Parallel Batch-Dynamic Algorithms for Spanners, and Extensions

2025-07-08

AI Summary

This paper addresses the **parallel efficient maintenance** of graph spanners and sparsifiers under **batched edge updates** (insertions and deletions) in dynamic graphs. We propose the first parallel dynamic framework supporting **batched, bidirectional updates**, built upon the distributed pointer machine model and integrating multi-level graph decomposition, dynamic clustering, lazy propagation, and Laplacian solvers for spectral sparsification. Our theoretical contributions are: (1) the first parallel dynamic algorithm for $(2k-1)$-spanners with $ ilde{O}(n^{1+1/k})$ edges; (2) an $O(n)$-edge spanner with stretch $ ilde{O}(log n)$; and (3) a novel $t$-bundle spanner unifying maintenance of cut- and spectral-sparsifiers. The algorithm achieves **near-linear amortized work** and **low parallel depth**, making it suitable for real-time analysis of large-scale graphs.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Data Mining & Knowledge Management: Scalability, Parallel & Distributed Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

This paper presents the first parallel batch-dynamic algorithms for computing spanners and sparsifiers. Our algorithms process any batch of edge insertions and deletions in an $n$-node undirected graph, in $ ext{poly}(log n)$ depth and using amortized work near-linear in the batch size. Our concrete results are as follows: - Our base algorithm maintains a spanner with $(2k-1)$ stretch and $ ilde{O}(n^{1+1/k})$ edges, for any $kgeq 1$. - Our first extension maintains a sparse spanner with only $O(n)$ edges, and $ ilde{O}(log n)$ stretch. - Our second extension maintains a $t$-bundle of spanners -- i.e., $t$ spanners, each of which is the spanner of the graph remaining after removing the previous ones -- and allows us to maintain cut/spectral sparsifiers with $ ilde{O}(n)$ edges.

Problem

• Develops parallel batch-dynamic algorithms for spanners
• Maintains spanners with near-linear work in batch size
• Extends to sparse spanners and cut/spectral sparsifiers

Innovation

• Parallel batch-dynamic algorithms for spanners
• Poly-log depth with near-linear work
• Maintains spanners and sparsifiers efficiently