Distance-Preserving Graph Compression Techniques

TL;DR

This paper tackles distance-preserving graph compression by focusing on optimally redistributing edge weights after contracting a given set of edges, with a rigorous emphasis on weighted paths and trees. It develops tight bounds and exact optimal strategies for a variety of contraction scenarios on paths (single edge, independent edges, contiguous subpaths) and trees (single edge via a marking problem, with equal-sized and varying-size subtree analyses; fractional versus integral markings). A key contribution is the demonstration that, in many cases, optimality is achieved by marking only neighboring edges on one side of the contracted edge, and in trees the optimal choice reduces to a side-selective, linear-time procedure. The results advance distance-preserving compression by providing provable guarantees on the error and efficient algorithms, while outlining open problems for general graphs and alternative error models with potential practical impact on scalable graph analytics.

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.

Figures (17)

• Figure 1: The path used as the running example in Section \ref{['prel']}: (a) A path of 8 vertices, with regular edges denoted by $e_i, w_i=w(e_i)$, and the contracted edge highlighted in red and denoted by $e^*=(u_1,v_1), w^*=w(e^*)$, and (b) The same path after contracting $e^*$ and marking $e_3$ by setting $w^{\prime} (e_3)=w_3+w^*$. In this example, $n_L=n_R$, this is not always the case.
• Figure 2: Merging a single edge $e^*=(v_2,v_3)$ with weight $B$ in a path of $n$ vertices, with $n_L \geq 0$, $n_R \geq 0$ vertices on the left of $v_1$ and the right of $v_4$ respectively ($n_L +n_R=n-2$) (a) The original graph before merging $v_2$ and $v_3$ into a supernode. The neighbouring edges of $e^*$ have weights $B$ and $C$. (b) The modified path after merging $v_2$ and $v_3$ into a supernode.
• Figure 3: The figure used in the proof of Lemma \ref{['construction']}, the vertices of $V_L$ and $V_R$ are depicted in red and blue respectively. (a) The original graph, before contracting $e^*= (v_{n_1 +1}, v_{n_1 +2}$) with weight $w^*$, (b) An arbitrary weight redistribution which is transformed in the proof of Lemma \ref{['construction']} to the redistribution of Figure \ref{['fig12']}-(b).
• Figure 4: The construction method of Lemma \ref{['construction']}, (a) Before applying the construction and (b) After applying the construction
• Figure 5: Case 2 in the proof of Lemma \ref{['construction']}: (a) The original graph (b) The original weight redistribution (c) After applying the construction method of Lemma \ref{['construction']}. The affected shortest path of Case 2 is highlighted in red.

Theorems & Definitions (62)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Definition 7
• Definition 8
• Lemma 1
• proof