共查询到20条相似文献,搜索用时 796 毫秒
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Annamria Kovcs 《Journal of Discrete Algorithms》2009,7(3):327-340
denotes the problem of scheduling n jobs on m machines of different speeds such that the makespan is minimized. In the paper two special cases of are considered: case I, when machine speeds are equal, and there is only one faster machine; and case II, when machine speeds are all powers of 2 (2-divisible machines). Case I has been widely studied in the literature, while case II is significant in an approach to design so called monotone algorithms for the scheduling problem.We deal with the worst case approximation ratio of the classic list scheduling algorithm ‘Largest Processing Time (LPT)’. We provide an analysis of this ratio for both special cases: For ‘one fast machine’, a tight bound of is given. For 2-divisible machines, we show that in the worst case . Besides, we prove another lower bound of when LPT breaks ties arbitrarily.To our knowledge, the best previous lower and upper bounds were in case I [T. Gonzalez, O.H. Ibarra, S. Sahni, Bounds for LPT schedules on uniform processors, SIAM Journal on Computing 6 (1) (1977) 155–166], respectively in case II [R.L. Graham, Bounds on multiprocessing timing anomalies, SIAM Journal on Applied Mathematics 17 (1969) 416–429; A. Kovács, Fast monotone 3-approximation algorithm for scheduling related machines, in: Proc. 13th Europ. Symp. on Algs. (ESA), in: LNCS, vol. 3669, Springer, 2005, pp. 616–627]. Moreover, Gonzalez et al. conjectured the lower bound 4/3 to be tight in the ‘one fast machine’ case [T. Gonzalez, O.H. Ibarra, S. Sahni, Bounds for LPT schedules on uniform processors, SIAM Journal on Computing 6 (1) (1977) 155–166]. 相似文献
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Let be a field of q elements, where q is a power of an odd prime p. The polynomial defined by has the property that where ρ is the quadratic character on . This univariate identity was applied to prove a recent theorem of N. Katz. We formulate and prove a bivariate extension, and give an application to quadratic residuacity. 相似文献
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Greg Aloupis Mirela Damian Robin Flatland Matias Korman Özgür Özkan David Rappaport Stefanie Wuhrer 《Computational Geometry》2013,46(3):328-339
Given a set S of points in the plane representing wireless devices, each point equipped with a directional antenna of radius r and aperture angle , our goal is to find orientations and a minimum r for these antennas such that the induced communication graph is strongly connected. We show that if , if , if , and if suffices to establish strong connectivity, assuming that the longest edge in the Euclidean minimum spanning tree of S is 1. These results are worst-case optimal and match the lower bounds presented in [I. Caragiannis, C. Kaklamanis, E. Kranakis, D. Krizanc, A. Wiese, Communication in wireless networks with directional antennae, in: Proc. of the 20th Symp. on Parallelism in Algorithms and Architectures, 2008, pp. 344–351]. In contrast, is sometimes necessary when . 相似文献
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Let χ range over the even Dirichlet characters mod , a prime. Let be the associated theta series. It is known that the square mean value of is asymptotic to as p goes to infinity. We prove that the fourth mean value of is asymptotic to as p goes to infinity. We give similar results for mean values of odd Dirichlet characters mod p. 相似文献
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Gábor Korchmáros Maria Montanucci Pietro Speziali 《Journal of Pure and Applied Algebra》2018,222(7):1810-1826
Let be the algebraic closure of a finite field of odd characteristic p. For a positive integer m prime to p, let be the transcendence degree 1 function field defined by . Let and . The extension is a non-Galois extension. Let K be the Galois closure of F with respect to H. By Stichtenoth [20], K has genus , p-rank (Hasse–Witt invariant) and a -automorphism group of order at least . In this paper we prove that this subgroup is the full -automorphism group of K; more precisely where Δ is an elementary abelian p-group of order and D has an index 2 cyclic subgroup of order . In particular, , and if K is ordinary (i.e. ) then . On the other hand, if G is a solvable subgroup of the -automorphism group of an ordinary, transcendence degree 1 function field L of genus defined over , then ; see [15]. This shows that K hits this bound up to the constant .Since has several subgroups, the fixed subfield of such a subgroup N may happen to have many automorphisms provided that the normalizer of N in is large enough. This possibility is worked out for subgroups of Δ. 相似文献
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