Distance-Preserving Graph Compression Techniques

TL;DR

This paper introduces optimal distance-preserving graph compression methods for weighted paths and trees, focusing on minimizing shortest path length errors after edge contraction.

Contribution

It presents novel optimal algorithms for edge contraction in weighted paths and trees to preserve shortest path distances.

Findings

Optimal edge contraction algorithms for paths and trees.

Minimized error in shortest path length after compression.

Focused on weighted graphs with non-negative weights.

Abstract

We study the problem of distance-preserving graph compression for weighted paths and trees. The problem entails a weighted graph $G = (V, E)$ with non-negative weights, and a subset of edges $E^{\prime} \subset E$ which needs to be removed from G (with their endpoints merged as a supernode). The goal is to redistribute the weights of the deleted edges in a way that minimizes the error. The error is defined as the sum of the absolute differences of the shortest path lengths between different pairs of nodes before and after contracting $E^{\prime}$. Based on this error function, we propose optimal approaches for merging any subset of edges in a path and a single edge in a tree. Previous works on graph compression techniques aimed at preserving different graph properties (such as the chromatic number) or solely focused on identifying the optimal set of edges to contract. However, our focus in this paper is on achieving optimal edge contraction (when the contracted edges are provided as input) specifically for weighted trees and paths.