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A radio network is a synchronous network of processors that communicate by transmitting messages to their neighbors, where a processor receives a message in a given step if and only if it is silent in this step and precisely one of its neighbors transmits. In this paper we prove the existence of a family of radius-2 networks on n vertices for which any broadcast schedule requires at least sZ(log* n) rounds of transmissions. This matches an upper bound of O(log* n) rounds for networks of radius 2 proved earlier by Bar-Yehuda, Goldreich, and Itai, in "Proceedings of the 4th ACM Symposium on Principles of Distributed Computing, 1986," pp. 98-107.

We present a general framework for solving resource allocation and scheduling problems. Given a resource of fixed size, we present algorithms that approximate the maximum throughput or the minimum loss by a constant factor. Our approximation factors apply to many problems, among which are: (i) real-time scheduling of jobs on parallel machines; (ii) bandwidth allocation for sessions between two endpoints; (iii) general caching; (iv) dynamic storage allocation; (v) bandwidth allocation on optical line and ring topologies. For some of these problems we provide the first constant factor approximation algorithm. Our algorithms are simple and efficient. They use the local-ratio technique and can be equivalently interpreted within the primal-dual schema.

This paper is concerned with the solvability of the problem of processor renaming in unreliable, completely asynchronous distributed systems. Fischer et al. prove in [S] that "nontrivial consensus" cannot be attained in such systems, even when only a single, benign processor failure is possible. In contrast, this paper shows that problems of processor renaming can be solved even in the presence of up to t c n/2 faulty processors, contradicting the widely held belief that no nontrivial problem can be solved in such a system. The problems deal with renaming processors so as to reduce the size of the initial name space. When only uniqueness of the new names is required, we present a lower bound of n + 1 on the size of the new name space, and a renaming algorithm that establishes an upper bound on n + t. If the new names are required also to preserve the original order, a tight bound of 2'(n -t + 1) -1 is obtained.

We study the problem of scheduling activities of several types under the constraint that, at most, a fixed number of activities can be scheduled in any single time slot. Any given activity type is associated with a service cost and an operating cost that increases linearly with the number of time slots since the last service of this type. The problem is to find an optimal schedule that minimizes the long-run average cost per time slot. Applications of such a model are the scheduling of maintenance service to machines, multi-item replenishment of stock, and minimizing the mean response time in Broadcast Disks. Broadcast Disks recently gained a lot of attention because they were used to model backbone communications in wireless systems, Teletext systems, and Web caching in satellite systems.The first contribution of this paper is the definition of a general model that combines into one several important previous models. We prove that an optimal cyclic schedule for the general problem exists, and we establish the NP-hardness of the problem. Next, we formulate a nonlinear program that relaxes the optimal schedule and serves as a lower bound on the cost of an optimal schedule. We present an efficient algorithm for finding a near-optimal solution to the nonlinear program. We use this solution to obtain several approximation algorithms.(1) A 9/8 approximation for a variant of the problem that models the Broadcast Disks application. The algorithm uses some properties of "Fibonacci sequences." Using this sequence, we present a 1 57-approximation algorithm for the general problem.(2) A simple randomized algorithm and a simple deterministic greedy algorithm for the problem. We prove that both achieve approximation factor of 2. To the best of our knowledge this is the first worst-case analysis of a widely used greedy heuristic for this problem.1. Introduction. We study a problem of scheduling activities of several types over an infinite number of time slots. We describe the model in terms of a generalized version of the maintenance service scheduling problem studied in Anily et al. (1998). In this formulation, there are m machines 1 m that are to be scheduled for maintenance over an infinite discrete time horizon. In each time slot, at most M machines can be scheduled for maintenance. The cost of operating a machine at any given time slot depends on the number of time slots since the last maintenance of that machine. We assume that each machine i is associated with a constant a i > 0 and the cost of operating the machine in the hth time slot after the last maintenance of that machine is h + b a i , for h ≥ 0 and integer b ≥ 0. (The value of b is determined by the application.) We assume that the cost associated with the maintenance service of the ith machine is c i ≥ 0. The problem is to find an optimal schedule specifying at which time slots to maintain each of the machines to minimize the long-run average cost per time slot.More formally, a schedule for the Generalized Maintenance Scheduling Problem (GMSP) with m machines ...