! Insert a value with specified key. ! Search for value given key. ! Delete value with given key. ! O(log N) time per op.

Transcription

1 Symbol Table Review 4.4 Balanced Trees Symbol table: key-value pair abstraction.! a value with specified key.! for value given key.! value with given key. Randomized BST.! O(log ) time per op. unless you get ridiculously unlucky! Store subtree count in each node.! Generate random numbers for each insert/delete op. Reference: Chapter 3, Algorithms in Java, 3 rd dition, Robert Sedgewick. This lecture.! Splay trees.! trees.! Red-black trees.! B-trees. Robert Sedgewick and Kevin Wayne Copyright Trees Trees: and tree..! Scheme to keep tree balanced.! Compare search key against keys in node.! Generalize node to allow multiple keys.! Find interval containing search key.! Follow associated link (recursively). Allow, 2, or 3 keys per node.! 2-node: one key, two children..! 3-node: two keys, three children.! to bottom for key.! 4-node: three keys, four children. F R! 2-node at bottom: convert to 3-node.! 3-node at bottom: convert to 4-node. F R! 4-node at bottom:?? k! F F! k! R R! k k! F F! k! R R! k A C H I U A C H I U 3 4

2 2-3-4 Trees: Splitting Four odes Trees: Splitting a Four ode Transform tree on the way down.! nsures last node is not a 4-node.! Local transformation to split 4-nodes: Splitting a four node: move middle key up. D D Q K Q W K W Invariant: current node is not a 4-node.! One of two above transformations must apply at next node.! ion at bottom is easy since it's not a 4-node. A-C -J L-P R-V X-Z A-C -J L-P R-V X-Z Trees Balance in Trees Tree grows up from the bottom. Property. All paths from top to bottom have exactly the same length. X M A P Tree height.! Worst case: lg [all 2-nodes]! Best case: log 4 = /2 lg [all 4-nodes]! Between 0 and 20 for a million nodes.! Between 5 and 30 for a billion nodes. L ote. Comparison within nodes not accounted for. 7 8

3 2-3-4 Trees: Implementation? Red-Black Trees Direct implementation complicated because of:! Maintaining multiple node types.! Implementation of getchild.! Large number of cases for split. Represent trees as binary trees.! Use "internal" edges for 3- and 4- nodes. red private ode insert(ode h, Key key, Value val) { ode x = h; while (x!= null) { x = x.getchild(key); if (x.is4ode()) x.split(); if (x.is2ode()) x.make3ode(key, val); else if (x.is3ode()) x.make4ode(key, val); fantasy code! Correspondence between trees and red-black trees.! ot - because 3-nodes swing either way. 9 0 Splitting odes in Red-Black Trees Red-Black Tree ode Split xample Two easy cases: switch colors. inserting G change colors Two hard cases: use rotations. right rotate R " do single rotation left rotate " do double rotation 2

4 Red-Black Tree Construction Balance in Red-Black Trees Property. Length of longest path is at most twice the length of shortest path. X A M P Tree height. Worst case: 2 lg. L ote. Comparison within nodes are counted. 3 4 Symbol Table: Implementations Cost Summary Red-Black Trees in Practice Red-black trees vs. splay trees. Worst Case Average Case! Fewer rotations than splay trees. at most 2 per insertion Implementation! One extra bit per node for color. possible to eliminate Sorted array Unsorted list BST Randomized BST Splay log log log log log log Hashing * * * log log log * assumes hash map is random for all keys is the number of nodes ever inserted probabilistic guarantee amortized guarantee log log log log log log Red-Black log log log log log log log log Red-black trees vs. hashing.! Hashing code is simpler and usually faster: arithmetic to compute hash vs. comparison.! Hashing performance guarantee is weaker.! BSTs have more flexibility and can support wider range of ops. Red-black trees are widely used as system symbol tables.! Java: TreeMap, TreeSet.! C++ STL: map, multimap, multiset. 5 6

5 Red Black Tree: Java Library B-Trees Java has built-in libraries for symbol tables.! TreeMap = red black tree implementation. import java.util.treemap; public class TreeMapDemo { public static void main(string[] args) { TreeMap<String, String> st = new TreeMap <String, String>(); st.put(" " "); st.put(" " "); System.out.println(st.get(" Duplicate policy.! Java HashMap allows null values.! Our implementations forbid null values. B-Tree. Generalizes trees by allowing up to M links per node. Main application: file systems.! Reading a page into memory from disk is expensive.! Accessing info on a page in memory is free.! Goal: minimize # page accesses.! ode size M = page size. Space-time tradeoff.! M large # only a few levels in tree.! M small # less wasted space.! Typical M = 000, < trillion. Bottom line: number of page accesses is log M per op.! 3 or 4 in practice! 7 8 B-Tree xample B-Tree xample (cont) Page insert 275 M = 5 Key = Value = int 9 20

6 Symbol Table: Implementations Cost Summary B-Trees in the Wild File systems. Worst Case Average Case! Windows: HPFS. Implementation Sorted array log log / 2 / 2! Mac: HFS, HFS+.! Linux: ReiserFS, XFS, xt3fs, JFS. journaling Unsorted list BST Randomized BST Splay log log Hashing * * * log log log log log log log log log log log Red-Black log log log log log log log log B-Tree page accesses Databases.! Most common index type in modern databases.! ORACL, DB2, IGRS, SQL, PostgreSQL,... Variants.! B trees: Bayer-McCreight (972, Boeing)! B+ trees: all data in external nodes.! B* trees: keeps pages at least 2/3 full.! R-trees for spatial searching: GIS, VLSI. B-Tree: umber of PAG accesses is log M per op Summary Goal: ST implementation with log guarantee for all ops.! Probabilistic: randomized BST.! Amortized: splay tree.! Worst-case: red-black tree.! Algorithms are variations on a theme: rotations when inserting. Abstraction extends to give search algorithms for huge files.! B-tree. 23