Binary Search Trees Definition Of Binary Search Tree

Binary Search Trees Definition Of Binary Search Tree

10 Pages · 2004 · 66 KB · English

Binary Search Trees. • Dictionary Operations: ƒ get(key) ƒ put(key, value) ƒ remove(key). • Additional operations: ƒ ascend() ƒ get(index) (indexed binary search tree) ƒ remove(index) (indexed binary search tree). Complexity Of Dictionary Operations get(), put() and remove(). Data Structu

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Binary Search Trees • Dictionary Operations: ƒget(key) ƒ put(key, value) ƒ remove(key) •Additional operations: ƒascend() ƒ get(index) (indexed binary search tree) ƒremove(index) (indexed binary search tree) Complexity Of Dictionary Operations get(), put() and remove() Data Structure Worst Case Expected Hash Table O(n) O(1) Binary Search Tree O(n) O(log n) Balanced Binary Search TreeO(log n) O(log n) nis number of elements in dictionary Complexity Of Other Operations ascend(), get(index), remove(index) Data Structure ascend get and remove Hash Table O(D + n log n) O(D + n log n) Indexed BST O(n) O(n) Indexed Balanced BST O(n) O(log n) Dis number of buckets Definition Of Binary Search Tree • A binary tree •Each node has a (key, value) pair •For every node x, all keys in the left subtree of xare smaller than that in x •For every node x, all keys in the right subtree of xare greater than that in x Example Binary Search Tree 20 10 6 2 8 15 40 30 25 Only keys are shown The Operation ascend() 20 10 6 2 8 15 40 30 25 Do an inorder traversal O(n) time The Operation get() 20 10 6 2 8 15 40 30 25 Complexity is O(height) = O(n) , where n is number of nodes/elements The Operation put() 20 10 6 2 8 15 40 30 25 Put a pair whose key is 35 35 The Operation put() Put a pair whose key is 7 20 10 6 2 8 15 40 30 25 35 7 The Operation put() 20 10 6 2 8 15 40 30 25 Put a pair whose key is 18 35 7 18 The Operation put() 20 10 6 2 8 15 40 30 25 Complexity of put() is O(height) 35 7 18 The Operation remove() Three cases: Element is in a leaf Element is in a degree 1 node Element is in a degree 2 node Remove From A Leaf Remove a leaf element key = 7 20 10 6 2 8 15 40 30 25 35 7 18 Remove From A Leaf (contd) Remove a leaf element key = 35 20 10 6 2 8 15 40 30 25 35 7 18 Remove From A Degree 1 Node Remove from a degree 1 node key = 40 20 10 6 2 8 15 40 30 25 35 7 18 Remove From A Degree 1 Node (contd) Remove from a degree 1 node key = 15 20 10 6 2 8 15 40 30 25 35 7 18 Remove From A Degree 2 Node Remove from a degree 2 node key = 10 20 10 6 2 8 15 40 30 25 35 7 18 Remove From A Degree 2 Node 20 10 6 2 8 15 40 30 25 Replace with largest key in left subtree (or smallest in right subtree) 35 7 18 Remove From A Degree 2 Node 20 10 6 2 8 15 40 30 25 Replace with largest key in left subtree (or smallest in right subtree) 35 7 18 Remove From A Degree 2 Node 20 8 6 2 8 15 40 30 25 Replace with largest key in left subtree (or smallest in right subtree) 35 7 18 Remove From A Degree 2 Node 20 8 6 2 8 15 40 30 25 Largest key must be in a leaf or degree 1 node 35 7 18 Another Remove From A Degree 2 Node Remove from a degree 2 node key = 20 20 10 6 2 8 15 40 30 25 35 7 18 Remove From A Degree 2 Node 20 10 6 2 8 15 40 30 25 Replace with largest in left subtree 35 7 18 Remove From A Degree 2 Node 20 10 6 2 8 15 40 30 25 Replace with largest in left subtree 35 7 18 Remove From A Degree 2 Node 18 10 6 2 8 15 40 30 25 Replace with largest in left subtree 35 7 18 Remove From A Degree 2 Node 18 10 6 2 8 15 40 30 25 Complexity is O(height) 35 7 Indexed Binary Search Tree • Binary search tree •Each node has an additional field ƒ leftSize = number of nodes in its left subtree Example Indexed Binary Search Tree 20 10 6 2 8 15 40 30 25 35 7 18 0 0 1 1 4 0 0 7 0 0 1 3 leftSize values are in red leftSize And Rank Rank of an element is its position in inorder (inorder = ascending key order) [2,6,7,8,10,15,18,20,25,30,35,40] rank(2) = 0 rank(15) = 5 rank(20) = 7 leftSize(x) = rank(x) with respect to elements in subtree rooted at x leftSize And Rank 20 10 6 2 8 15 40 30 25 35 7 18 0 0 1 1 4 0 0 7 0 0 1 3 sorted list = [2,6,7,8,10,15,18,20,25,30,35,40] get(index) And remove(index) 720 10 6 2 8 15 40 30 25 35 7 18 0 0 1 1 4 0 0 0 0 1 3 sorted list = [2,6,7,8,10,15,18,20,25,30,35,40] get(index) And remove(index) • if index = xleftSize desired element is xelement •if index < xleftSize desired element is index ’th element in left subtree of x •if index > xleftSize desired element is (index xleftSize1) ’th element in right subtree of x Applications (Complexities Are For Balanced Trees) Bestfit bin packing in O(n log n) time Representing a linear list so that get(index) , add(index, element) , and remove(index) run in O(log(list size)) time (uses an indexed binary tree, not indexed binary search tree) Can’t use hash tables for either of these applications Linear List As Indexed Binary Tree h e b a d f l j i k c g 0 0 1 1 4 0 0 7 0 0 1 3 list = [a,b,c,d,e,f,g,h,i,j,k,l] add(5,’m’) h e b a d f l j i k c g 0 0 1 1 4 0 0 7 0 0 1 3 list = [a,b,c,d,e,f,g,h,i,j,k,l] add(5,’m’) h e b a d f l j i k c g 0 0 1 1 4 0 0 7 0 0 1 3 list = [a,b,c,d,e, m, f,g,h,i,j,k,l] find node with element 4 (e) add(5,’m’) h e b a d f l j i k c g 0 0 1 1 4 0 0 7 0 0 1 3 list = [a,b,c,d,e, m, f,g,h,i,j,k,l] find node with element 4 (e)

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