Parallel Batch-Dynamic Algorithms for Spanners, and Extensions

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 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 Õ(n1+1/k) edges, for any k ≥ 1. • Our first extension maintains a sparse spanner with only O(n) edges, and Õ(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 Õ(n) edges.