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feb18 - Illinois Institute of Technology Department of...

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Illinois Institute of Technology Department of Computer Science Lecture 9: February 18, 2009 CS 430 Introduction to Algorithms Spring Semester, 2009 1 Augmenting Balanced Trees For particular applications, it is useful to modify a standard data structure to support additional function- ality. Two examples are presented here, order-statistic trees and interval trees. 1.1 Order-Statistic Trees In Lecture 5 (February 4) we discussed the selection problem: given an array, find the k th largest element. Can we modify red-black trees to allow us to solve this problem? Indeed we can, with order-statistic trees. 1 We augment the each node in the red-black tree with its size, where size ( x ) is defined as the number of items in the subtree rooted at x . With this new information, finding the k th largest element of a tree is a simple process which can be performed in O ( height ( T )) time—and since red-black trees are of height O (log n ), the running time of the selection algorithm is O (log n ) 2 .
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