A New 0(klog n) Algorithm for Josephus Problem

TL;DR

This paper introduces a non-recursive, space-efficient O(k log n) algorithm for the Josephus problem, leveraging a relation with Nim from combinatorial game theory, especially effective for small k and large n.

Contribution

It presents a novel non-recursive algorithm with improved space complexity for the Josephus problem, connecting it to Nim theory.

Findings

Algorithm has O(k log n) time complexity.

Space complexity is reduced to O(1).

Performance is better for small k and large n.

Abstract

We present a new O(k log n) algorithm of the Josephus problem. The time complexity of our algorithm is O(k log n), and this time complexity is on a par with the existing O(k log n) algorithm. We do not have any recursion overhead or stack overflow because we do not use any recursion. Therefore, the space complexity of our algorithm is O(1), and ours is better than the existing O(k log n) algorithm in this respect. When k is small and n is large, our algorithm is better than the existing O(k log n) algorithm. This new algorithm is based on a relation between the Josephus problem and a maximum Nim of combinatorial game theory.