lecture11 - Introduction to Algorithms...

Info iconThis preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon
Introduction to Algorithms 6.046J/18.401J/SMA5503 Lecture 11 Prof. Erik Demaine
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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:
Background image of page 2
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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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.)
Background image of page 4
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 9 i = 5 k = 2 i = 3 k = 2 i = 1 k = 1 Running time = O ( h ) = O (lg n ) for red-black trees.
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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.
Background image of page 6
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
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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.
Background image of page 8
Image of page 9
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 07/09/2009 for the course CSE 6.046J/18. taught by Professor Piotrindykandcharlese.leiserson during the Fall '04 term at MIT.

Page1 / 25

lecture11 - Introduction to Algorithms...

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online