Complexity and Approximations for Multimessage Multicasting

Abstract

We consider multimessage multicasting over thenprocessor complete (or fully connected) static network (MMC). First we present a linear time algorithm that constructs for every degreedproblem instance a communication schedule with total communication time at mostd2, wheredis the maximum number of messages that each processor may send or receive. Then we present degreedproblem instances such that all their communication schedules have total communication time at leastd2. We observe that our lower bound applies when the fan-out (maximum number of processors receiving any given message) is huge, and thus the number of processors is also huge. Since this environment is not likely to arise in the near future, we turn our attention to the study of important subproblems that are likely to arise in practice. We show that when each message has fan-outk=1 theMMCproblem corresponds to the makespan openshop preemptive scheduling problem which can be solved in polynomial time and show that fork⩾2 our problem is NP-complete and remains NP-complete even when forwarding is allowed. We present an algorithm to generate a communication schedule with total communication time 2d−1 for any degreedproblem instance with fan-outk=2. Our main result is anO(q·d·e) time algorithm, wheree⩽nd(the input length), with an approximation bound ofqd+k1/q(d−1), for any integerqsuch thatk>q⩾2. Our algorithms are centralized and require all the communication information ahead of time. Applications where all of this information is readily available include iterative algorithms for solving linear equations, and most dynamic programming procedures. The Meiko CS-2 machine and computer systems with processors communicating via dynamic permutation networks whose basic switches can act as data replicators (e.g.,nbynBenes network with 2 by 2 switches that can also act as data replicators) will also benefit from our results at the expense of doubling the number of communication phases.