Tight bounds for a distributed selection game with applications to fixed-connection machines

We define a distributed selection game that generalizes a selection problem considered by S.R. Kosaraju (1989). We offer a tight analysis of our distributed selection game, and show that the lower bound for this abstract communication game directly implies near-tight lower bounds for certain selection problems on fixed-connection machines. For example, we prove that any deterministic comparison-based selection algorithm on an (n/log n)-processor bounded-degree hypercubic machine requires Ω(log^3/2n) steps in the worst case. This lower bound implies a non-trivial separation between the power of bounded-degree hypercubic and expander-based machines. Furthermore, we show that the algorithm underlying our tight upper bound for the distributed selection game can be adapted to run in O((log^3/2n) (log log n)²) steps on any (n/log n)-processor hypercubic machine