11-Augmenting_Data_Structures

11-Augmenting_Data_Structures - Algorithms LECTURE 11...

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Algorithms L11.1 Professor Ashok Subramanian L ECTURE 11 Augmenting Data Structures Dynamic order statistics Methodology Interval trees Algorithms
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Algorithms 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 Notation for nodes:
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Algorithms L11.3 Example of an OS-tree M 9 C 5 A 1 F 3 N 1 Q 1 P 3 H 1 D 1 size [ x ] = size [ left [ x ]] + size [ right [ x ]] + 1
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Algorithms L11.4 Selection 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 ) Implementation trick: Use a sentinel (dummy record) for NIL such that size [ NIL ] = 0 . (OS-R ANK is in the textbook.)
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Algorithms L11.5 Example A 1 N 1 Q 1 P 3 D 1 OS-S ELECT ( root , 5) i = 5 k = 6 M 9 C 5 i = 5 k = 2 i = 3 k = 2 F 3 i = 1 k = 1 H 1 Running time = O ( h ) = O (lg n ) for red-black trees.
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Algorithms 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.
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Algorithms L11.7 Example of insertion A 1 N 1 Q 1 P 3 D 1 I NSERT (“K”) M 10 C 6 F 4 H 2 K 1
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Algorithms L11.8 Handling rebalancing Don’t forget that RB-I NSERT
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This note was uploaded on 04/29/2011 for the course IT 201 taught by Professor K.v.arya during the Spring '11 term at IIT Kanpur.

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11-Augmenting_Data_Structures - Algorithms LECTURE 11...

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