Maintenance of a Minimum Spanning Forest in a Dynamic Plane Graph

We give an eecient algorithm for maintaining a minimum spanning forest of a plane graph subject to on-line modiications. The modii-cations supported include changes in the edge weights, and insertion 1 and deletion of edges and vertices which are consistent with the given embedding. To implement the algorithms, we develop a data structure called an edge-ordered dynamic tree, which is a variant of the dynamic tree data structure of Sleator and Tarjan. Using this data structure, our algorithm runs in O(log n) time per operation and O(n) space. The algorithm can be used to maintain the connected components of a dynamic planar graph in O(log n) time per operation. We also show that any algorithm will need (log n) amortized time per operation, given a set of machine operations that is fairly general.