Radix Sorting With No Extra Space

TL;DR

This paper introduces a space-efficient, stable radix sorting algorithm for integers of size O(log n) that operates in linear time with only constant extra space, and extends the approach to read-only keys and arbitrary ranges.

Contribution

It presents the first in-place, linear-time radix sorting algorithm for integers of size O(log n), including read-only keys, and provides a transformation for any RAM sorting algorithm to be space-efficient.

Findings

Achieves O(n) time with O(1) extra space for integers of size O(log n).

Extends to read-only keys with the same bounds.

Provides a black-box transformation for arbitrary RAM sorting algorithms.

Abstract

It is well known that n integers in the range [1,n^c] can be sorted in O(n) time in the RAM model using radix sorting. More generally, integers in any range [1,U] can be sorted in O(n sqrt{loglog n}) time. However, these algorithms use O(n) words of extra memory. Is this necessary? We present a simple, stable, integer sorting algorithm for words of size O(log n), which works in O(n) time and uses only O(1) words of extra memory on a RAM model. This is the integer sorting case most useful in practice. We extend this result with same bounds to the case when the keys are read-only, which is of theoretical interest. Another interesting question is the case of arbitrary c. Here we present a black-box transformation from any RAM sorting algorithm to a sorting algorithm which uses only O(1) extra space and has the same running time. This settles the complexity of in-place sorting in terms of the complexity of sorting.