Algorithms ROBERT SEDGEWICK | KEVIN WAYNE 3.2 BINARY SEARCH TREES ‣ BSTs ‣ ordered operations Algorithms ‣ deletion FOURTH EDITION ROBERT SEDGEWICK | KEVIN WAYNE http://algs4.cs.princeton.edu 3.2 BINARY SEARCH TREES ‣ BSTs ‣ ordered operations Algorithms ‣ deletion ROBERT SEDGEWICK | KEVIN WAYNE http://algs4.cs.princeton.edu Binary search trees Definition. A BST is a binary tree in symmetric order. root a left link a subtree A binary tree is either: Empty. ・ right child ・Two disjoint binary trees (left and right). of root null links Anatomy of a binary tree parent of A and R Symmetric order. Each node has a key, key left link S and every node’s key is: of E E X ・Larger than all keys in its left subtree. A R 9 value C H associated ・Smaller than all keys in its right subtree. with R keys smaller than E keys larger than E Anatomy of a binary search tree 3 Binary search tree demo Search. If less, go left; if greater, go right; if equal, search hit. successful search for H S E X A R C H M 4 Binary search tree demo Insert. If less, go left; if greater, go right; if null, insert. insert G S E X A R C H G M 5 BST representation in Java Java definition. A BST is a reference to a root Node. A Node is composed of four fields: ・A Key and a Value. ・A reference to the left and right subtree. smaller keys larger keys private class Node { private Key key; BST private Value val; Node key private Node left, right; val public Node(Key key, Value val) { left right this.key = key; this.val = val; } BST with smaller keys BST with larger keys } Binary search tree Key and Value are generic types; Key is Comparable 6 BST implementation (skeleton) public class BST<Key extends Comparable<Key>, Value> { private Node root; root of BST private class Node { /* see previous slide */ } public void put(Key key, Value val) { /* see next slides */ } public Value get(Key key) { /* see next slides */ } public void delete(Key key) { /* see next slides */ } public Iterable<Key> iterator() { /* see next slides */ } } 7 BST search: Java implementation Get. Return value corresponding to given key, or null if no such key. public Value get(Key key) { Node x = root; while (x != null) { int cmp = key.compareTo(x.key); if (cmp < 0) x = x.left; else if (cmp > 0) x = x.right; else if (cmp == 0) return x.val; } return null; } Cost. Number of compares is equal to 1 + depth of node. 8 BST insert Put. Associate value with key. inserting L S E X A R Search for key, then two cases: C H M ・Key in tree ⇒ reset value. search for L ends P at this null link ・Key not in tree ⇒ add new node. S E X A R C H M create new node L P S E X A R C H M reset links L P on the way up Insertion into a BST 9 BST insert: Java implementation Put. Associate value with key. concise, but tricky, public void put(Key key, Value val) recursive code; read carefully! { root = put(root, key, val); } private Node put(Node x, Key key, Value val) { if (x == null) return new Node(key, val); int cmp = key.compareTo(x.key); if (cmp < 0) x.left = put(x.left, key, val); else if (cmp > 0) x.right = put(x.right, key, val); else if (cmp == 0) x.val = val; return x; } Cost. Number of compares is equal to 1 + depth of node. 10 best case H C S Tree shape A E R X ・Many BSTs correspond to same set of keys. typical case S Number of compares for bestsearch/insert case is equal to 1 + depth of node. ・ H E X C S A R A E R X C H A best case typical case S worst case H C C S E X E A E R X A R H C H R S X typical case S A worst case E X C BST possibilities A R E C H H R S X Bottom Aline. Treeworst shape case depends on order of insertion. C E BST possibilities H 11 R S X BST possibilities BST insertion: random order visualization Ex. Insert keys in random order. 12 Sorting with a binary heap Q. What is this sorting algorithm? 0. Shuffle the array of keys. 1. Insert all keys into a BST. 2. Do an inorder traversal of BST. A. It's not a sorting algorithm (if there are duplicate keys)! Q. OK, so what if there are no duplicate keys? Q. What are its properties? 13 Correspondence between BSTs and quicksort partitioning P H T D O S U A E I Y C M L Remark. Correspondence is 1–1 if array has no duplicate keys. 14 BSTs: mathematical analysis Proposition. If N distinct keys are inserted into a BST in random order, the expected number of compares for a search/insert is ~ 2 ln N. Pf. 1–1 correspondence with quicksort partitioning. Proposition. [Reed, 2003] If N distinct keys are inserted in random order, expected height of tree is ~ 4.311 ln N. How Tall is a Tree? How Tall is a Tree? Bruce Reed Bruce Reed CNRS, Paris, France CNRS, Paris, France [email protected] [email protected] ABSTRACT purpose of this note to prove that for /3 -- ½ + ~,3 we ABSTRACT Let H~ be the height of a randompurpose binary of search this note tree toon prove n thathave: for /3 -- ½ + ~,3 we Let H~ be the height of a randomnodes. binary We search show thattree onthere n existshave: constants a = 4.31107... THEOREM 1. E(H~) = ~logn -/31oglogn + O(1) nodes. We show that there existsand/3 constants = 1.95... a = 4.31107... such that E(H~) = c~logn -/31oglogn + and Var(Hn) = O(1) . and/3 = 1.95... such that E(H~)O(1), = c~logn We also -/31oglogn show that +Var(H~) =THEOREM O(1). 1. E(H~) = ~logn -/31oglogn + O(1) and O(1), We also show that Var(H~) = O(1). Var(Hn) = O(1) . Remark By the definition of a, /3 = 3~ Categories and Subject Descriptors 7"g~" The first defi- Remark By the definition of a, nition/3 = 7"g~"given3~ Theis more first suggestive defi- of why this value is correct, Categories and Subject Descriptors as we will see. But… Worst-case height is N. E.2 [Data Structures]: Trees nition given is more suggestive of why this value is correct, E.2 [Data Structures]: Trees as we will see. For more information on random binary search trees, one 1. THE RESULTS For more information on randommay binary consult search [6],[7], trees, [1], one [2], [9], [4], and [8]. may consult [6],[7], [1], [2], [9], [4],Remark and [8]. After I announced these results, Drmota(unpublished) [ exponentially small chance when keys are Ainserted binary search tree inis a binaryrandom tree to each ordernode of which ] 1. 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