Quasirandom load balancing

TLDR: It is shown that first-passage probabilities of a random walk on a path with arbitrary weights can be expressed as a convolution of independent geometric probability distributions and the quasirandom algorithm is the first known algorithm for this setting which is optimal both in time and achieved smoothness.

Abstract: We propose a simple distributed algorithm for balancing indivisible tokens on graphs The algorithm is completely deterministic, though it tries to imitate (and enhance) a random algorithm by keeping the accumulated rounding errors as small as possibleOur new algorithm approximates the idealized process (where the tokens are divisible) on important network topologies surprisingly closely On d-dimensional torus graphs with n nodes it deviates from the idealized process only by an additive constant In contrast to that, the randomized rounding approach of Friedrich and Sauerwald [8] can deviate up to Ω(polylog n) and the deterministic algorithm of Rabani, Sinclair and Wanka [23] has a deviation of Ω(n1/d) This makes our quasirandom algorithm the first known algorithm for this setting which is optimal both in time and achieved smoothness We further show that also on the hypercube our algorithm has a smaller deviation from the idealized process than the previous algorithmsTo prove these results, we derive several combinatorial and probabilistic results that we believe to be of independent interest In particular, we show that first-passage probabilities of a random walk on a path with arbitrary weights can be expressed as a convolution of independent geometric probability distributions