Sorting processes with energy-constrained comparisons ^*

Abstract

We study very simple sorting algorithms based on a probabilistic comparator model. In this 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 repeatedly compare and swap pairs of randomly chosen elements, and they correspond to natural Markovian processes. The study of these Markov chains reveals an interesting phenomenon. Namely, in several cases, the algorithm that repeatedly compares only adjacent elements is better than the one making arbitrary comparisons: in the long-run, the former algorithm produces sequences that are "better sorted". The analysis of the underlying Markov chain poses interesting questions as the latter algorithm yields a nonreversible chain, and therefore its stationary distribution seems difficult to calculate explicitly. We nevertheless provide bounds on the stationary distributions and on the mixing time of these processes in several restrictions.

Sorting by Swaps with Noisy Comparisons

“…However, we will see that both extreme cases have their drawbacks in terms of speed of convergence (for r = 1) and quality of the final solution (for r = n). A similar trade-off has also been observed in [18] in a slightly different noise model. Thus we also include general values of r in our analysis to interpolate between both cases.…”

“…Proof. The first equality has been shown in [18]: In the sum i<j : π(i)>π(j) j − i, each element i is added s i and subtracted l i times, where s i is the number of smaller elements on the right of i and l i the number of larger elements on its left. The difference d i = s i − l i is equal to i's dislocation to the left, i.e., d i = i − π(i).…”

“…For 0 < p < 1/2 and 1 ≤ r ≤ n we will study the algorithm SwapSort p,r as given by Algorithm 1, which produces a random offspring by choosing uniformly at random two elements in distance at most r and swapping them. The comparison between offspring and parent yields the correct answer with probability 1 − p, and the wrong answer with probability p. While the extreme cases r = 1 (adjacent swaps) [18,7] and r = n (arbitrary swaps) [18,37] have been studied before in related settings, this is the first time that general r is considered. However, we will see that both extreme cases have their drawbacks in terms of speed of convergence (for r = 1) and quality of the final solution (for r = n).…”

“…• W (π) is the weighted number of inversions, where each inversion (i, j) is weighted by j − i [18].…”

Optimal Dislocation with Persistent Errors in Subquadratic Time

“…Alonso et al [2] and Hadjicostas and Lakshamanan [11] studied the classical Quicksort and recursive Mergesort algorithms, respectively. Sorting by repeatedly performing random swaps results in Markovian processes which have been studied by Geissmann et al [8,10].…”