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Parallel machine covering with limited number of preemptions

Applied Mathematics(A Journal of Chinese Universities) 2014年第1期29卷 18-28页

作者： JIANG Yi-wei HU Jue-liang WENG Ze-wei ZHU Yu-qing Department of Mathematics Zhejiang Sci-Tech University Department of Computer Science The University of Texas at Dallas

In this paper, we investigate the/-preemptive scheduling on parallel machines to maximize the minimum machine completion time, i.e., machine covering problem with limited number of preemptions. It is aimed to obtain the worst case ratio of the objective value of the optimal schedule with unlimited preemptions and that of the schedule allowed to be preempted at most i times. For the m identical machines case, we show the worst case ratio is 2m-i-1/m and we present a polynomial time algorithm which can guarantee the ratio for any 0 〈 i 〈2 m - 1. For the /-preemptive scheduling on two uniform machines case, we only need to consider the cases of i = 0 and i = 1. For both cases, we present two linear time algorithms and obtain the worst case ratios with respect to s, i.e., the ratio of the speeds of two machines.

关键词： 90B35 90C27 68q25 i-preemptive scheduling machine covering approximation algorithm worst case ratio

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Fast rectangular matrix multiplication and some applications

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Science China Mathematics 2008年第3期51卷 389-406页

作者： Victor Y PAN Department of Mathematics and Computer Science.Lehman College City University of New York BronxNew York 10468 USA

We study asymptotically fast multiplication algorithms for matrix pairs of arbitrary dimensions, and optimize the exponents of their arithmetic complexity bounds. For a large class of input matrix pairs, we improve the known exponents. We also show some applications of our results: (i) we decrease from O(n 2 + n 1+o(1)logq) to O(n 1.9998 + n 1+o(1)logq) the known arithmetic complexity bound for the univariate polynomial factorization of degree n over a finite field with q elements; (ii) we decrease from 2.837 to 2.7945 the known exponent of the work and arithmetic processor bounds for fast deterministic (NC) parallel evaluation of the determinant, the characteristic polynomial, and the inverse of an n × n matrix, as well as for the solution to a nonsingular linear system of n equations; (iii) we decrease from O(m 1.575 n) to O(m 1.5356 n) the known bound for computing basic solutions to a linear programming problem with m constraints and n variables.

关键词： rectangular matrix multiplication asymptotic arithmetic complexity bilinear algorithm polynomial factorization over finite fields 68q25 11Y16

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