Any Load-Balancing Regimen for Evolving Tree Computations on Circulant Graphs is Asymptotically Optimal Rolf Wanka ⋆ Dept. of Mathematics & Computer Science and Heinz Nixdorf Institute Paderborn University, 33095 Paderborn, Germany, wanka@upb.de Abstract. We analyze evolving tree computations on circulant (rings with “regular” chords) and related graphs. In an evolving α-ary tree computation, a complete tree grows level by level, i. e., every leaf gener- ates α new nodes that become the new leaves. The load balancing task is to spread the new nodes on a network of processors in the moment they were created in such a way that the accumulated number of nodes per processor, i. e., its load, is as close as possible to the average number of nodes per processor. Gao/Rosenberg [2] introduced evolving compu- tations and investigated the growth of complete binary trees on rings of processors. They showed that the so-called ks-regimen behaves opti- mally in the course of long computations. In this paper, we generalize evolving computations to trees of arbitrary degree and we generalize the regimen notion. We show that any regimen behaves optimally. For this purpose, we model the actual load distribution, the generation process, and the distribution regimen by formal infinite polynomials. Then we show that evaluating these polynomials for certain inputs leads to the analysis of these regimens on circulant and related graphs. It is shown that any regimen leads to a close to optimal load distribution. 1 Introduction Background. In the standard abstract formulation of load balancing in a dis- tributed network, processors are modeled as the vertices of a graph and links between them as edges. Each processor initially has a collection of unit-size jobs which we call tokens. In a dynamic setting, some of these tokens generate new tokens, so we distinguish between generating and non-generating tokens. The object is to balance the number of tokens by transmitting the new tokens along edges according to some local scheme. This problem has obvious applications in job scheduling and other coordination tasks in parallel and distributed systems. It also arises in the context of finite element computations, and in simulations of physical phenomena. ⋆ Partially supported by DFG SFB 376 “Massively Parallel Computation” and by the Future and Emerging Technologies programme of the EU under contract number IST-1999-14186 (ALCOM-FT).