A Curious Link Between Prime Numbers, the Maundy Cake Problem and Parallel Sorting

TL;DR

This paper introduces new theoretical algorithms for sorting sequences using n-ary comparators, linking the process to prime numbers, the Maundy cake problem, and a novel counting sequence for comparators.

Contribution

It presents a divide and conquer approach that leverages prime divisors and sums comparator outputs, connecting sorting algorithms to number theory and combinatorics.

Findings

The number of elements processed by special comparators divides the total sequence length.

Sorting can be expressed as summing outputs of prime-sized comparators.

A new sequence counts the number of comparators used in the algorithms.

Abstract

We present new theoretical algorithms that sums the n-ary comparators output in order to get the permutation indices in order to sort a sequence. By analysing the parallel ranking algorithm, we found that the special comparators number of elements it processes divide the number of elements to be sorted. Using the divide and conquer method, we can express the sorting problem into summing output of comparators taking a prime number of elements, given that this prime number divides the initial disordered sequence length. The number of sums is directly related to the Maundy cake problem. Furthermore, we provide a new sequence that counts the number of comparators used in the algorithms.