Searching and Sorting with O(n^2) processors in O(1) time

TL;DR

This paper introduces a novel parallel architecture using simple crosspoint switches that enables constant-time searching and sorting of $n$ elements, significantly reducing interprocessor communication overhead in large-scale systems.

Contribution

It proposes a new architecture with a linear crosspoint array and demonstrates how to perform fundamental operations like search and sort in constant or near-constant time.

Findings

Searching and sorting can be achieved in O(1) time with elementary logic gates.

The architecture reduces interprocessor communication by replacing complex networks.

Sorting time is O(1) with threshold logic gates and O(log n log log n) with simple gates.

Abstract

The proliferation of number of processing elements (PEs) in parallel computer systems, along with the use of more extensive parallelization of algorithms causes the interprocessor communications dominate VLSI chip space. This paper proposes a new architecture to overcome this issue by using simple crosspoint switches to pair PEs instead of a complex interconnection network. Based on the cyclic permutation wiring idea described in \cite{oruc2016self}, this pairing leads to a linear crosspoint array of $n(n-1)/2$ processing elements and as many crosspoints. We demonstrate the versatility of this new parallel architecture by designing fast searching and sorting algorithms for it. In particular, we show that finding a minimum, maximum, and searching a list of $n$ elements can all be performed in $O(1)$ time with elementary logic gates with $O(n)$ fan-in, and in $O(\lg n)$ time with $O(1)$ fan-in. We further show that sorting a list of $n$ elements can also be carried out in $O(1)$ time using elementary logic gates with $O(n)$ fan-in and threshold logic gates. The sorting time increases to $O(\lg n\lg\lg n)$ if only elementary logic gates with $O(1)$ fan-in are used. The algorithm can find the maximum among $n$ elements in $O(1)$ time, and sort $n$ elements in $O(\lg n (\lg\lg n))$ time. In addition, we show how other fundamental queries can be handled within the same order of time complexities.