Sort well with energy-constrained comparisons

TL;DR

This paper investigates simple energy-aware probabilistic sorting algorithms, revealing that adjacent comparisons often outperform arbitrary ones in producing better sorted sequences over time.

Contribution

It introduces a probabilistic comparator model and analyzes Markovian sorting algorithms, highlighting the advantage of adjacent comparisons in energy-constrained settings.

Findings

Adjacent comparison algorithms outperform arbitrary comparison algorithms in long-term sorting quality.

The study reveals non-reversible Markov chains in the analysis, posing new theoretical challenges.

Long-run behavior favors simpler, energy-efficient comparison strategies.

Abstract

We study very simple sorting algorithms based on a probabilistic comparator model. In our model, errors in comparing two elements are due to (1) the energy or effort put in the comparison and (2) the difference between the compared elements. Such algorithms keep comparing pairs of randomly chosen elements, and they correspond to Markovian processes. The study of these Markov chains reveals an interesting phenomenon. Namely, in several cases, the algorithm which repeatedly compares only adjacent elements is better than the one making arbitrary comparisons: on the long-run, the former algorithm produces sequences that are "better sorted". The analysis of the underlying Markov chain poses new interesting questions as the latter algorithm yields a non-reversible chain and therefore its stationary distribution seems difficult to calculate explicitly.