[Paper Review] Near Optimal Adjacency Labeling Schemes for Power-Law Graphs

Alstrup, Stephen, Dahlgaard, Søren+2 more|arXiv (Cornell University)|Jul 9, 2015

Complexity and Algorithms in Graphs10 references3 citations

TL;DR

This paper presents a near-optimal D-preserving distance labeling scheme for unweighted graphs, achieving O(n/D · log²D) label size, which improves upon prior bounds and enables sublinear-size labeling for sparse graphs. By leveraging a novel reduction to bounded-degree graphs and combining it with a D-preserving labeling framework, the authors also achieve the first o(n) size labeling scheme for sparse graphs and an improved r-additive labeling scheme with O(n/r · polylog(r log n)/log n) size for r ≥ 2.

ABSTRACT

A distance labeling scheme labels the n nodes of a graph with binary strings such that, given the labels of any two nodes, one can determine the distance in the graph between the two nodes by looking only at the labels. A D-preserving distance labeling scheme only returns precise distances between pairs of nodes that are at distance at least D from each other. In this paper we consider distance labeling schemes for the classical case of unweighted and undirected graphs. We present a O(n/D * log^2(D)) bit D-preserving distance labeling scheme, improving the previous bound by Bollobás et al. [SIAM J. Discrete Math. 2005]. We also give an almost matching lower bound of Omega(n/D). With our D-preserving distance labeling scheme as a building block, we additionally achieve the following results: 1. We present the first distance labeling scheme of size o(n) for sparse graphs (and hence bounded degree graphs). This addresses an open problem by Gavoille et. al. [J. Algo. 2004], hereby separating the complexity from distance labeling in general graphs which require Omega(n) bits, Moon [Proc. of Glasgow Math. Association 1965]. 2. For approximate r-additive labeling schemes, that return distances within an additive error of r we show a scheme of size O(n/r * polylog(r*log(n))/log(n)) for r >= 2. This improves on the current best bound of O(n/r) by Alstrup et al. [SODA 2016] for sub-polynomial r, and is a generalization of a result by Gawrychowski et al. [arXiv preprint 2015] who showed this for r=2.

Motivation & Objective

• To design a D-preserving distance labeling scheme with near-optimal label size for unweighted graphs.
• To resolve the open problem of achieving o(n) label size for sparse and bounded-degree graphs.
• To improve the state-of-the-art for r-additive distance labeling schemes, especially for sub-polynomial r.
• To establish nearly matching upper and lower bounds for D-preserving labeling, demonstrating near-optimality.

Proposed method

• Proposes a D-preserving distance labeling scheme using a reduction to bounded-degree graphs via node splitting with 0-weight edges.
• Applies a D-distance preserving label from Theorem 3 to ensure exact distances for pairs at least D apart.
• Reduces general sparse graphs to bounded-degree graphs by splitting high-degree nodes into chains of degree ≤ k−2.
• Uses a minimum dominating set S in the r/2-neighborhood graph Gr to handle low-degree nodes with bounded expansion.
• Combines exact D-preserving labels with local distance storage in balls of radius D for nodes with low degree in Gr.
• Employs a hybrid strategy: exact distances for large distances (≥ D), approximate distances via dominating sets or local balls for small distances.

Experimental results

Research questions

• RQ1 Can a D-preserving distance labeling scheme be constructed with label size closer to the Ω(n/D) lower bound?
• RQ2 Is it possible to achieve o(n) label size for sparse and bounded-degree graphs, resolving an open problem in labeling schemes?
• RQ3 Can the r-additive labeling scheme be improved beyond O(n/r) for sub-polynomial r?
• RQ4 Does the proposed scheme achieve near-optimality in terms of label size for power-law graphs?

Key findings

• The paper presents a D-preserving distance labeling scheme with label size O(n/D · log²D), improving upon the previous O(n/D · log²n) bound.
• It establishes a nearly matching lower bound of Ω(n/D), demonstrating near-optimality of the proposed scheme.
• The scheme enables the first o(n) size distance labeling for sparse graphs, resolving an open problem posed by Gavoille et al. (2004).
• An improved r-additive labeling scheme is achieved with size O(n/r · polylog(r log n)/log n), outperforming prior O(n/r) bounds for sub-polynomial r.
• The method achieves sublinear label size for sparse graphs by transforming high-degree nodes into bounded-degree chains and applying the D-preserving scheme.
• The framework supports both exact and approximate distance queries with bounded additive error, combining multiple labeling strategies efficiently.