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

TL;DR

This work investigates sorting n elements using t-comparators under strict round constraints, focusing on non-adaptive single-round and randomized two-round settings. It establishes a tight information-theoretic lower bound for one-round deterministic sorting, and shows optimal constructions for certain parameter families via affine/Steiner design theory, with a composition approach that extends to $n=t^{2^k}$. For the two-round randomized model, it delivers a Las-Vegas algorithm that is asymptotically optimal in comparator count, running in $Oig( ext{max}(rac{n^{3/2}}{t^2}, rac{n}{t})ig)$ comparators with high probability, and introduces the binary tree of deferred randomness to handle the probabilistic analysis. The results have implications for hardware-oriented and distributed computing contexts where multi-element comparisons are available but inter-round communication is costly, and they open avenues for exploring Steiner-system-based optimal sorting in broader parameter regimes.

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.

Figures (4)

• Figure 1: The figure shows the distribution of pivots (balls) on the tree of deferred randomness, marked as the numbers in each node. Here we have $\sqrt{n}=1600$ pivots and 16 bins ($c=100$). In the first two levels, the distribution is about even. The node $u_{11}$ receives too few balls and so the event $\mathcal{E}_{u_5}$ holds. Similarly, $b_9$ gets too few balls (bin's balls are not shown in the figure), causing $\mathcal{E}_{u_{12}}$ to happen. The nodes in gray portray the set $\mathsf{END}$ described in Section \ref{['sec:random:analysis']}, which includes the nodes where our deferred randomness revelation process stops. The cost of the scheme scales with the number of balls reaching nodes in this set.
• Figure 2: The matrices $M_0$, $M_1$
• Figure 3: The matrix $M_0$
• Figure 4: A histogram of the number of rounds required to the completion of the algorithm in beigel1990sorting for the case of $n=t^2$ with (a) $t=10$ and (b) $t=100$. Each histogram is based on $100$ repeated independent instances. In both $t$ values, the average number of rounds is above $4$.

Theorems & Definitions (42)

• Theorem 1.1: main, deterministic
• Theorem 1.2: main, randomized
• Definition 2.1
• Lemma 3.1
• proof
• Corollary 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4