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.

翻译：我们研究了加权路径和树的距离保持图压缩问题。该问题涉及一个带非负权重的加权图 $G = (V, E)$，以及需要从 $G$ 中删除的边子集 $E^{\prime} \subset E$（其端点合并为一个超节点）。目标是重新分配被删除边的权重，以最小化误差。误差定义为在收缩 $E^{\prime}$ 前后，不同节点对之间最短路径长度绝对差值的总和。基于该误差函数，我们提出了针对路径中任意边子集和树中单条边的最优合并方法。以往关于图压缩技术的研究旨在保持不同的图属性（如图的色数），或仅专注于识别最优的边收缩集合。然而，本文的重点是当被收缩的边作为输入提供时，针对加权树和路径实现最优的边收缩。