6.006 4. Balanced Binary Search Trees

MIT OpenCourseWare http://ocw.mit.edu6.006Introduction to AlgorithmsSpring 2008For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Lecture 4 Balanced Binary Search Trees 6.006 Spring 2008Lecture 4: Balanced Binary Search Trees Lecture Overview • The importance of being balanced AVL trees • – Definition – Balance – Insert Other balanced trees • • Data structures in general Readings CLRS Chapter 13. 1 and 13. 2 (but different approach: red-black trees) Recall: Binary Search Trees (BSTs) • rooted binary tree each node has • – key – left pointer – right pointer – parent pointer See Fig. 1 654120115029263211Figure 1: Heights of nodes in a BST 1 Lecture 4 Balanced Binary Search Trees 6.006 Spring 2008xxxFigure 2: BST property • BST property (see Fig. 2). • height of node = length (� edges) of longest downward path to a leaf (see CLRS B.5 for details). The Importance of Being Balanced: • BSTs support insert, min, delete, rank, etc. in O(h) time, where h = height of tree (= height of root). • h is between lg(n) and n: Fig. 3). vs.Perfectly BalancedPathFigure 3: Balancing BSTs balanced BST maintains h = O(lg n) all operations run in O(lg n) time. • ⇒ 2 Lecture 4 Balanced Binary Search Trees 6.006 Spring 2008AVL Trees: Definition AVL trees are self-balancing binary search trees. These trees are named after their two inventors G.M. Adel’son-Vel’skii and E.M. Landis 1 An AVL tree is one that requires heights of left and right children of every node to differ by at most ±1. This is illustrated in Fig. 4) Figure 4: AVL Tree Concept In order to implement an AVL tree, follow two critical steps: • Treat nil tree as height −1. • Each node stores its height. This is inherently a DATA STRUCTURE AUGMENTATION procedure, similar to augmenting subtree size. Alternatively, one can just store differrenc in heights. A good animation applet for AVL trees is available at this link .To compare Binary Search Trees and AVL balancing of trees use code provided here .1Original Russian article: Adelson-Velskii, G.; E. M. Landis (1962). ”An algorithm for the organization of information”. Proceedings of the USSR Academy of Sciences 146: 263266. (English translation by Myron J. Ricci in Soviet Math. Doklady, 3:12591263, 1962.) kk-1 3Lecture 4 Balanced Binary Search Trees 6.006 Spring 2008Balance: The balance is the worst when every node differs by 1. Let Nh = min (� nodes). ⇒ Nh = Nh−1 + Nh−2 +1 > 2Nh−2 ⇒ Nh > 2h/2 1 = h< lg h⇒ 2 Alternatively: Nh >Fn (nth Fibonacci number) In fact,Nh = Fn+2 − 1 (simple induction) φh Fh = √5 (rounded to nearest integer) 1+ √5where φ =2 ≈ 1.618 (golden ratio) = ⇒ maxh ≈ logφ(n) ≈ 1.440 lg(n) AVL Insert: 1. insert as in simple BST 2. work your way up tree, restoring AVL property (and updating heights as you go). Each Step: • suppose x is lowest node violating AVL • assume x is right-heavy (left case symmetric) • if x’s right child is right-heavy or balanced: follow steps in Fig. 5 • else follow steps in Fig. 6 • then continue up to x’s grandparent, greatgrandparent . . . 4Lecture 4 Balanced Binary Search Trees 6.006 Spring 2008xyABCk+1kk-1k-1xzABCk+1k-1Left-Rotate(x)kkyxCABk+1kkk-1yxCABkkk-1k-1Left-Rotate(x)Figure 5: AVL Insert Balancing xzADk+1k-1Left-Rotate(x)k-1yxABkk-1yBCkk-1 ork-2Right-Rotate(z)zCDkk-1k+1k-1 ork-2Figure 6: AVL Insert Balancing 5Lecture 4 Balanced Binary Search Trees 6.006 Spring 2008Example: An example implementation of the AVL Insert process is illustrated in Fig. 76541201150292632116541201150292621123Insert(23)x = 29: left-left case65412011502623321165412011501DoneInsert(55)293226231296541201150222623129x=65: left-right case551554120115012262312965Done3Figure 7: Illustration of AVL Tree Insert Process Comment 1. In general, process may need several rotations before an Insert is completed. Comment 2. Delete(-min) harder but possible. 6Lecture 4 Balanced Binary Search Trees 6.006 Spring 2008Balanced Search Trees: There are many balanced search trees. AVL Trees Adel’son-Velsii and Landis 1962B-Trees/2-3-4 Trees Bayer and McCreight 1972 (see CLRS 18)BB[α] Trees Nievergelt and Reingold 1973Red-black Trees CLRS Chapter 13Splay-Trees Sleator and Tarjan 1985Skip Lists Pugh 1989Scapegoat Trees Galperin and Rivest 1993Treaps Seidel and Aragon 1996Note 1. Skip Lists and Treaps use random numbers to make decisions fast with highprobability.Note 2. Splay Trees and Scapegoat Trees are “amortized”: adding up costs for severaloperations = fast on average.⇒ 7Lecture 4 Balanced Binary Search Trees 6.006 Spring 2008Splay Trees Upon access (search or insert), move node to root by sequence of rotations and/or double-rotations (just like AVL trees). Height can be linear but still O(lg n) per operation “on average” (amortized) Note: We will see more on amortization in a couple of lectures. Optimality • For BSTs, cannot do better than O(lg n) per search in worst case. • In some cases, can do better e.g. – in-order traversal takes Θ(n) time for n elements. – put more frequent items near rootA Conjecture: Splay trees are O(best BST) for every access pattern.• With fancier tricks, can achieve O(lg lg u) performance for integers 1 ··· u [Van Ernde Boas; see 6.854 or 6.851 (Advanced Data Structures)] Big Picture: Abstract Data Type(ADT): interface spec. e.g. Priority Queue: • Q = new-empty-queue() • Q.insert(x) • x = Q.deletemin()vs.Data Structure (DS): algorithm for each op.There are many possible DSs for one ADT. One example that we will discuss much later in the course is the “heap” priority queue. 8