Nearly optimal perfectly periodic schedules

  • Authors:

  • Amotz Bar-Noy;Aviv Nisgav;Boaz Patt-Shamir

  • Affiliations:

  • Department of Computer and Information Science, CUNY, Brooklyn, NY;Department of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel;Department of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel

  • Venue:
  • Distributed Computing - Special issue: Selected papers from PODC '01
  • Year:
  • 2002

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Abstract

We consider the problem of scheduling a set of pages on a single broadcast channel using time-multiplexing. In a perfectly periodic schedule, time is divided into equal size slots, and each page is transmitted in a time slot precisely every fixed interval of time (the period of the page). We study the case in which each page i has a given demand probability wi, and the goal is to design a perfectly periodic schedule that minimizes the average time a random client waits until its page is transmitted. We seek approximate polynomial solutions. Approximation bounds are obtained by comparing the costs of a solution provided by an algorithm and a solution to a relaxed (non-integral) version of the problem. A key quantity in our methodology is a fraction we denote by a1, that depends on the maximum demand probability: a1 =def√maxi {wi}/Σ√wi. The best known polynomial algorithm to date guarantees an approximation of 3/2 + 3/2 a1. In this paper, we develop a tree-based methodology for perfectly periodic scheduling, and using new techniques, we derive algorithms with better bounds. For small a1 values, our best algorithm guarantees approximation of 1 + 3√3a1/1-3√3a1. On the other hand, we show that the integrality gap between the cost of any perfectly periodic schedule and the cost of the fractional problem is at least 1 + a12. We also provide algorithms with good performance guarantees for large values of a1.