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Algorithms for dynamically maintaining minimum spanning trees (MSTs) havereceived much attention in both the parallel and sequential settings. Whileprevious work has given optimal algorithms for dense graphs, all existingparallel batch-dynamic algorithms perform polynomial work per update in theworst case for sparse graphs. In this paper, we present the firstwork-efficient parallel batch-dynamic algorithm for incremental MST, which caninsert $\ell$ 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 analgorithm for constructing a compressed path tree of an edge-weighted tree,which is a smaller tree that contains all pairwise heaviest edges between agiven set of marked vertices. Using our batch-incremental MST algorithm, wedemonstrate a range of applications that become efficiently solvable inparallel in the sliding-window model, such as graph connectivity, approximateMSTs, testing bipartiteness, $k$-certificates, cycle-freeness, and maintainingsparsifiers.

Work-efficient Batch-incremental Minimum Spanning Trees with Applications to the Sliding Window Model