Red black trees1109

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A red-black tree is a type of self balancing binary search tree, with one extra attribute for each node: the colour, which is either red or black.

A red–black tree is a binary search tree that inserts and deletes in such a way that the tree is always reasonably balanced.

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Red black trees1109

  1. 1. PRAVIN D’SILVA 1109 MCA , GOA UNIVERSITY
  2. 2. Acknowledgement I would like express my sincere gratitude to Dr. Jyoti Pawar, for the support and guidance as well as the feedback she has provided throughout this study.
  3. 3. Definition • A red-black tree is a type of self balancing binary search tree, with one extra attribute for each node: the colour, which is either red or black. • A red–black tree is a binary search tree that inserts and deletes in such a way that the tree is always reasonably balanced.
  4. 4. Balanced binary tree • A non-empty binary tree T is balanced if: 1) Left subtree of T is balanced 2) Right subtree of T is balanced 3) The difference between heights of left subtree and right subtree is not more than 1.
  5. 5. Red-Black tree • Recall binary search tree ▫ Key values in the left subtree <= the node value ▫ Key values in the right subtree >= the node value • Operations: ▫ insertion, deletion ▫ Search, maximum, minimum, successor, predecessor. ▫ O(h), h is the height of the tree.
  6. 6. Red-black trees • Definition: a binary tree, satisfying: 1. 2. 3. 4. 5. • • Every node is red or black The root is black Every leaf is NIL and is black If a node is red, then both its children are black For each node, all paths from the node to descendant leaves contain the same number of black nodes. Purpose: keep the tree balanced. Other balanced search tree: ▫ AVL tree, 2-3-4 tree, Splay tree, Treap
  7. 7. Example of red black trees Each Red Node can have only Black children
  8. 8. Not a red black tree
  9. 9. Black height of a red black tree • Black height does not count the root itself. • we use "nil leaves" or "null leaves", which contain no data and merely serve to indicate where the tree ends 7 BH=2 BH=2 3 BH=1 BH=0 22 BH=1 8 10 BH=1, ignore red nodes!! NIL pointer
  10. 10. Complexity • The persistent version of red-black trees requires O(log n) space for each insertion or deletion.
  11. 11. Some operations in log(n) • Search, minimum, maximum, successor, predecessor. • Let us discuss insert and delete.
  12. 12. Inserting in a red black tree • Let k be the key being inserted. • As in case of a BST, we first search for k; this gives us the place where we have to insert k. • We create a new node with key k and insert it at this place • The new node is colored red.
  13. 13. • Since inserted node is Red , the black height of the tree remains unchanged. • However if the parent node is also red, then we have a double red problem k k NO PROBLEM DOUBLE RED PROBLEM
  14. 14. INSERTION : CASE 1 • Parent of node (a) must be black (b). • The other child of (b) is black (c) . • Rotation corrects the defect. b c a b a k c k
  15. 15. Insertion case 2 • Parent of node (a) is red • Parent of (a) must be black (b) • The other child of (b) is also red (c) c|b|a b b h+1 c h c a k h k h h h h+1 a h h h h h
  16. 16. Deletion • To delete a node we proceed as in a BST • Hence the node which is deleted is the parent of an external node • Hence it is either a leaf or parent of a leaf • Steps: ▫ ▫ ▫ ▫ Search Identify Leaf, then delete If internal node, find successor or predecessor, swap, then delete the successor or predecessor
  17. 17. Consider a red black a a b c b c
  18. 18. Case 1 a c b a h-1 h-1 h-1 h-1 h-1 h-1 h-1 h-1 b a a b h-1 h-1 h-1 h-1 c | h-1 h-1 h-1 h-1 | b| c | a
  19. 19. Case 2 • • • • If parent is a red node (a) Then it has a child (b) which must be black If (b) has no red child Recoloring solves the problem a a b b h-1 h-1 h-1 h-1 h-1 h-1
  20. 20. Case 3 a a b b h-1 h-1 h-1 h-1 h-1 h-1
  21. 21. Deletion Summary • In most cases, deletion can be completed by simple rotation/ coloring. • In case 3, the height of the subtree reduces and so we need to proceed up the tree • But in case 3, we only recolor the nodes • Thus, if we proceed up the tree, then we only need to recolor. Eventually we would do a rotation.
  22. 22. References • "Introduction to Algorithms, Third Edition," by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest and Clifford Stein. • Lecture Series on Data Structures and Algorithms by Dr. Naveen Garg, Department of Computer Science and Engineering, IIT Delhi. http://nptel.iitm.ac.in • Lecture 10: Red-black Trees, Rotations, Insertions, Deletions: http://videolectures.net/mit6046jf05_demaine_lec10/ • http://mitpress.mit.edu/algorithms/solutions/chap13-solutions.pdf • http://www.cs.purdue.edu/homes/ayg/CS251/slides/chap13c.pdf • Left leaning Red black trees • http://www.cs.princeton.edu/~rs/talks/LLRB/RedBlack.pdf • http://www.cs.princeton.edu/~rs/talks/LLRB/LLRB.pdf
  23. 23. Thank You

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