Variable-Length Insertion-Based Noisy Sorting

In this work, we study the problem of sorting n elements with pairwise comparisons under the presence of observation noise. We consider variable-length algorithms with a random number of queries M, and attempt to characterize the noisy sorting capacity defined as the maximal ratio $\frac{{n\log n}}{{{\text{E}}[M]}}$ such that the ordering can be correctly estimated with a vanishing error probability. This can be viewed as a generalization of the framework introduced in [1] to allow variable-length algorithms. We provide upper and lower bounds for the noisy sorting capacity. The proposed algorithm attaining the lower bound is based on the insertion sort algorithm for the sorting problem in the noiseless case and the variable-length version of the Burnashev–Zigangirov algorithm for coding over channels with feedback. Moreover, we also derive an upper bound on the maximal ratio that can be achieved by noisy sorting algorithms that are based on insertion sort.