balanced

balanced - Symbol Table Review 4.4 Balanced Trees Symbol...

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Robert Sedgewick and Kevin Wayne • Copyright © 2006 • http://www.Princeton.EDU/~cos226 4.4 Balanced Trees Reference: Chapter 13, Algorithms in Java, 3 rd Edition, Robert Sedgewick. 2 Symbol Table Review Symbol table: key-value pair abstraction. ! Insert a value with specified key. ! Search for value given key. ! Delete value with given key. Randomized BST. ! O(log N) time per op. [unless you get ridiculously unlucky] ! Store subtree count in each node. ! Generate random numbers for each insert/delete op. This lecture. 2-3-4 trees, red-black trees, B-trees. Robert Sedgewick and Kevin Wayne • Copyright © 2006 • http://www.Princeton.EDU/~cos226 2-3-4 Trees 4 2-3-4 Tree 2-3-4 tree. Generalize node to allow multiple keys; keep tree balanced. Perfect balance. Every path from root to leaf has same length. Allow 1, 2, or 3 keys per node. ! 2-node: one key, two children. ! 3-node: two keys, three children. ! 4-node: three keys, four children. M O K R W C E < K K R > R D F G J A S V Y Z Q N L
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5 2-3-4 Tree: Search Search. ! Compare search key against keys in node. ! Find interval containing search key. ! Follow associated link (recursively). M O K R W C E < K K R > R D F G J A S V Y Z Q N L 6 2-3-4 Tree: Insert Insert. ! Search to bottom for key. ! 2-node at bottom: convert to 3-node. ! 3-node at bottom: convert to 4-node. ! 4-node at bottom: ?? M O K R W C E < K K R > R D F G J A S V Y Z Q N L 7 2-3-4 Tree: Splitting Four Nodes Transform tree on the way down. ! Ensures last node is not a 4-node. ! Local transformation to split 4-nodes: Invariant. Current node is not a 4-node. Consequence. Insertion at bottom is easy since it's not a 4-node. 8 2-3-4 Tree: Splitting a Four Node Ex. To split a four node, move middle key up. A-C K Q W D E-J L-P R-V X-Z A-C K D Q E-J L-P R-V X-Z W
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9 2-3-4 Tree Tree grows up from the bottom. E A P E X M L 10 2-3-4 Tree: Balance Property. All paths from root to leaf have same length. Tree height. ! Worst case: lg N [all 2-nodes] ! Best case: log 4 N = 1/2 lg N [all 4-nodes] ! Between 10 and 20 for a million nodes. ! Between 15 and 30 for a billion nodes.
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balanced - Symbol Table Review 4.4 Balanced Trees Symbol...

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