Stephen Alstrup

Nearest common ancestors: a survey and a new distributed algorithm

Several papers describe linear time algorithms to preprocess a tree, such that one can answer subsequent nearest common ancestor queries in constant time. Here, we survey these algorithms and related results. A common idea used by all the algorithms for the problem is that a solution for complete binary trees is straightforward. Furthermore, for complete binary trees we can easily solve the problem in a distributed way by labeling the nodes of the tree such that from the labels of two nodes alone one can compute the label of their nearest common ancestor. Whether it is possible to distribute the data structure into short labels associated with the nodes is important for several applications such as routing. Therefore, related labeling problems have received a lot of attention recently.Previous optimal algorithms for nearest common ancestor queries work using some mapping from a general tree to a complete binary tree. However, it is not clear how to distribute the data structures obtained using these mappings. We conclude our survey with a new simple algorithm that labels the nodes of a rooted tree such that from the labels of two nodes alone one can compute in constant time the label of their nearest common ancestor. The labels assigned by our algorithm are of size

Maintaining information in fully dynamic trees with top trees

O(log n)

Optimal static range reporting in one dimension

bits where

Nearest Common Ancestors: A Survey and a New Algorithm for a Distributed Environment

n

Improved Algorithms for Finding Level Ancestors in Dynamic Trees

is the number of nodes in the tree. The algorithm runs in

Maintaining Center and Median in Dynamic Trees

O(n)

Optimal Pointer Algorithms for Finding Nearest Common Ancestors in Dynamic Trees

time.

Generalized Dominators for Structured Programs

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 Fredericksons [1997a] topology trees or with Sleator and Tarjans [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.

Time and Space Efficient Multi-method Dispatching

We consider static one dimensional range searching problems. These problems are to build static data structures for an integer set <italic>S</italic> subseteq <italic>U</italic>, where <italic>U</italic> = {0,1,dots,2^<italic>w</italic>-1}, which support various queries for integer intervals of <italic>U</italic>. For the query of reporting all integers in <italic>S</italic> contained within a query interval, we present an optimal data structure with linear space cost and with query time linear in the number of integers reported. This result holds in the unit cost RAM model with word size <italic>w</italic> and a standard instruction set. We also present a linear space data structure for approximate range counting. A range counting query for an interval returns the number of integers in <italic>S</italic> contained within the interval. For any constant ε>0, our range counting data structure returns in constant time an approximate answer which is within a factor of at most 1+ε of the correct answer.

Improved labeling scheme for ancestor queries

Several papers describe linear time algorithms to preprocess a tree, in order to answer subsequent nearest common ancestor queries in constant time. Here, we survey these algorithms and related results. Whereas previous algorithms produce a linear space data structure, in this paper we address the problem of distributing the data structure into short labels associated with the nodes. Localized data structures received much attention recently as they play an important role for distributed applications such as routing. We conclude our survey with a new simple algorithm that labels in O(n) time all the nodes of an n-node rooted tree such that from the labels of any two nodes alone one can compute in constant time the label of their nearest common ancestor. The labels assigned by our algorithm are of size O(log n) bits.