operations insertion Red-black_treesver1.ppt
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RandB Trees
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Left rotation:
y=right[x]; right[x] left[y]; If(left[y]!=nil) p[left[y]]=x; p[y]=p[x]; if(p[x]==nil)
{root=y;} else if (left[p[x]]==x) left[p[x]]=y; else right[p[x]]=y;
left[y]=x; p[x]=y;
Right rotation:
x=left[y]; left[y]=right[x];
If(right[x]!=nil) p[right[x]]=y;
p[x]=p[y]; if(p[y]==nil) root=x;
If(left[p[y]]=y) left[p[y]]=x;
else right[p[y]]=x;
right[x]=y; p[y]=x;](https://image.slidesharecdn.com/red-blacktreesver1-250804091651-8f571683/85/operations-insertion-Red-black_treesver1-ppt-6-320.jpg)
![Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Case 1,2,3: p[z] is the left child of p[p[z]].
Correspondingly, there are 3 other cases,
In which p[z] is the right child of p[p[z]]](https://image.slidesharecdn.com/red-blacktreesver1-250804091651-8f571683/85/operations-insertion-Red-black_treesver1-ppt-10-320.jpg)
operations insertion Red-black_treesver1.ppt
- 1. 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.
- 2. Red-black trees • Definition: a binary tree, satisfying: 1. Every node is red or black 2. The root is black 3. Every leaf is NIL and is black 4. If a node is red, then both its children are black 5. 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
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- 4. Fields and property • Left, right, ,parent, color, key • bh(x), black-height of x, the number of black nodes on any path from x (excluding x) to a leaf. • A red-black tree with n internal nodes has height at most 2log(n+1). – Note: internal nodes: all normal key-bearing nodes. External nodes: Nil nodes or the Nil Sentinel. – A subtree rooted at x contains at least 2bh(x) -1 internal nodes. – By property 4, bh(root)≥h/2. – n ≥ 2h/2 -1
- 5. Some operations in log(n) • Search, minimum, maximum, successor, predecessor. • Let us discuss insert or delete.
- 6. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Left rotation: y=right[x]; right[x] left[y]; If(left[y]!=nil) p[left[y]]=x; p[y]=p[x]; if(p[x]==nil) {root=y;} else if (left[p[x]]==x) left[p[x]]=y; else right[p[x]]=y; left[y]=x; p[x]=y; Right rotation: x=left[y]; left[y]=right[x]; If(right[x]!=nil) p[right[x]]=y; p[x]=p[y]; if(p[y]==nil) root=x; If(left[p[y]]=y) left[p[y]]=x; else right[p[y]]=x; right[x]=y; p[y]=x;
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- 9. Properties violations • Property 1 (each node black or red): hold • Proper 3: (each leaf is black sentinel): hold. • Property 5: same number of blacks: hold • Property 2: (root is black), not, if z is root (and colored red). • Property 4: (the child of a red node must be black), not, if z’s parent is red.
- 10. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Case 1,2,3: p[z] is the left child of p[p[z]]. Correspondingly, there are 3 other cases, In which p[z] is the right child of p[p[z]]
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- 12. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. case 1: z’s uncle is red.
- 13. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. What is the running time of RB_INSERT_FIX? And RB_INSERT? Case 2: z’s uncle is black and z is a right child. Case 3: z’s uncle is black and z is a left child
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