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CS 332

by: Mrs. Carolyne Abbott
Mrs. Carolyne Abbott
GPA 3.71

David Luebke

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David Luebke
Class Notes
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This 24 page Class Notes was uploaded by Mrs. Carolyne Abbott on Monday September 21, 2015. The Class Notes belongs to CS 332 at University of Virginia taught by David Luebke in Fall. Since its upload, it has received 16 views. For similar materials see /class/209679/cs-332-university-of-virginia in ComputerScienence at University of Virginia.


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Date Created: 09/21/15
CS 332 Algorithms Go over exam Binary Search Trees Davi ddddd ke Exam a Hand back go over exam David Luebke 2 7262007 Review Dynamic Sets 0 Next few lectures will focus on data structures rather than straight algorithms 0 In particular structures for dynamic sets I Elements have a key and satellite data I Dynamic sets support queries such as o SearchS k MinimumS MaximumS Success0rS x Predecess0rS x I They may also support modifying operations like 0 InsertS x DeleteS x David Luebke 3 7262007 Review Binary Search Trees o Binary Search Trees BSTS are an important data structure for dynamic sets 0 In addition to satellite data eleements have I key an identifying eld inducing a total ordering I left pointer to a left child may be NULL I right pointer to a right child may be NULL I p pointer to a parent node NULL for root David Luebke 4 7262007 Review Binary Search Trees 0 BST property key eftSubtree x Skeyx SkeyrightSubtree x 0 Example Davi ddddd ke Inorder Tree Walk o What does the following code d0 TreeWalkx TreeWalkleftx printx TreeWalkrightx o A prints elements in sorted increasing order 0 This is called an inorder tree walk I Preora er tree walk print root then left then right I Postora er tree walk print left then right then root David Luebke 6 7262007 Inorder Tree Walk 0 Example o How long will a tree walk take 0 Prove that inora er walk prints in monotonically increasing order Davi ddddd ke Operations on BSTs Search 0 Given a key and a pointer to a node returns an element with that key or NULL TreeSearchx k if x NULL or k keyx return x if k lt keyx return TreeSearchleftx k else return TreeSearchrightx k David Luebke 8 7262007 BST Search Example 0 Search for D and C Davi ddddd ke Operations on BSTs Search 0 Here s another function that does the same TreeSearchx k while x NULL if k lt X and k keyx keyx leftx else x rightx return x 0 Which of these two functions is more e ciem David Luebke 10 7262007 Operations of BSTs Insert o Adds an element X to the tree so that the binary search tree property continues to hold 0 The basic algorithm I Like the search procedure above I Insert X in place of NULL I Use a trailing pointer to keep track of Where you came from like inserting into singly linked list David Luebke 11 7262007 BST Insert Example 0 Example Insert C Davi ddddd ke BST SearchInsert Running Time o What is the running time of T reeSearchO or T reelhsertO o A Oh where h height of tree 0 What is the height of a binary search tree 0 A worst case h 0a when tree is just a linear string of left or right Children I We ll keep all analysis in terms of h for now I Later we ll see how to maintain h Olg n Davi ddddd ke Sorting With Binary Search Trees o Informal code for sorting array A of length n BSTSortA for i1 to n TreeInsertAi InorderTreewalkroot o Argue that this is 201 lg n o What will be the running time in the llewtame lxhwwagecawehh ren ndmntqfahy ng2 David Luebke 14 7262007 Sorting With BSTs 0 Average case analysis for i1 to n TreeInsert Ai l It s a fOI39IIl Of qUICkSOFt InorderTreeWalk root a David Luebke 15 7262007 Sorting with BSTs 0 Same partitions are done as with quicksort but in a different order I In previous example 0 Everything was compared to 3 once 0 Then those items lt 3 were compared to 1 once 0 Etc I Same comparisons as quicksort different order 0 Example consider inserting 5 David Luebke 16 7262007 Sorting with BSTs 0 Since run time is proportional to the number of comparisons same time as quieksort On lg n 0 Which do you think is better quicksort 0r BSTsort Why Davi ddddd ke Sorting with BSTs 0 Since run time is proportional to the number of comparisons same time as quicksort On lg n 0 Which do you think is better quicksort 0r BSTSort Why 0 A quicksort I Better constants I Sorts in place I Doesn t need to build data structure David Luebke 18 7262007 More BST Operations o BSTs are good for more than sorting For example can implement a priority queue o What Operations must a priority queue have I Insert I Minimum I ExtractMin David Luebke 19 7262007 BST Operations Minimum 0 HOW can we implement a Minimum query a What is the running time 72 2007 BST Operations Successor o For deletion we will need a Suooessor operation 0 Draw Fig 132 o What is the successor of node 3 Node 5 Node 3 o What are the general rules for nding the successor of node x hint two cases Davi ddddd ke BST Operations Successor 0 Two cases I X has a right subtree successor is minimum node in right subtree I X has no right subtree successor is rst ancestor of X Whose left child is also ancestor of X o Intuition As long as you move to the left up the tree you re Visiting smaller nodes 0 Predecessor similar algorithm David Luebke 22 7262007 BST Operations Delete o Deletion is a bit tricky o 3 cases I X has no children 0 Remove X I X has one child Example delete K o Spl1ce outX or H or B I X has two children 0 Swap X with successor 0 Perform case 1 or 2 to delete it David Luebke 23 7262007 BST Operations Delete 0 Why will case 2 always go to case 0 or case 1 o A because when X has 2 Children its successor is the minimum in its right subtree 0 Could we swap x with predecessor instead of successor o A yes Would it be a good idea 0 A might be good to alternate Davi ddddd ke


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