3 research outputs found
Parallel Batch-Dynamic Graph Connectivity
In this paper, we study batch parallel algorithms for the dynamic
connectivity problem, a fundamental problem that has received considerable
attention in the sequential setting. The most well known sequential algorithm
for dynamic connectivity is the elegant level-set algorithm of Holm, de
Lichtenberg and Thorup (HDT), which achieves amortized time per
edge insertion or deletion, and time per query. We
design a parallel batch-dynamic connectivity algorithm that is work-efficient
with respect to the HDT algorithm for small batch sizes, and is asymptotically
faster when the average batch size is sufficiently large. Given a sequence of
batched updates, where is the average batch size of all deletions, our
algorithm achieves expected amortized work per
edge insertion and deletion and depth w.h.p. Our algorithm
answers a batch of connectivity queries in expected
work and depth w.h.p. To the best of our knowledge, our algorithm
is the first parallel batch-dynamic algorithm for connectivity.Comment: This is the full version of the paper appearing in the ACM Symposium
on Parallelism in Algorithms and Architectures (SPAA), 201
Fully-dynamic minimum spanning forest with improved worst-case update time
We give a Las Vegas data structure which maintains a minimum spanning forest
in an n-vertex edge-weighted dynamic graph undergoing updates consisting of any
mixture of edge insertions and deletions. Each update is supported in O(n^{1/2
- c}) expected worst-case time for some constant c > 0 and this worst-case
bound holds with probability at least 1 - n^{-d} where d is a constant that can
be made arbitrarily large. This is the first data structure achieving an
improvement over the O(n^{1/2}) deterministic worst-case update time of
Eppstein et al., a bound that has been standing for nearly 25 years. In fact,
it was previously not even known how to maintain a spanning forest of an
unweighted graph in worst-case time polynomially faster than Theta(n^{1/2}).
Our result is achieved by first giving a reduction from fully-dynamic to
decremental minimum spanning forest preserving worst-case update time up to
logarithmic factors. Then decremental minimum spanning forest is solved using
several novel techniques, one of which involves keeping track of
low-conductance cuts in a dynamic graph. An immediate corollary of our result
is the first Las Vegas data structure for fully-dynamic connectivity where each
update is handled in worst-case time polynomially faster than Theta(n^{1/2})
w.h.p.; this data structure has O(1) worst-case query time.Comment: Small changes to Section 1.1 and a minor fix of the analysis for
maintaining an MSF of small clusters. 61 pages, 7 figures, 3 with pseudocode.
Submitted to STOC'17. Builds on an earlier (unpublished, submitted to
FOCS'16) version by the same author which had a similar bound for
fully-dynamic connectivit