A Randomised Approach to Distributed Sorting

TL;DR

This work introduces a lightweight, graph-based randomised sorting framework that is well suited to distributed and fault-tolerant environments. By assigning edge weights $w(\{i,j\})$ and repeatedly sorting endpoints of randomly chosen edges, the harmonic sorter achieves $\Theta\bigl(n (\log n)^2\bigr)$ comparisons and supports linear speedups in parallel settings, with both asynchronous and synchronous implementations. The authors establish a rigorous reduction to balanced 0--1 inputs, analyze a Uniform Sorter as a warm-up, and then develop two analyses for the Harmonic Sorter: a $(\log n)^2$-time bound and a refined $\log n$-time bound, along with parallelisation results. The framework offers robustness, simplicity, and natural scalability for distributed sorting tasks, potentially outperforming deterministic networks in large-scale, fault-prone environments.

Abstract

We introduce and analyse a new, extremely simple, randomised sorting algorithm: - choose a pair of indices $\{i, j\}$ according to some distribution $q$; - sort the elements in positions $i$ and $j$ of the array in ascending order. Choosing $q_{\{i,j\}} \propto 1/|j - i|$ yields an order-$n (\log n)^2$ sorting time. We call it the harmonic sorter. The sorter trivially parallelises in the asynchronous setting, yielding a linear speed-up. We also exhibit a low-communication, synchronous version with a linear speed-up. We compare and contrast this algorithm with other sorters, and discuss some of its benefits, particularly its robustness and amenability to parallelisation and distributed computing.

Figures (2)

• Figure 1: Two examples of collections $(M_0, M_1, M_2, M_3)$ of matchings, both with $n = 64$. Above. Edge-lengths $D = 11$, so $K = \floor{\lg(n/D)} = 2$. Each of the four matchings has $2^{K-1} = 2$ 'blocks', each containing $n/2^{K+1} = 8$ edges of length $D = 11$.Below. Edge-lengths $D = 6$, so $K = \floor{\lg(n/D)} = 3$. Each of the four matchings has $2^{K-1} = 4$ 'blocks', each containing $n/2^{K+1} = 4$ edges of length $D = 6$.The blocks are all rotations of the 'fundamental block' $E_{D,K}$; the bisecting straight lines indicate the starting point of this rotated block. Each matching contains $n/4 = 16$ edges
• Figure 2: Branching process from recursion. Vertex labels indicate which binary interval has just been split; eg, label $01$ indicates that $X_t(.010\mathord\star) = .010\mathord\star$ and $X_t(.011\mathord\star) = .011\mathord\star$ at this point. Edge labels indicate the passage time; eg, label $T_{01}$ indicates that it took time $T_{01}$ between achieving $\set{ X_t(.01\mathord\star) = .01\mathord\star }$ and $\set{ X_t(.010\mathord\star) = .010\mathord\star } \cap \set{ X_t(.011\mathord\star) = .011\mathord\star }$