We present the first sublinear-time algorithm for a distributed
message-passing network sto compute its edge connectivity λ exactly in
the CONGEST model, as long as there are no parallel edges. Our algorithm takes
O~(n1−1/353D1/353+n1−1/706) time to compute λ and a
cut of cardinality λ with high probability, where n and D are the
number of nodes and the diameter of the network, respectively, and O~
hides polylogarithmic factors. This running time is sublinear in n (i.e.
O~(n1−ϵ)) whenever D is. Previous sublinear-time
distributed algorithms can solve this problem either (i) exactly only when
λ=O(n1/8−ϵ) [Thurimella PODC'95; Pritchard, Thurimella, ACM
Trans. Algorithms'11; Nanongkai, Su, DISC'14] or (ii) approximately [Ghaffari,
Kuhn, DISC'13; Nanongkai, Su, DISC'14].
To achieve this we develop and combine several new techniques. First, we
design the first distributed algorithm that can compute a k-edge connectivity
certificate for any k=O(n1−ϵ) in time O~(nk​+D).
Second, we show that by combining the recent distributed expander decomposition
technique of [Chang, Pettie, Zhang, SODA'19] with techniques from the
sequential deterministic edge connectivity algorithm of [Kawarabayashi, Thorup,
STOC'15], we can decompose the network into a sublinear number of clusters with
small average diameter and without any mincut separating a cluster (except the
`trivial' ones). Finally, by extending the tree packing technique from [Karger
STOC'96], we can find the minimum cut in time proportional to the number of
components. As a byproduct of this technique, we obtain an O~(n)-time
algorithm for computing exact minimum cut for weighted graphs.Comment: Accepted at 51st ACM Symposium on Theory of Computing (STOC 2019