Constant-time dynamic weight approximation for minimum spanning forest

We give two fully dynamic algorithms that maintain a (1 + epsilon)-approximation of the weight M of a minimum spanning forest (MSF) of an n-node graph G with edges weights in [1, W], for any epsilon > 0. (1) Our deterministic algorithm takes O(W2 log W /epsilon(3)) worst-case update time, which is O (1) if both W and E are constants. (2) Our randomized (Monte -Carlo style) algorithm works with high probability and runs in worst-case O (log W /epsilon(4)) update time if W = O((m*)(1/6)/log(2/3) n), where m* is the minimum number of edges in the graph throughout all the updates. It works even against an adaptive adversary. We complement our algorithmic results with two cell-probe lower bounds for dynamically maintaining an approximation of the weight of an MSF of a graph. (C) 2021 Elsevier Inc. All rights reserved.