Optimal embedding of hypercube into cylinder

TL;DR

This paper investigates the optimal way to embed high-dimensional hypercubes into cylindrical structures, demonstrating that Gray code embedding minimizes wirelength and is the optimal strategy for such embeddings.

Contribution

It provides a formula for wirelength in hypercube-to-cylinder embeddings and proves Gray code embedding as the optimal method.

Findings

Derived the wirelength formula for hypercube embedding into cylinders.

Proved Gray code embedding is optimal for minimizing wirelength.

Established theoretical bounds for embedding strategies.

Abstract

We study the problem of Embedding Wirelength of $n$-dimensional Hypercube $Q_n$ into Cylinder $C_{2^{n_1}}\times P_{2^{n_2}}$, where $n_1+ n_2=n$, called EWHC. We show that such wirelength corresponding to Gray code embedding is $2^{n_2}(3\cdot 2^{2n_1-3}-2^{n_1-1})+2^{n_1} (2^{2n_2-1}-2^{n_2-1})$. In addition, we prove that Gray code embedding is an optimal strategy of EWHC.