Idempotent permutations

TL;DR

This paper explores the combinatorial properties of idempotent permutations and demonstrates their application in achieving optimal in-place sorting of integer keys within linear time and minimal space, approaching theoretical bounds.

Contribution

It provides a new combinatorial interpretation of idempotent permutations and applies them to develop an efficient in-place sorting method with minimal space usage.

Findings

In-place linear time sorting using idempotent permutations is possible within 4log(n) bits.

Theoretical lower bounds for sorting time and space are approached using these permutations.

A variant requires additional bits for tagging keys outside the original range.

Abstract

Together with a characteristic function, idempotent permutations uniquely determine idempotent maps, as well as their linearly ordered arrangement simultaneously. Furthermore, in-place linear time transformations are possible between them. Hence, they may be important for succinct data structures, information storing, sorting and searching. In this study, their combinatorial interpretation is given and their application on sorting is examined. Given an array of n integer keys each in [1,n], if it is allowed to modify the keys in the range [-n,n], idempotent permutations make it possible to obtain linearly ordered arrangement of the keys in O(n) time using only 4log(n) bits, setting the theoretical lower bound of time and space complexity of sorting. If it is not allowed to modify the keys out of the range [1,n], then n+4log(n) bits are required where n of them is used to tag some of the keys.