Naively Sorting Evolving Data is Optimal and Robust

We study comparison sorting in the evolving data model, introduced by Anagnostopoulos, Kumar, Mah-dian and Upfal (2011), 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$b$</tex> of evolution steps take place between two comparisons. In fact, the only previous result achieving optimal linear total deviation, by Besa Vial, Devanny, Eppstein, Goodrich and Johnson (2018a), applies just for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$b=1$</tex>. We analyze a very simple sorting algorithm suggested by Mahdian (2014), 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, with high probability, optimal total deviation, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(n)$</tex>, and optimal maximum deviation, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(\log n)$</tex>, under very general model settings. Namely, the perturbation introduced by each evolution step is sampled from a general distribution of bounded moment generating function, and we just require that the average number of evolution steps between two sorting steps be bounded by an (arbitrary) constant, where the average is over a linear number of steps. The key ingredients of our proof are a novel potential function argument that inserts “gaps” in the list of items, and a general analysis 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 and open problems in the three aforementioned works, and provide theoretical support that simple quadratic algorithms are optimal and robust for sorting evolving data, as empirically observed by Besa Vial, Devanny, Eppstein, Goodrich and Johnson (2018b).