Fully Dynamic Connectivity: Upper and Lower Bounds

Keywords and Synonyms

Dynamic connected components; Dynamic spanning forests        

Problem Definition

The problem is concerned with efficiently maintaining information about connectivity in a dynamically changing graph. A dynamic graph algorithm maintains a given property $\mathcal{P}$ on a graph subject to dynamic changes, such as edge insertions, edge deletions and edge weight updates. A dynamic graph algorithm should process queries on property $\mathcal{P}$ quickly, and perform update operations faster than recomputing from scratch, as carried out by the fastest static algorithm. An algorithm is said to be fully dynamic if it can handle both edge insertions and edge deletions. A partially dynamic algorithm can handle either edge insertions or edge deletions, but not both: it is incremental if it supports insertions only, and decremental if it supports deletions only.

In the fully dynamic connectivity problem, one wishes to maintain an undirected graph $G = (V, E)$under an...