Rate of convergence of major cost incurred in the in-situ permutation algorithm

TL;DR

This paper analyzes the convergence rate of the major cost in the in-situ permutation algorithm, showing it converges at a rate of () () () in the Zolotarev metric, and builds on previous work relating to permutation costs and distribution convergence.

Contribution

It establishes the precise rate of convergence of the algorithm's cost to its limiting distribution, extending prior results with a detailed rate analysis.

Findings

Convergence rate is () in the Zolotarev metric.

Major cost follows a recurrence similar to Quicksort comparisons.

Normalized cost converges in distribution as shown by Hwang.

Abstract

The in-situ permutation algorithm due to MacLeod replaces $(x_{1},\cdots,x_{n})$ by $(x_{p(1)},\cdots,x_{p(n)})$ where $\pi=(p(1),\cdots,p(n))$ is a permutation of $\{1,2,\cdots,n\}$ using at most $O(1)$ space. Kirshenhofer, Prodinger and Tichy have shown that the major cost incurred in the algorithm satisfies a recurrence similar to sequence of the number of key comparisons needed by the Quicksort algorithm to sort an array of $n$ randomly permuted items. Further, Hwang has proved that the normalized cost converges in distribution. Here, following Neininger and R\"uschendorf, we prove the that rate of convergence to be of the order $\Theta(\ln(n)/n)$ in the Zolotarev metric.