Embedding of Hypercube into Cylinder

TL;DR

This paper addresses the problem of embedding hypercube graphs into cylindrical graphs to minimize wirelength, providing an exact formula for specific cases, which advances understanding in graph embeddings for parallel computing.

Contribution

It derives the exact wirelength formula for embedding hypercubes into cylinders, a problem previously open for this graph combination.

Findings

Exact wirelength formula for hypercube into cylinder embedding.

Provides theoretical foundation for optimizing task mapping in parallel computers.

Advances understanding of graph embedding problems in high-performance computing.

Abstract

Task mapping in modern high performance parallel computers can be modeled as a graph embedding problem, which simulates the mapping as embedding one graph into another and try to find the minimum wirelength for the mapping. Though embedding problems have been considered for several regular graphs, such as hypercubes into grids, binary trees into grids, et al, it is still an open problem for hypercubes into cylinders. In this paper, we consider the problem of embedding hypercubes into cylinders to minimize the wirelength. We obtain the exact wirelength formula of embedding hypercube $Q^r$ into cylinder $C_{2^3}\times P_{2^{r-3}}$ with $r\ge3$.