Outline. Computer Science 331. Cost of Binary Search Tree Operations. Bounds on Height: Worst- and Average-Case

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1 Outline Computer Science Average Case Analysis: Binary Search Trees Mike Jacobson Department of Computer Science University of Calgary Lecture #7 Motivation and Objective Definition 4 Mike Jacobson (University of Calgary) Computer Science Lecture #7 / 8 Mike Jacobson (University of Calgary) Computer Science Lecture #7 / 8 Motivation and Objective Cost of Binary Search Tree Operations Motivation and Objective Bounds on Height: Worst- and Average-Case Operations on a Binary Search Tree T... Require a walk down (part of) a path from the root to a leaf of the tree Constant time is required for each node that is visited If a binary search tree T has size n and height h then n h+, so that h log (n + ) and n h +, so that h n. Thus, the worst-case time of each operation is: linear in the height of T Worst Case: Average Case: These bounds cannot be improved. In particular, h = n in some cases. It seems that h Θ(log n) most of the time. Mike Jacobson (University of Calgary) Computer Science Lecture #7 / 8 Mike Jacobson (University of Calgary) Computer Science Lecture #7 4 / 8

2 Motivation and Objective Objective, and Difficulty Useful Property of Shape Problem: There are infinitely many binary search trees of a given size! Objective: Prove that the height of a binary search tree really is logarithmic in its size, most of the time. Consider the following binary search trees, each obtained by inserting four elements into an empty tree. Difficulty: b This or any other average case analysis requires an assumption about how frequently each binary search tree (of a given size) occurs. 4 k z If our assumption is inaccurate then so is our analysis! f Insertion Order:, 4,, Insertion Order: b, z, k, f Mike Jacobson (University of Calgary) Computer Science Lecture #7 5 / 8 Mike Jacobson (University of Calgary) Computer Science Lecture #7 6 / 8 Useful Property of Shape (cont.) Assumption for Analysis If T is generated by inserting a sequence of values x, x,..., x n into an initially empty tree, and T is generated by inserting a sequence of values y, y,..., y n into an initially empty tree, and for all i, j such that i, j n, x i x j if and only if y i y j then T and T have the same shape and the same height. Conclusion: It is sufficient to consider the relative order of the inserted keys when considering the height of a binary search tree. Condition and Assumption for Analysis: Condition: We will consider binary search trees of size n, produced by inserting,,..., n into an empty tree in some order Fact: There are n = n! possible relative orders of these values Assumption: We will assume that each of these relative orders is equally likely. Mike Jacobson (University of Calgary) Computer Science Lecture #7 7 / 8 Mike Jacobson (University of Calgary) Computer Science Lecture #7 8 / 8

3 Possible Relative Orders and Trees When n = Definition Insertion order appears above each tree. If a binary search tree has height h, its exponential-height is h. T :,, T :,, T 5 :,, Heights and Exponential Heights of Previous Trees i height(t i ) exp-height(t i ) T :,, T 4 :,, T 6 :,, Average Exponential Height if n = (Written as Y n ): E(exp-height) = Y = 6 ( ) = 0 Note: Tree shapes do not all occur with the same probability (under our assumption). Goal: determine an upper bound on Y n, derive bound on avg. height Mike Jacobson (University of Calgary) Computer Science Lecture #7 9 / 8 Mike Jacobson (University of Calgary) Computer Science Lecture #7 0 / 8 Trees with Root i Exponential Height with Root i The trees with root i are as follows: R i i R n i Bounds on height and exponential height: If a tree T has a left subtree with height h L and a right subtree with height h R, then height of T is + max(h L, h R ) If a tree T has a left subtree with exp-height H L and a right subtree with exp-height H R, then the exp-height of T is R i : BST with i nodes,,..., i all (i )! relative orders equally likely R n i : BST with n i nodes i +, i +,..., n all (n i)! relative orders equally likely max(h L, H R ) (H L + H R ). Consequence: The average exponential-height of a binary search tree with n nodes (,,..., n) and root i is Y n,i = max(y i, Y n i ) (Y i + Y n i ) Relationship holds for i = and i = n if we define Y 0 to be 0. Mike Jacobson (University of Calgary) Computer Science Lecture #7 / 8 Mike Jacobson (University of Calgary) Computer Science Lecture #7 / 8

4 Recurrence for Y n Bounding Y n Using the Recurrence Since every binary search tree with size one has height zero, Y = 0 =. A binary search tree with n nodes,,..., n has root i with likelihood /n (under our assumption). Thus Y n = n n n i= Y n,i n (Y n i + Y i ) i= = 4 n Y i. n i=0 Mike Jacobson (University of Calgary) Computer Science Lecture #7 / 8 It is possible to use mathematical induction to show that 4 n ( ) i + = 4 ( ) ( ) n + n + = n n 4 i=0 ( ) n n! where the binomial coefficient = k k!(n k)!. It is also easily checked that Y = = ( ) +. 4 These can be used with the previous inequality to prove that Y n ( ) n + (n + )(n + )(n + ) = 4 4 for every integer n. Mike Jacobson (University of Calgary) Computer Science Lecture #7 4 / 8 Useful Property of f (x) = x Useful Property of f (x) = x (cont.) Consider the function f (x) = x : **x Theorem (Jensen s Inequality) For every integer m and positive values x, x,..., x m, f ( m (x + x + + x m ) ) m (f (x ) + f (x ) + + f (x m )) if the function f is convex This function is convex: If α 0, β 0, and α + β = then Can be proved by induction on m. Because x is convex, Jensen s Inequality is applicable f (αx + βx ) αf (x ) + βf (x ). Mike Jacobson (University of Calgary) Computer Science Lecture #7 5 / 8 Mike Jacobson (University of Calgary) Computer Science Lecture #7 6 / 8

5 Application of Property Simplification of Bound Let X n be the average height of a binary search tree with size n (under our assumption). Then where m = n! and h i = height(t i ). X n = m (h + h + + h m ) Consequence of Previous Inequality: X n ( m h + h + + h ) m = Y n. Note that this implies X n log Y n. Corollaries: Under Our Assumption about Construction of Trees Average height of a binary search tree of size n is X n log Y n log ( 4 ( n+ ) ), so that X n log n = log n for sufficiently large n. If c is a positive integer, n is sufficiently large, and T is a randomly constructed BST with size n, then the probability that is less than c. height(t ) c log n Mike Jacobson (University of Calgary) Computer Science Lecture #7 7 / 8 Mike Jacobson (University of Calgary) Computer Science Lecture #7 8 / 8