The Power of Two Choices in Graphical Allocation

SIAM Journal on Computing, Ahead of Print.
Abstract. The graphical balls-into-bins process is a generalization of the classical 2-choice balls-into-bins process, where the bins correspond to vertices of an arbitrary underlying graph [math]. At each time step an edge of [math] is chosen uniformly at random, and a ball must be assigned to either of the two endpoints of this edge. The standard 2-choice process corresponds to the case of [math]. For any [math]-edge-connected, [math]-regular graph on [math] vertices, and any number of balls, we give an allocation strategy that, with high probability, ensures a gap of [math] between the load of any two bins. In particular, this implies a polylogarithmic bound for natural graphs such as cycles and tori, for which the classical greedy allocation strategy is conjectured to have a polynomial gap between the bin loads. For every graph [math], we also show an [math] lower bound on the gap achievable by any allocation strategy. This implies that our strategy achieves the optimal gap, up to polylogarithmic factors, for every graph [math]. Our allocation algorithm is simple to implement and requires only [math] time per allocation. It can be viewed as a more global version of the greedy strategy that compares average load on certain fixed sets of vertices, rather than on individual vertices. A key idea is to relate the problem of designing a good allocation strategy to that of finding suitable multicommodity flows. To this end, we consider Räcke’s cut-based decomposition tree and define certain orthogonal flows on it.