Dynamic Trees:Conclusions.

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Conclusions

In this chapter we have surveyed data structures for maintaining properties of dynamically changing trees, focusing our attention on linking and cutting trees [15], topology trees [6], top trees [1], ET trees [10, 18], and reachability trees [5]. We have shown that these data structures typically support updates and many different kinds of queries in logarithmic (amortized or worst-case) time. Hence, problems such as tree membership, maintaining the center or diameter of a tree, finding the minimum cost on a given path can be solved in O(log n) time in a fully dynamic setting.

All the data structures for maintaining properties of dynamically changing trees are not only important and interesting on their own, but are often used as building blocks by many dynamic graph algorithms, as we will see in Chapter 36.

Some of these data structures, such as the union find data structures and the linking and cutting trees of Sleator and Tarjan have myriads of applications in other problems, such as implementing property grammars, computational geometry problems, testing equivalence of finite state machines, computing the longest common subsequence of two sequences, performing unification in logic programming and theorem proving, finding minimum spanning trees, and maximum flow algorithms. Since all these problems are outside the scope of this survey, we have not mentioned these applications and have restricted our attention to the applications of these data structures to dynamic tree and graph algorithms only.

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