[Paper Review] Near-Optimal $O(k)$-Robust Geometric Spanners

Buchin, Kevin, Har-Peled, Sariel+1 more|arXiv (Cornell University)|Dec 24, 2018

Computational Geometry and Mesh Generation9 references3 citations

TL;DR

This paper presents a near-optimal construction of $(1+\epsilon)k$-robust geometric spanners in $\mathbb{R}^d$ with $O(n\log^2 n \log\log n)$ edges, achieving near-linear sparsity while ensuring resilience to up to $k$ faulty nodes. The method combines well-separated pair decompositions, expander graphs, hierarchical centroid decomposition, and Willard's order-maintenance structure to bound the number of nodes affected by failures, improving upon prior $n^{\log^{O(d)} n}$ constructions.

ABSTRACT

For any constants $d\ge 1$, $ε>0$, $t>1$, and any $n$-point set $P\subset\mathbb{R}^d$, we show that there is a geometric graph $G=(P,E)$ having $O(n\log^2 n\log\log n)$ edges with the following property: For any $F\subseteq P$, there exists $F^+\supseteq F$, $|F^+| \le (1+ε)|F|$ such that, for any pair $p,q\in P\setminus F^+$, the graph $G-F$ contains a path from $p$ to $q$ whose (Euclidean) length is at most $t$ times the Euclidean distance between $p$ and $q$. In the terminology of robust spanners (Bose \et al, SICOMP, 42(4):1720--1736, 2013) the graph $G$ is a $(1+ε)k$-robust $t$-spanner of $P$. This construction is sparser than the recent constructions of Buchin, Olàh, and Har-Peled (arXiv:1811.06898) who prove the existence of $(1+ε)k$-robust $t$-spanners with $n\log^{O(d)} n$ edges.

Motivation & Objective

• To construct sparse geometric spanners resilient to node failures, where the number of affected nodes is bounded by a function of the number of failed nodes.
• To close the gap between the $\Omega(n\log n)$ lower bound and existing $n^{\log^{O(d)} n}$ upper bounds for $(1+\epsilon)k$-robust spanners.
• To achieve near-linear edge count while maintaining robustness in constant-dimensional Euclidean spaces.
• To develop a construction that is within a $O(\log n \log\log n)$ factor of optimal in any constant dimension $d \geq 1$.

Proposed method

• Construct a well-separated pair decomposition (WSPD) of the point set using a fair-split tree to partition points into well-separated pairs.
• Use expander graphs on subsets of the WSPD pairs to ensure that small subsets of nodes still have strong connectivity, enabling robustness.
• Apply hierarchical centroid decomposition (balanced separators) to recursively partition the tree structure and control the number of affected nodes.
• Integrate Willard's order-maintenance data structure to maintain dynamic orderings with stringent density constraints on smaller substructures.
• Construct two types of auxiliary graphs: one to ensure expansion from surviving nodes into larger sets (PR3), and another to maintain connectivity within subtrees (PR4).
• Combine the WSPD edges with the auxiliary expander graphs to form the final spanner, ensuring that for any failure set $F$, only $O((1+\epsilon)|F|)$ nodes are additionally affected.

Experimental results

Research questions

• RQ1 Can a $(1+\epsilon)k$-robust $t$-spanner be constructed with near-linear $O(n\log^2 n\log\log n)$ edges in constant-dimensional Euclidean space?
• RQ2 What is the minimal edge count achievable for a $k$-robust spanner in $\mathbb{R}^d$, and how close can it get to the $\Omega(n\log n)$ lower bound?
• RQ3 Can hierarchical decomposition and expander-based connectivity be combined to achieve robustness without incurring $\log^{O(d)} n$ factors in edge count?
• RQ4 How does the robustness of spanners relate to the intrinsic dimensionality of the point set, especially in dense or uniformly distributed configurations?
• RQ5 Is $O(n\log^2 n\log\log n)$ the tight upper bound for $(1+\epsilon)k$-robust spanners, or can the $\log\log n$ factor be removed?

Key findings

• The paper constructs a $(1+\epsilon)k$-robust $t$-spanner with $O(n\log^2 n\log\log n)$ edges for any $n$-point set in $\mathbb{R}^d$, $d \geq 1$, improving upon prior $n^{\log^{O(d)} n}$ constructions.
• The construction is within a factor of $O(\log n \log\log n)$ of the $\Omega(n\log n)$ lower bound for $1$-dimensional point sets, making it nearly optimal in any constant dimension.
• The method ensures that for any failure set $F$, the number of additional nodes $F^+$ affected is at most $(1+7\epsilon)|F|$, satisfying the $(1+\epsilon)k$-robustness condition.
• The use of expander graphs on hierarchical substructures ensures strong expansion properties even after failures, maintaining short paths between surviving nodes.
• The final spanner graph $G$ is formed by combining WSPD edges with auxiliary expander graphs, achieving both sparsity and robustness.
• The construction avoids the $\log^{O(d)} n$ factor by using a novel hierarchical decomposition with stricter density constraints on smaller substructures, differing from prior reductions to 1D problems.