Workefficient Batchincremental Minimum Spanning Trees with Applications to the Sliding Window Model
Abstract
Algorithms for dynamically maintaining minimum spanning trees (MSTs) have received much attention in both the parallel and sequential settings. While previous work has given optimal algorithms for dense graphs, all existing parallel batchdynamic algorithms perform polynomial work per update in the worst case for sparse graphs. In this paper, we present the first workefficient parallel batchdynamic algorithm for incremental MST, which can insert $\ell$ edges in $O(\ell \log(1+n/\ell))$ work in expectation and $O(\text{polylog}(n))$ span w.h.p. The key ingredient of our algorithm is an algorithm for constructing a compressed path tree of an edgeweighted tree, which is a smaller tree that contains all pairwise heaviest edges between a given set of marked vertices. Using our batchincremental MST algorithm, we demonstrate a range of applications that become efficiently solvable in parallel in the slidingwindow model, such as graph connectivity, approximate MSTs, testing bipartiteness, $k$certificates, cyclefreeness, and maintaining sparsifiers.
 Publication:

arXiv eprints
 Pub Date:
 February 2020
 arXiv:
 arXiv:2002.05710
 Bibcode:
 2020arXiv200205710A
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Distributed;
 Parallel;
 and Cluster Computing