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lecture11

# lecture11 - Introduction to Algorithms...

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Introduction to Algorithms 6.046J/18.401J/SMA5503 Lecture 11 Prof. Erik Demaine

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Introduction to Algorithms Day 20 L11.2 © 2001 by Charles E. Leiserson 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:
Introduction to Algorithms Day 20 L11.3 © 2001 by Charles E. Leiserson 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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Introduction to Algorithms Day 20 L11.4 © 2001 by Charles E. Leiserson 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.)
Introduction to Algorithms Day 20 L11.5 © 2001 by Charles E. Leiserson Example 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 OS-S ELECT ( root , 5) i = 5 k = 6 9 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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Introduction to Algorithms Day 20 L11.6 © 2001 by Charles E. Leiserson 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.
Introduction to Algorithms Day 20 L11.7 © 2001 by Charles E. Leiserson 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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Introduction to Algorithms Day 20 L11.8 © 2001 by Charles E. Leiserson Handling rebalancing Don’t forget that RB-I NSERT and RB-D ELETE may also need to modify the red-black tree in order to maintain balance.
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