# lec11 - Introduction to Algorithms 6.046J/18.401J LECTURE...

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October 24, 2005 Copyright © 2001-5 by Erik D. Demaine and Charles E. Leiserson L11.1 Introduction to Algorithms 6.046J/18.401J L ECTURE 11 Augmenting Data Structures Dynamic order statistics Methodology Interval trees Prof. Charles E. Leiserson

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October 24, 2005 Copyright © 2001-5 by Erik D. Demaine and Charles E. Leiserson L11.2 Dynamic order statistics OS-S ELECT ( i , S ) : returns the i th smallest element in the dynamic set S . OS-R ANK ( x , S ) : returns the rank of x S in the sorted order of S ’s elements. I DEA : Use a red-black tree for the set S , but keep subtree sizes in the nodes. key size key size Notation for nodes:
October 24, 2005 Copyright © 2001-5 by Erik D. Demaine and Charles E. Leiserson L11.3 Example of an OS-tree M 9 M 9 C 5 C 5 A 1 A 1 F 3 F 3 N 1 N 1 Q 1 Q 1 P 3 P 3 H 1 H 1 D 1 D 1 size [ x ] = size [ left [ x ]] + size [ right [ x ]] + 1

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October 24, 2005 Copyright © 2001-5 by Erik D. Demaine and Charles E. Leiserson L11.4 Selection Implementation trick: Use a sentinel (dummy record) for NIL such that size [ NIL ] = 0 . OS-S ELECT ( x , i ) i th smallest element in the subtree rooted at x k size [ left [ x ]] + 1 k = rank( x ) if i = k then return x if i < k then return OS-S ELECT ( left [ x ] , i ) else return OS-S ELECT ( right [ x ] , i – k ) (OS-R ANK is in the textbook.)
October 24, 2005 Copyright © 2001-5 by Erik D. Demaine and Charles E. Leiserson L11.5 Example OS-S ELECT ( root , 5) M 9 M 9 C 5 C 5 A 1 A 1 F 3 F 3 N 1 N 1 Q 1 Q 1 P 3 P 3 H 1 H 1 D 1 D 1 i = 5 k = 6 M 9 M i = 5 k = 2 i = 3 k = 2 i = 1 k = 1 Running time = O ( h ) = O (lg n ) for red-black trees.

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October 24, 2005 Copyright © 2001-5 by Erik D. Demaine and Charles E. Leiserson L11.6 Data structure maintenance Q. Why not keep the ranks themselves in the nodes instead of subtree sizes? A. They are hard to maintain when the red-black tree is modified. Modifying operations: I NSERT and D ELETE . Strategy: Update subtree sizes when inserting or deleting.
October 24, 2005 Copyright © 2001-5 by Erik D. Demaine and Charles E. Leiserson L11.7 Example of insertion M 9 M 9 C 5 C 5 A 1 A 1 F 3 F 3 N 1 N 1 Q 1 Q 1 P 3 P 3 H 1 H 1 D 1 D 1 I NSERT (“K”) 10 10 6 6 4 4 2 2 K 1 K 1

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lec11 - Introduction to Algorithms 6.046J/18.401J LECTURE...

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