Load Balanced Tree Embeddings

TL;DR: This work presents efficient embeddings with a balanced load for the case when both architectures are complete binary trees and considers the embedding problem when every edge of T has a weight associated with it.

Abstract: When an n-processor architecture T is embedded into an m-processor architecture H with n > m and every processor of H is assigned at least ⌞n/m⌟ and at most [n/m] processors of T, the embedding has a balanced processor load. We present efficient embeddings with a balanced load for the case when both architectures are complete binary trees. We show that T can be embedded into H with a dilation of 1 and congestion of at most min{[n/m], 2 log n}. We also consider embeddings that achieve a balanced l/i load; i.e. every processor of H simulates at most [(n + 1)/2m] leaves and at most [(n − 1)/2m] interior processors of T. We present an embedding that achieves a balanced l/i load, a dilation of 2[log log m] + 1 and a congestion of O(log n). We show that every embedding strategy achieving a balanced l/i load must have a dilation of at least 3. We also consider the embedding problem when every edge of T has a weight associated with it.