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Abstract. In a perfectly periodic schedule, each job must be scheduled precisely every some fixed number of time units after its previous occurrence. Traditionally, motivated by centralized systems, the perfect periodicity requirement is relaxed, the main goal being to attain the requested average rate. Recently, motivated by mobile clients with limited power supply, perfect periodicity seems to be an attractive alternative that allows clients to save energy by reducing their "busy waiting" time. In this case, clients may be willing to compromise their requested service rate in order to get perfect periodicity. In this paper we study a general model of perfectly periodic schedules, where each job has a requested period and a length; we assume that m jobs can be served in parallel for some given m. Job lengths may not be truncated, but granted periods may be different than the requested periods. We present an algorithm which computes schedules such that the worst-case proportion between the requested period and the granted period is guaranteed to be close to the lower bound. This algorithm improves on previous algorithms for perfect schedules in providing a worst-case guarantee rather than an average-case guarantee, in generalizing unit length jobs to arbitrary length jobs, and in generalizing the single-server model to multiple servers.

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 w i , 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 a 1 , that depends on the maximum demand probability:The best known polynomial algorithm to date guarantees an approximation of 3 2 + 3 2 a 1 . In this paper, we develop a treebased methodology for perfectly periodic scheduling, and using new techniques, we derive algorithms with better bounds. For small a 1 values, our best algorithm guarantees approximation of 1 +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 + a 2 1 . We also provide algorithms with good performance guarantees for large values of a 1 .

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We consider a system where users wish to find similar users. To model similarity, we assume the existence of a set of queries, and two users are deemed similar if their answers to these queries are (mostly) identical: each user has a vector of preferences, and two users are similar if their preference vectors differ in only a few coordinates. The preferences are unknown to the system initially, and the goal of the algorithm is to classify the users into classes of roughly the same preferences with the least possible number of queries presented to any user. We prove nearly matching lower and upper bounds on that problem. Specifically, we present an "anytime" algorithm that maintains a partition of the users, and the quality of the partition improves over time: let n be the number of users. At time T , groups ofÕ (n/T ) users with the same preferences will be separated (with high probability) if they differ in sufficiently many queries. We present a lower bound that matches the upper bound, up to a constant factor, for nearly all possible distances between user groups.