Sorting in One and Two Rounds Using t-Comparators

We examine sorting algorithms for n elements whose basic operation is comparing t elements simultaneously (a t-comparator). We focus on algorithms that use only a single round or two rounds – comparisons performed in the second round depend on the outcomes of the first round comparators. Algorithms with a small number of rounds are well-suited to distributed settings in which communication rounds are costly. We design deterministic and randomized algorithms. In the deterministic case, we show an interesting relation to design theory (namely, to 2-Steiner systems), which yields a single-round optimal algorithm for n = t^2k with any k ≥ 1 and a variety of possible values of t. For some values of t, however, no algorithm can reach the optimal (information-theoretic) bound on the number of comparators. For this case (and any other n and t), we show an algorithm that uses at most three times as many comparators as the theoretical bound. We also design a randomized Las-Vegas two-round sorting algorithm for any n and t. Our algorithm uses an asymptotically optimal number of (Formula presented) comparators, with high probability, i.e., with probability at least 1 - 1/n. The analysis of this algorithm involves the gradual unveiling of randomness, using a novel technique which we coin the binary tree of deferred randomness.