The communication complexity of distributed task allocation

We consider a distributed task allocation problem in which <i>m</i> players must divide a set of <i>n</i> tasks between them. Each player <i>i</i> receives as input a set <i>X_i</i> of tasks such that the union of all input sets covers the task set. The goal is for each player to output a subset <i>Y_i</i> ⊆ <i>X_i</i>, such that the outputs (<i>Y₁</i>,...,<i>Y_m</i>) form a partition of the set of tasks. The problem can be viewed as a distributed one-shot variant of the well-known <i>k</i>-server problem, and we also show that it is closely related to the problem of finding a rooted spanning tree in directed broadcast networks. We study the communication complexity and round complexity of the task allocation problem. We begin with the classical two-player communication model, and show that the randomized communication complexity of task allocation is Ω(<i>n</i>), even when the set of tasks is known to the players in advance. For the multi-player setting with <i>m</i> = <i>O</i>(<i>n</i>) we give two upper bounds in the shared-blackboard model of communication. We show that the problem can be solved in <i>O</i>(log <i>n</i>) rounds and <i>O</i>(<i>n</i> log <i>n</i>) total bits for arbitrary inputs; moreover, if for any set <i>X</i> of tasks, there are at least α|<i>X</i>| players that have at least one task from <i>X</i> in their inputs, then <i>O</i>((1/α + log <i>m</i>)log <i>n</i>) rounds suffice even if each player can only write <i>O</i>(log <i>n</i>) bits on the blackboard in each round. Finally, we extend our results to the case where the players communicate over an arbitrary directed communication graph instead of a shared blackboard. As an application of these results, we also consider the related problem of constructing a directed spanning tree in strongly-connected directed networks and we show lower and upper bounds for that problem.