Red-Black Tree Neil Tang 02/07/2008

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Red-Black Tree Neil Tang 02/07/2008 CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Class Overview Definition Tree height Rotation and color flip Insert Delete CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Definition A red-black tree is a special binary search tree in which every node is either red or black; the root is black; every leaf (null) is black; if a node is red, then both its children are black; every simple path from a node to a descendant leaf contains the same number of black nodes. CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Height Theorem: A red-black tree with N nodes has height at most 2log(N+1). (pp.274 Cormen’s book) Experiments suggests that the average red-black tree is about as deep as an average AVL tree. The rotation happens less frequently. CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms An Example CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Rotation if S is Black CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Color Flip if S is red CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Example: Insert 45 CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Insert CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Insert CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Insert CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Delete 15 15 5 3 10 6 7 23 16 18 20 12 5 16 Case 1: 3 12 20 18 23 10 13 6 7 15 5 3 10 6 7 23 16 18 20 13 12 15 5 3 10 6 7 23 18 20 13 12 Case 2: 15 5 3 10 6 7 23 16 18 20 13 12 15 5 3 6 23 16 18 20 13 12 10 7 Case 3: CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Delete In terms of deletion, everything boils down to delete a leaf. Deletion of a red leaf is trivial. How to delete a black leaf: Change the color of the leaf to be deleted to red by a top-down pass. CS223 Advanced Data Structures and Algorithms

Three Cases When X Has Two Black Children CS223 Advanced Data Structures and Algorithms