Constant-time dynamic weight approximation for minimum spanning forest

Abstract

We give two fully dynamic algorithms that maintain a (1+ε)-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 ε>0. (1) Our deterministic algorithm takes O(W2log⁡W/ε3)worst-case update time, which is O(1) if both W and ε are constants. (2) Our randomized (Monte-Carlo style) algorithm works with high probability and runs in worst-caseO(log⁡W/ε4) update time if W=O((m⁎)1/6/log2/3⁡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.