csc 213 – large scale programming lecture 21: red-black trees
TRANSCRIPT
CSC 213 –Large Scale
Programming
Lecture 21:
Red-Black Trees
Today’s Goal
Implement (2,4)-trees with a binary tree Colors nodes red & black Using colors, group Entrys into (2,4) nodes Provides O(log n) time for insert, remove, & find
(2,4) Trees: Pro & Con
Cons: Cannot use existing BST code
Pros: (2,4) Trees balance without rotations Fewer balancing cases than AVL or splay trees
Just sick, twisted, & wrong: n-node naming scheme is crime against humanity
Red-Black Trees
Binary tree representation of (2,4) tree Red node is part of parent’s node in (2,4) tree Black node is also child of parent’s node in (2,4) tree
As a BST, maximizes reuse of code2 6 73 54
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2 7
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5OR
Red-Black Tree Properties
Root Property: Root node is black External Property: Leaves are black Internal Property: Red nodes’ children are black Depth Property: Leaves have identical black depth
Black depth = number of black ancestors a node has
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62 12
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Insertion
Insert Entry into BST as usual If new node is leaf, color it red
If new Node is root, paint it black If node’s parent is red, violates internal property
Must reorganize tree to remove double reddouble red Example: insert(3)
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Insertion
Insert Entry into BST as usual If new node is leaf, color it red
If new Node is root, paint it black
If node’s parent is red, violates internal property Must reorganize tree to remove double reddouble red
Example: insert(4) needs balancing
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Remedying Double Red
If aunt of node is red Double redDouble red denotes creation of 5-node Perform recoloring (equivalent to (2,4) tree split)
3 4 6 86
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Recoloring
Remedies double reddouble red when uncle is red Paint parent & uncle black, grandparent red
If grandparent is root, must also paint it black As in splitting, double reddouble red may propagate up
2 4 6 76 7
… 4 …
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Remedying a Double Red
Otherwise, node’s aunt is black Double redDouble red is poorly balanced 4-node Restructure the 3 nodes like an AVL tree Preserves overall balance of the tree
3 4 66
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Restructuring
Remedies double reddouble red when uncle is black No further propagation -- just an AVL balance
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4 6 7
.. 2 ..
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2
4 6 7
.. 2 ..
Restructuring (cont.)
Four ways to perform restructuring Differ only in 3 nodes relationships All end with identical result!
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Deletion
Start with normal BST deletion If removed Entry or external node’s sibling
red Paint external node’s sibling black
Example: remove(1)6
3 8
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Deletion
If removed Entry & external node’s sibling are black Paint sibling double blackdouble black This causes violation of internal property
Example: remove(8) causes double blackdouble black
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3 8
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Remedying Double BlackDouble Black
Solution depends on sibling Case 1: sibling is black with red child
Restructure nodes like in an AVL tree Case 2: sibling and its children are black
Equal to (2,4) tree underflow; perform recoloring Case 3: sibling is red
adjust so as to represent 3-node better After adjustment, can then apply case 1 or case 2
(2,4) Tree Transfer
Restructuring double blackdouble black is like transfer
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6 8 10
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6 9
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6 10
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(2,4) Tree Fusion
Recoloring double blackdouble black is like fusion
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6 10
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… 6 9
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…
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…
(2,4) Tree Fusion
Recoloring double blackdouble black is like fusion
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1 7 1 4
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1 7
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Adjustment
Adjusting double blackdouble black stalls for time Transforms into situation we can fix
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For Friday
Friday is BST problem day Bring any questions you may have to answer Will have variety of BST-related problems Gives you last chance to work on these ideas