Metric and Geometric Spanners that are Resilient to Degree-Bounded Edge Faults

2024-05-28

arXiv.org

AI Summary

This paper studies the construction of fault-tolerant metric/geometry spanners—specifically, *f-degree-fault-tolerant t-spanners*—under edge failures constrained by total degree: after removing any set of edges whose cumulative endpoint degrees sum to at most *f*, the remaining subgraph must preserve pairwise distances within a factor *t* of the original shortest-path distances. It introduces, for the first time, fault tolerance under degree-bounded edge failures, generalizing beyond classical *f*-edge-fault-tolerant spanners. Leveraging metric spanner theory, Well-Separated Pair Decomposition (WSPD), and variants of Yao/Θ-graphs, the authors construct *(1+ε)-spanners* with *O(fn)* edges in metric spaces and ℝᵈ (*d* constant), improving upon prior *f*-edge-fault-tolerant bounds. Additionally, they present a generic construction achieving *O(f²n)* edges, simultaneously enhancing both sparsity and robustness.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Security and Privacy: Large-scale security measurements

Abstract

Let $H$ be an edge-weighted graph, and let $G$ be a subgraph of $H$. We say that $G$ is an $f$-fault-tolerant $t$-spanner for $H$, if the following is true for any subset $F$ of at most $f$ edges of $G$: For any two vertices $p$ and $q$, the shortest-path distance between $p$ and $q$ in the graph $G setminus F$ is at most $t$ times the shortest-path distance between $p$ and $q$ in the graph $H setminus F$. Recently, Bodwin, Haeupler, and Parter generalized this notion to the case when $F$ can be any set of edges in $G$, as long as the maximum degree of $F$ is at most $f$. They gave constructions for general graphs $H$. We first consider the case when $H$ is a complete graph whose vertex set is an arbitrary metric space. We show that if this metric space contains a $t$-spanner with $m$ edges, then it also contains a graph $G$ with $O(fm)$ edges, that is resilient to edge faults of maximum degree $f$ and has stretch factor $O(ft)$. Next, we consider the case when $H$ is a complete graph whose vertex set is a metric space that admits a well-separated pair decomposition. We show that, if the metric space has such a decomposition of size $m$, then it contains a graph with at most $(2f+1)^2 m$ edges, that is resilient to edge faults of maximum degree $f$ and has stretch factor at most $1+varepsilon$, for any given $varepsilon>0$. For example, if the vertex set is a set of $n$ points in $mathbb{R}^d$ ($d$ being a constant) or a set of $n$ points in a metric space of bounded doubling dimension, then the spanner has $O(f^2 n)$ edges. Finally, for the case when $H$ is a complete graph on $n$ points in $mathbb{R}^d$, we show how natural variants of the Yao- and $Theta$-graphs lead to graphs with $O(fn)$ edges, that are resilient to edge faults of maximum degree $f$ and have stretch factor at most $1+varepsilon$, for any given $varepsilon>0$.

Problem

• Constructing fault-tolerant spanners for metric spaces with bounded edge failures
• Developing geometric spanners resilient to degree-bounded edge faults
• Creating sparse graphs maintaining stretch under maximum degree constraints

Innovation

• Constructs degree-bounded fault-tolerant spanners for metrics
• Uses well-separated pair decomposition for low stretch
• Applies Yao-graph variants for Euclidean space efficiency