Naively Sorting Evolving Data is Optimal and Robust

TL;DR

This work studies sorting under an evolving data model in which the true total order changes while the algorithm processes input. It analyzes a simple Naïve Sort algorithm that interleaves random adjacent swaps with local rank perturbations, proving it achieves an optimal linear total deviation $O(n)$ and an optimal maximum deviation $O(\log n)$ with high probability, under a general perturbation distribution with bounded MGF and bounded average evolution rate $b$. The proof introduces a gap-augmented list and two exponential potential functions, $\Phi$ and $\Psi$, to separately analyze sorting and mixing steps, plus a phase-based reset scheme and theta-filtering to control drift and ensure convergence. A key technical contribution is a decoupled analysis framework that separates sorting from evolution, enabling robust bounds in settings where prior methods fail. The results settle longstanding conjectures for the general $b\ge 2$ regime, provide theoretical support for empirical observations of simple quadratic algorithms under evolving ranks, and establish concentration and lower bounds that underscore the tightness and robustness of Naïve Sort in evolving environments.

Abstract

We study sorting in the evolving data model, introduced by [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 linear total deviation, by [BvDEGJ18a], applies just for $b=1$. We analyze a very simple sorting algorithm suggested by [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, with high probability, 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 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 aforementioned works, and provide theoretical support for empirical observations in [BvDEGJ18b].

Figures (6)

• Figure 1: An example of a left block (shown in red), a right block (shown in green), and a stationary block (shown in gray) of list $l$, where $d = 4$ and $\tau = \mathop{\mathrm{\mathtt{id}}}\nolimits_n$. The head of the left block is denoted by darker red and the head of the right block with darker green.
• Figure 2: The five cases of swaps between adjacent element for which the value of the potential remains the same (all of which correspond to mixing steps). The figures show the following three stages: $(i)$ the original list $l_t$ (with targets $\tau_t$), $(ii)$ the list $l_t'$ after the mixing step (with targets $\tau_t'$), and $(iii)$ the list $l_{t+1}$ after the $\mathop{\mathrm{\mathtt{adm}}}\nolimits$ operation (with targets $\tau_{t+1}$).
• Figure 3: The three cases of sorting steps involving swapping the head of a right block.
• Figure 4: Arrows showing the possible displacements of element $i$ by $D_{t+1}$. Red arrows represent an increase on $\delta_t(i)$, while green represent a decrease. Dashed arrows correspond to reaching a destination that is out of boundaries and in all cases lead to an increase (and their lengths upper bound the true increase). Solid black arrows represent the process corresponding to the additive term $\operatorname{\mathbb{E}}\left[ e^{\alpha |D_{t+1}|} \right]$ used to upper bound the contributions for when $i$ surpasses the position $\sigma_t^{-1}(i)$.
• Figure 5: Visualizations for Case 2, where element $j \neq i$ was selected in step $t+1$.

Theorems & Definitions (52)

• theorem 1
• definition 2: Permutation Distances
• definition 3: Local Rank Perturbation
• definition 4: Bounded Average Rate of Mixing Steps
• definition 5: $d$-Padding & Local Optimimality
• definition 6: Displacements
• lemma 6
• definition 7: Admissibility
• definition 8: Auxiliary Quantities $l_t,\tau_t,\sigma_t$
• definition 9: Potential Functions