Range Minimum Query and Lowest Common Ancestor

Range Minimum Query and Lowest Common Ancestor

Introduction
Notations
Range Minimum Query (RMQ)
Trivial algorithms for RMQ
A solution
Sparse Table (ST) algorithm
Segment Trees
Lowest Common Ancestor (LCA)
A solution
Another easy solution in
Reduction from LCA to RMQ
From RMQ to LCA
An algorithm for the restricted RMQ
Conclusion

Introduction
The problem of finding the Lowest Common Ancestor (LCA) of a pair of nodes in a rooted tree has been studied more carefully in the second part of the 20th century and now is fairly basic in algorithmic graph theory. This problem is interesting not only for the tricky algorithms that can be used to solve it, but for its numerous applications in string processing and computational biology, for example, where LCA is used with suffix trees or other tree-like structures. Harel and Tarjan were the first to study this problem more attentively and they showed that after linear preprocessing of the input tree LCA, queries can be answered in constant time. Their work has since been extended, and this tutorial will present many interesting approaches that can be used in other kinds of problems as well.

Let's consider a less abstract example of LCA: the tree of life. It's a well-known fact that the current habitants of Earth evolved from other species. This evolving structure can be represented as a tree, in which nodes represent species, and the sons of some node represent the directly evolved species. Now species with similar characteristics are divided into groups. By finding the LCA of some nodes in this tree we can actually find the common parent of two species, and we can determine that the similar characteristics they share are inherited from that parent.

Range Minimum Query (RMQ) is used on arrays to find the position of an element with the minimum value between two specified indices. We will see later that the LCA problem can be reduced to a restricted version of an RMQ problem, in which consecutive array elements differ by exactly 1.

However, RMQs are not only used with LCA. They have an important role in string preprocessing, where they are used with suffix arrays (a new data structure that supports string searches almost as fast as suffix trees, but uses less memory and less coding effort).

In this tutorial we will first talk about RMQ. We will present many approaches that solve the problem -- some slower but easier to code, and others faster. In the second part we will talk about the strong relation between LCA and RMQ. First we will review two easy approaches for LCA that don't use RMQ; then show that the RMQ and LCA problems are equivalent; and, at the end, we'll look at how the RMQ problem can be reduced to its restricted version, as well as show a fast algorithm for this particular case.