Maintaining information in fully dynamic trees with top trees

Abstract

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.This article includes work presented at ICALP'97 [Alstrup et al. 1997] and SWAT'00 [Alstrup et al. 2000].

Efficient bufferless packet switching on trees and leveled networks

“…This explains the interest that trees have generated in the literature (see, for example, [2,3,30,37,41,46,54]). Furthermore, as articulated by Leighton [30], a spanning tree can be used to send packets in an arbitrary network.…”

“…Hot-potato algorithms have been studied for specific network multiprocessor architectures such as the 2-dimensional mesh and torus [6,9,17,19,20,23,24,28,29,39,49], the ddimensional mesh [10,11,15], the hypercube [14,16,24,26,42], trees [3,22,46], and Vertex symmetric networks [35]. (Multiprocessor architectures are extensively covered in [30].)…”

LS(Graph): a constraint-based local search for constraint optimization on trees and paths

“…For facilitating manipulations on dynamic trees, the trees are implicitly stored as rooted trees. Several well-known data structures have been proposed for representing dynamic trees, for instance, ST-trees [57,58], topology trees [33], ET-trees [36], top trees [6,59], and RC-trees [1] (and the references therein). These data structures maintain a forest of dynamic rooted trees, supporting update actions (e.g., link and cut) and some queries (e.g., minimum (maximum) cost edge, node on a path, nearest common ancestors of two nodes, medians, centers of a tree) in O(log n) time per operation where n is the number of vertices of the given graph.…”

“…Lines 3-4 compute the set of connected components CC of the graph generated from the current graph by removing all edges E of paths of S i . For each graph G i of these connected components and each set of commodities T i that belong to G i , we perform recursively the LS-Recursive method (see lines [6][7][8].…”

“…We have e 2 belongs to pathtr(sv 2 , su 2 ) (7). From (6) and (7) we have that e 2 belongs to pathtr 1 (x, su 2 ) (8). From (4) and (8), we have e 2 ∈ path tr 1 (nca tr 1 (s, v 2 ), nca tr 1 (s, u 2 ).…”