09-Randomly-Built-BST

09-Randomly-Built-BST - Algorithms LECTURE 9 Randomly built...

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Algorithms L7.1 Professor Ashok Subramanian L ECTURE 9 Randomly built binary search trees Expected node depth Analyzing height Convexity lemma Jensen’s inequality Exponential height Post mortem Algorithms
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Algorithms L7.2 3 Binary-search-tree sort T Create an empty BST for i = 1 to n do T REE -I NSERT ( T , A [ i ]) Perform an inorder tree walk of T . Example: A = [3 1 8 2 6 7 5] 8 1 2 6 5 7 Tree-walk time = O ( n ) , but how long does it take to build the BST?
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Algorithms L7.3 Analysis of BST sort BST sort performs the same comparisons as quicksort, but in a different order! 3 1 8 2 6 7 5 1 2 8 6 7 5 2 6 7 5 7 5 The expected time to build the tree is asymptot- ically the same as the running time of quicksort.
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Algorithms L7.4 Node depth The depth of a node = the number of comparisons made during T REE -I NSERT . Assuming all input permutations are equally likely, we have ( 29 ) (lg ) lg ( 1 node insert to s comparison # 1 1 n O n n O n i E n n i = = = = Average node depth . (quicksort analysis)
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Algorithms L7.5 Expected tree height But, average node depth of a randomly built BST = O (lg n ) does not necessarily mean that its expected height is also O (lg n ) (although it is). Example. lg n n h = ) (lg 2 lg 1 n O n n n n n = + Ave. depth
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Algorithms L7.6 Height of a randomly built binary search tree Prove Jensen’s inequality , which says that f ( E [ X ]) E [ f ( X )] for any convex function f and random variable X . Analyze the exponential height of a randomly built BST on n nodes, which is the random variable Y = 2 , where X is the random Outline of the analysis:
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L7.7 Convex functions A function f : R R is convex if for all α , β 0 such that α + β = 1 , we have f ( α x + β y ) α f ( x ) + β f ( y ) for all x , y R . α
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09-Randomly-Built-BST - Algorithms LECTURE 9 Randomly built...

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