Optimally Sorting Evolving Data

TL;DR

This paper introduces optimal algorithms for maintaining an approximate sorted order in evolving data, demonstrating that a simple insertion sort approach is asymptotically optimal in a dynamic comparison model.

Contribution

The paper proves that a repeated insertion sort algorithm maintains an optimal O(n) Kendall tau distance in an evolving data setting, closing the gap with the theoretical lower bound.

Findings

Insertion sort maintains O(n) Kendall tau distance with high probability.

The result is asymptotically optimal, matching the Omega(n) lower bound.

Insertion sort outperforms quicksort in this evolving data model.

Abstract

We give optimal sorting algorithms in the evolving data framework, where an algorithm's input data is changing while the algorithm is executing. In this framework, instead of producing a final output, an algorithm attempts to maintain an output close to the correct output for the current state of the data, repeatedly updating its best estimate of a correct output over time. We show that a simple repeated insertion-sort algorithm can maintain an O(n) Kendall tau distance, with high probability, between a maintained list and an underlying total order of n items in an evolving data model where each comparison is followed by a swap between a random consecutive pair of items in the underlying total order. This result is asymptotically optpimal, since there is an Omega(n) lower bound for Kendall tau distance for this problem. Our result closes the gap between this lower bound and the previous best algorithm for this problem, which maintains a Kendall tau distance of O(n log log n) with high probability. It also confirms previous experimental results that suggested that insertion sort tends to perform better than quicksort in practice.