.
We present DRR-gossip, an energy-efficient and robust aggregate computation algorithm in wireless sensor networks. We prove that the DRR-gossip algorithm requires O(n) messages and O(n(3/2)/log(1/2)n) one-hop wireless...
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ISBN:
(纸本)9781605583969
We present DRR-gossip, an energy-efficient and robust aggregate computation algorithm in wireless sensor networks. We prove that the DRR-gossip algorithm requires O(n) messages and O(n(3/2)/log(1/2)n) one-hop wireless transmissions to obtain aggregates on a random geometric graph. This reduces the energy consumption by at least a factor of 1/log(n) over the standard uniform gossip algorithm. Experiments validate the theoretical results and show that DRR-gossip needs much less transmissions than other gossip-based schemes.
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