[Paper Review] Diameter Spanner, Eccentricity Spanner, and Approximating Extremal Graph Distances: Static, Dynamic, and Fault Tolerant.

Keerti Choudhary, Omer Gold|arXiv (Cornell University)|Dec 4, 2018

Complexity and Algorithms in Graphs30 references6 citations

TL;DR

This paper presents the first non-trivial dynamic and fault-tolerant algorithms for maintaining a $(1.5 + \epsilon)$-approximation of graph diameter and $2$-approximation of vertex eccentricities in directed weighted graphs. It introduces extremal-distance spanners, including $(1.5)$-diameter-spanners with $O(n^{1.5})$ edges, and achieves near-optimal amortized update times, nearly matching static bounds for $1.5$-approximation of diameter.

ABSTRACT

The diameter, vertex eccentricities, and the radius of a graph are some of the most fundamental graph parameters. Roditty and Williams [STOC 2013] gave an $O(m\sqrt{n})$ time algorithm for computing a 1.5 approximation of graph diameter. We present the first non-trivial algorithm for maintaining `< 2'- approximation of graph diameter in dynamic setting. Our algorithm maintain a $(1.5+\epsilon)$ approximation of graph diameter that takes amortized update time of $O(\epsilon^{-1}n^{1.25})$ in partially dynamic setting. For graphs whose diameter remains bounded by some large constant, the total amortized time of our algorithm is $O(\epsilon^{-2}\sqrt{n})$, which almost matches the best known bound for static $1.5$-approximation of diameter. Backurs et al. [STOC 2018] gave an $ ilde O(m\sqrt{n})$ time algorithm for computing 2-approximation of eccentricities. They also showed that no $O(n^{2-o(1)})$ time algorithm can achieve an approximation factor better than $2$ for graph eccentricities, unless SETH fails. We present the $ ilde O(m)$ time algorithm for computing $2$-approximation of vertex eccentricities in directed weighted graphs. We also present fault tolerant data-structures for maintaining $1.5$-diameter and $2$-eccentricities. We initiate the study of Extremal Distance Spanners. Given a graph G=(V,E), a subgraph H=(V,E0) is defined to be a {\em $t$-diameter-spanner} if the diameter of $H$ is at most $t$ times the diameter of $G$. We show that for any $n$-vertex directed graph $G$ we can compute a sparse subgraph $H$ which is a $(1.5)$-diameter-spanner of $G$ and contains at most $O(n^{1.5})$ edges. We also show that this bound is tight for graphs whose diameter is bounded by $n^{1/4-\epsilon}$. We present several other extremal-distance spanners with various size-stretch trade-offs. Finally, we extensively study these objects in the dynamic and fault-tolerant settings.

Motivation & Objective

• To develop the first dynamic and fault-tolerant algorithms for maintaining approximate graph diameter and vertex eccentricities.
• To address the gap in dynamic algorithms for extremal graph parameters like diameter and eccentricity, which had not been studied beyond static settings.
• To introduce and study extremal-distance spanners—subgraphs that preserve approximate extremal distances—with tight size-stretch trade-offs.
• To achieve near-optimal amortized update times for dynamic diameter and eccentricity approximation, matching known static bounds in key regimes.

Proposed method

• Designs a dynamic algorithm that maintains a $(1.5 + \epsilon)$-approximation of graph diameter in partially dynamic graphs with amortized update time $O(\epsilon^{-1}n^{1.25})$.
• Introduces a novel construction of $(1.5)$-diameter-spanners with $O(n^{1.5})$ edges for any $n$-vertex directed graph, proven to be tight for graphs with diameter $O(n^{1/4 - \epsilon})$.
• Employs a sparse subgraph construction based on hopsets and distance oracles to maintain approximate distances under edge updates.
• Adapts techniques from static $1.5$-approximation of diameter and $2$-approximation of eccentricities to the dynamic and fault-tolerant settings.
• Uses fault-tolerant data structures to maintain spanners under edge or vertex failures, ensuring resilience in dynamic environments.
• Leverages the tight connection between extremal-distance spanners and dynamic approximation algorithms to achieve near-optimal bounds.

Experimental results

Research questions

• RQ1 Can we maintain a $(1.5 + \epsilon)$-approximation of graph diameter in dynamic graphs with near-optimal update time?
• RQ2 What is the smallest possible size of a subgraph that preserves a $1.5$-approximation of the original graph's diameter?
• RQ3 Can we achieve $\tilde{O}(m)$ time for $2$-approximation of vertex eccentricities in directed weighted graphs, even in dynamic settings?
• RQ4 How do extremal-distance spanners—such as $t$-diameter-spanners—behave under dynamic updates and faults?
• RQ5 What are the tight size-stretch trade-offs for extremal-distance spanners in directed graphs?

Key findings

• The paper presents a dynamic algorithm that maintains a $(1.5 + \epsilon)$-approximation of graph diameter with amortized update time $O(\epsilon^{-1}n^{1.25})$ in partially dynamic graphs.
• For graphs with bounded diameter (by a large constant), the total amortized time is $O(\epsilon^{-2}\sqrt{n})$, nearly matching the best known static bound for $1.5$-approximation of diameter.
• A novel $(1.5)$-diameter-spanner construction is introduced, with $O(n^{1.5})$ edges, which is proven tight for graphs with diameter $O(n^{1/4 - \epsilon})$.
• The paper gives an $\tilde{O}(m)$-time algorithm for computing a $2$-approximation of vertex eccentricities in directed weighted graphs.
• Extremal-distance spanners are formally introduced, with a hierarchy of size-stretch trade-offs, including $t$-diameter-spanners and $t$-eccentricity-spanners.
• Fault-tolerant data structures are developed that maintain $1.5$-diameter and $2$-eccentricity approximations under edge or vertex failures.