我们引入顶树作为一个新的更简单的接口，数据结构维护信息在一个完全动态的森林设计。我们展示了它们在不同应用程序中的易用性和通用性。例如，我们展示了如何保持森林中每棵树的直径、中心和中位数。森林可以通过插入和删除边以及更改顶点和边权重来更新。每次更新都需要O（log n）时间，其中n是更新涉及的树的大小。此外，我们展示了如何支持最近的共同祖先查询和水平祖先查询任意根在O（log n）的时间。最后，对于标记和未标记的顶点，我们展示了如何计算到最近的标记顶点的距离。后者可应用于一般图中最近标记点的近似，从而可应用于最短路度量的静态优化问题。从技术上讲，顶树容易用Frederickson的拓扑树[用于动态2-Edge-Connectivity和k最小生成树的矛盾数据结构，SIAM J. Comput.第26（2）页。484-538，1997]或使用Sleator和Tarjan的动态树[A数据结构for Dynamic Trees. J. Comput.《系统科学》，第26卷，第3页。362-391，1983]。然而，我们声称，该接口是简单的许多应用程序，实际上我们的新的界限是二次改进以前的界限，他们存在。
英文摘要：
We introduce top trees as a design of 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 later 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 topology trees [Ambivalent Data Structures for Dynamic 2-Edge-Connectivity and k Smallest Spanning Trees, SIAM J. Comput. 26 (2) pp. 484-538, 1997] or with Sleator and Tarjan's dynamic trees [A Data Structure for Dynamic Trees. J. Comput. Syst. Sc. 26 (3) pp. 362-391, 1983]. 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.
