Naively Sorting Evolving Data is Optimal and Robust

We study comparison sorting in the evolving data model [AKMU11], where the true total order changes while the sorting algorithm is processing the input. More precisely, each comparison operation of the algorithm is followed by a sequence of evolution steps, where an evolution step perturbs the rank of a random item by a "small" random value. The goal is to maintain an ordering that remains close to the true order over time. Previous works have analyzed adaptations of classic sorting algorithms, assuming that an evolution step changes the rank of an item by just one, and that a fixed constant number $b$ of evolution steps take place between two comparisons. In fact, the only previous result achieving optimal $O(n)$ total deviation from the true order, where $n$ is the number of items, applies just for $b=1$ [BDEGJ18]. We analyze a very simple sorting algorithm suggested in [M14], which samples a random pair of adjacent items in each step and swaps them if they are out of order. We show that the algorithm achieves and maintains, w.h.p., optimal total deviation, $O(n)$, and optimal maximum deviation, $O(\log n)$, under very general model settings. Namely, the perturbation introduced by each evolution step follows a distribution of bounded moment generating function, and over a linear number of steps, on average the number of evolution steps between two sorting steps is bounded by an arbitrary constant. Our proof consists of a novel potential function argument that inserts "gaps" in the list of items, and a general framework which separates the analysis of sorting from that of the evolution steps, and is applicable to a variety of settings for which previous approaches do not apply. Our results settle conjectures by [AKMU11] and [M14], and provide theoretical support for the empirical evidence that simple quadratic algorithms are optimal and robust for sorting evolving data [BDEGJ18].