The Undirected Optical Indices of Complete $m$-ary Trees

TL;DR

This paper determines the exact undirected optical index of complete m-ary trees and explores the difference between this index and the edge-forwarding index, contributing to optimal wavelength assignment in optical networks.

Contribution

The paper provides the exact value of the undirected optical index for complete m-ary trees and analyzes the gap with the edge-forwarding index, advancing understanding of wavelength allocation.

Findings

Exact optical index for complete m-ary trees derived

Identified the gap between optical index and edge-forwarding index

Enhanced understanding of wavelength assignment in optical networks

Abstract

The routing and wavelength assignment problem arises from the investigation of optimal wavelength allocation in an optical network that employs Wavelength Division Multiplexing (WDM). Consider an optical network that is represented by a connected, simple graph $G$. An all-to-all routing $R$ in $G$ is a set of paths connecting all pairs of vertices of $G$. The undirected optical index of $G$ is the minimum integer $k$ to guarantee the existence of a mapping $\phi:R\to\{1,2,\ldots,k\}$, such that $\phi(P)\neq\phi(P')$ if $P$ and $P'$ have common edge(s), over all possible routings $R$. A natural lower bound of the undirected optical index of $G$ is the (undirected) edge-forwarding index, which is defined to be the minimum of the maximum edge-load over all possible all-to-all routings. In this paper, we first derive the exact value of the optical index of the complete $m$-ary trees, and then investigate the gap between undirected optical and edge-forwarding indices.