We design top trees as a new simpler interface for data structures maintaining information in a fully dynamic forest. We demonstrate how easy and versatile they are to use on a host of different applications. For example, we show how to maintain the diameter, center, and median of each tree in the forest. The forest can be updated by insertion and deletion of edges and by changes to vertex and edge weights. Each update is supported in O (log n ) time, where n is the size of the tree(s) involved in the update. Also, we show how to support nearest common ancestor queries and level ancestor queries with respect to arbitrary roots in O (log n ) time. Finally, with marked and unmarked vertices, we show how to compute distances to a nearest marked vertex. The latter has applications to approximate nearest marked vertex in general graphs, and thereby to static optimization problems over shortest path metrics.Technically speaking, top trees are easily implemented either with Frederickson's [1997a] topology trees or with Sleator and Tarjan's [1983] dynamic trees. However, we claim that the interface is simpler for many applications, and indeed our new bounds are quadratic improvements over previous bounds where they exist.

Deterministic fully dynamic graph algorithms are presented for connectivity, minimum spanning tree, 2-edge connectivity, and biconnectivity. Assuming that we start with no edges in a graph with n vertices, the amortized operation costs are O (log 2 n ) for connectivity, O (log 4 n ) for minimum spanning forest, 2-edge connectivity, and O (log 5 n ) biconnectivity.

Deterministic fully dynamic graph algorithms are presented for connectivity, minimum spanning forest, a-edge connectivity, and biconnectivity.Assuming that we start with no edges in a graph with n vertices, the amortized operation cost0 arc O(log2 n) for connectivity and O(log4 n) for minimum spanning forest, 2+dgeconnectivity, and biconnectity.