Semi-online scheduling on two uniform machines with the known largest size

  • Authors:

  • Sheng-Yi Cai;Qi-Fan Yang

  • Affiliations:

  • Department of Mathematics, Zhejiang University, Hangzhou, P.R. China 310027 and School of Mathematics & Information Science, Wenzhou University, Wenzhou, P.R. China 325035;Department of Mathematics, Zhejiang University, Hangzhou, P.R. China 310027

  • Venue:
  • Journal of Combinatorial Optimization
  • Year:
  • 2011

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Abstract

This paper investigates semi-online scheduling on two uniform machines with the known largest size. Denote by s j the speed of each machine, j=1,2. Assume 0s 1驴s 2, and let s=s 2/s 1 be the speed ratio. First, for the speed ratio $s\in [1,\sqrt{2}]$ , we present an optimal semi-online algorithm $\mathcal{LSMP}$ with the competitive ratio $\mathrm{max}\{\frac {2(s+1)}{2s+1},s\}$ . Second, we present a semi-online algorithm $\mathcal{HSMP}$ . And for $s\in(\sqrt{2},1+\sqrt{3})$ , the competitive ratio of $\mathcal{HSMP}$ is strictly smaller than that of the online algorithm $\mathcal{LS}$ . Finally, for the speed ratio s驴s *驴3.715, we show that the known largest size cannot help us to design a semi-online algorithm with the competitive ratio strictly smaller than that of $\mathcal{LS}$ . Moreover, we show a lower bound for $s\in(\sqrt{2},s^{*})$ .