Sorting in One and Two Rounds using $t$-Comparators

TL;DR

This paper explores efficient sorting algorithms using $t$-comparators in one or two rounds, establishing optimal and near-optimal bounds through combinatorial design theory and probabilistic methods.

Contribution

It introduces deterministic algorithms linked to 2-Steiner systems and a randomized two-round algorithm with asymptotically optimal comparator complexity.

Findings

Deterministic algorithms optimal for $n=t^{2^k}$ using design theory.

A two-round randomized algorithm with $O(rac{n^{3/2}}{t^2})$ comparators.

Some cases where optimal bounds are unattainable, with algorithms using up to three times the theoretical minimum.

Abstract

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. 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 $n=t^{2^k}$ with any $k\ge 1$ 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-rounds sorting algorithm for any $n$ and $t$. Our algorithm uses an asymptotically optimal number of $O(\max(\frac{n^{3/2}}{t^2},\frac{n}{t}))$ 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.