A maximal independent set (MIS) can be maintained in an evolving m-edge graph by simply recomputing it from scratch in O(m) time after each update. But can it be maintained in time sublinear in m in fully dynamic graphs? We answer this fundamental open question in the affirmative. We present a deterministic algorithm with amortized update time O(min{∆, m 3/4 }), where ∆ is a fixed bound on the maximum degree in the graph and m is the (dynamically changing) number of edges. We further present a distributed implementation of our algorithm with O(min{∆, m 3/4 }) amortized message complexity, and O(1) amortized round complexity and adjustment complexity (the number of vertices that change their output after each update). This strengthens a similar result by Censor-Hillel, Haramaty, and Karnin (PODC'16) that required an assumption of a non-adaptive oblivious adversary.
For instance, the data structure in REF maintains the maximal independent set of the graph deterministically in O(m 3/4 ) amortized update time (m being the dynamic number of edges).
Fully dynamic maximal independent set with sublinear update time
STOC 2018
Computer Science Mathematics