Workload Balancing on Trees, Grids, Hypercubes and Cliques

We introduce a simple communication scheme for doing complete workload balancing (WLB) on diierent kind of networks with n processors. Considering cliques we show that 1-WLB can be decided in time O(n log n) whereas 2-WLB is NP-complete. Another presented result shows that the WLB-problem can only be solved in log n steps on networks with degree d log n. Therefore we present optimal algorithms for doing WLB on trees and grids using a diierent communication strategy. The tree algorithm performs WLB by moving a minimal amount of workload. A simple and fast algorithm working on hypercubes is derived. 1 Notions and deenitions Let N = (V; E) be a processor network consisting of a set of processors (nodes) V = fp 0 ; : : :; p n?1 g and a set of communication edges E = ffa; bg j a; b 2 V; a 6 = bg. All processors have only got a local memory and work simultaneously at the same program. The processors run asynchronously, but 2 processors synchronize if they communicate with each other. This means that the sender will wait for the receiver and vice versa until communication takes place. This principle is known as synchronized message passing and was introduced by Hoare 3] in 1978. We consider a function : f0; : : :; n ? 1g ! IR 0 , where (k) denotes the workload of a processor p k 2 V at a xed moment in time. Let opt := P n?1 i=0 (i)=n the optimal workload of a processor and ij := (i)?(j) 2 , where i; j 2 f0; : : :; n ? 1g. The problem to generate the situation (i) = opt for all p i 2 V by moving workload across the network is called the complete workload balancing-problem (WLB-problem). We assume that any division of workload is possible so that the WLB-problem is always solvable. Potentially (k) is an estimation for the time required to solve problems placed on p k. In this case moving workload means to move unsolved problems. In every communication step each processor is only allowed to communicate with at least one other processor, either sending or receiving. In a communication between a pair of processors (p i ; p j) either ij 0 workload is moved from p i to p j or ? ij > 0 workload is moved from …