On the GI-Completeness of a Sorting Networks Isomorphism

TL;DR

This paper proves that determining isomorphism between sorting networks under the BZ model is GI-Complete, establishing its computational difficulty and contrasting it with the polynomial-time solvable CM case.

Contribution

It classifies the complexity of the BZ sorting network isomorphism problem as GI-Complete, linking it to the graph isomorphism problem and extending the understanding of itemset isomorphism.

Findings

BZ sorting network isomorphism is GI-Complete

No polynomial-time algorithm currently exists for BZ isomorphism

CM sorting network isomorphism can be solved in polynomial time

Abstract

The subitemset isomorphism problem is really important and there are excellent practical solutions described in the literature. However, the computational complexity analysis and classification of the BZ (Bundala and Zavodny) subitemset isomorphism problem is currently an open problem. In this paper we prove that checking whether two sorting networks are BZ isomorphic to each other is GI-Complete; the general GI (Graph Isomorphism) problem is known to be in NP and LWPP, but widely believed to be neither P nor NP-Complete; recent research suggests that the problem is in QP. Moreover, we state the BZ sorting network isomorphism problem as a general isomorphism problem on itemsets --- because every sorting network is represented by Bundala and Zavodny as an itemset. The complexity classification presented in this paper applies sorting networks, as well as the general itemset isomorphism problem. The main consequence of our work is that currently no polynomial-time algorithm exists for solving the BZ sorting network subitemset isomorphism problem; however the CM (Choi and Moon) sorting network isomorphism problem can be efficiently solved in polynomial time.