ITEC2620 Introduction to Data Structures

Transcription

1 9//207 ITEC2620 Introduction to Data Structures Lecture b Recursion and Binary Tree Operations Divide and Conquer Break a problem into smaller subproblems that are easier to solve What happens when the sub-problems have the same form as the original problem? Solve smaller versions of the same problem repeatedly Recursion! Basics of Recursion Solve a problem by solving smaller versions Each smaller version is the recursive subproblem When do you stop? Need a base case that can be solved trivially Factorial Example I n! = n * n- * n-2 * * 3 * 2 * n factorial is the product of all numbers from to n (inclusive) The above shows the standard (nonrecursive) formulation

2 9//207 Factorial Example II A recursive formulation n! = n * n-! if n > Recursive sub-problem Factorial is defined by another factorial n! = if n =, 0 Base problem Trivial no calculations required Factorial Example III Standard (non-recursive) code public static int factorial (int n) { int nfactorial = ; for (int i = n; i > 0; i--) nfactorial *= i; // i = n, n-, n-2,..., 2, return nfactorial; } Factorial Example IV Factorial Example V Recursive code public static int factorial2 (int n) { if ( n <= ) } return ; // recursive sub-problem return n * factorial2( n- ); // base problem The function calls int result = factorial2(); return * factorial2(); return * factorial2(3); return 3 * factorial2(2); return 2 * factorial2(); return ; 2

3 9//207 Factorial Example VI Base case then unwinds the calls return ; return 2 * ; // 2 return 3 * 2; // 6 return * 6; // 2 return * 2; // 20 Binary Search Example I Recursive sub-problem if not found, do binary search on the remaining half to search Base problem return found (array index), or return failure (-) Binary Search Example II public static int binarysearch2 (int K, int[] ar, int lower, int upper) { // base problem (for failure no search range left) if (lower+ >= upper) return - // search in the middle of remaining search range int middle = (lower + upper)/2; Binary Search Example III } // recursive sub-problem ( lower ) if (K < ar[middle]) return binarysearch2 (K, ar, lower, middle); // base problem (for success found it!) else if (K == ar[middle]) return middle; // recursive sub-problem ( higher ) else // K > ar[middle] return binarysearch2 (K, ar, middle, upper); 3

4 9//207 Recursive code The recursive function calls itself over and over Each unit of computation happens in a new function call Cannot store data in local variables Local variables become parameter variables E.g. lower and upper in binary search Tree Traversals I We have stored our data into a Binary Search Tree (BST) We can perform binary search on our data We can update the data in the tree efficiently How do we print the data in order? Tree Traversals II Tree Traversals III Easy to print this in order for (int i = 0; i < ar.length; i++) System.out.println (ar[i]); How do we print this in order?

5 9//207 Tree Traversals IV Print (in order) everything before the root node, print the root node, print everything after the root node Print left sub-tree in order, 2, 3, Print the root node Print the right sub-tree in order 6, 7, 8, 9 How to print the sub-trees in order? Recursion! Tree Traversals V Recursive sub-problem Print left (sub) sub-tree in order, print (sub) tree root, print right (sub) sub-tree Base problem When can you stop? What s a trivial BST to print? One node, or Empty Tree Traversals VI Code Trace I // a method within public class BinaryNode public void printtree() { if (left!= null) left.printtree(); System.out.println(key); if (right!= null) right.printtree(); }.printtree()

6 9//207 Code Trace II Code Trace III left.printtree() left.printtree() Code Trace IV Code Trace V left == null System.out.println(key) 6

7 9//207 Code Trace VI Code Trace VII right.printtree() left == null Code Trace VIII Code Trace IX System.out.println(key) 2 right == null 7

8 9//207 Code Trace X Code Trace XI System.out.println(key) 3 right.printtree() Code Trace XII Code Trace XIII left = null System.out.println(key) 8

9 9//207 Code Trace XIV Code Trace XV right = null System.out.println(key) Code Trace XV right.printtree() Code Trace XVI.printTree(); node3.printtree(); node.printtree(); System.out.println(); node2.printtree(); System.out.println(3); node.printtree(); 9

10 9//207 Tree Traversals Three orders for traversals In-order Print in sorted order Pre-order Search on non-sorted key Post-order Delete a tree Pre-Order Traversal I public void preorder() { System.out.println(key); if (left!= null) left.preorder(); if (right!= null) right.preorder(); } Pre-Order Traversal II Post-Order Traversal I public void postorder() { if (left!= null) left.postorder(); if (right!= null) right.postorder(); System.out.println(key); } 0

11 9//207 Post-Order Traversal II Tree Traversals Summary I preorder postorder inorder Tree Traversals Summary II Non-Recursive Tree Traversal I What is the first node to print? Follow left till null, print that node We can do this

12 9//207 Non-Recursive Tree Traversal II What is the second node to print? Follow left till null if right!= null, print left-most node of right sub-tree else, print parent node Not too bad Non-Recursive Tree Traversal III What is the third node to print? Follow left till null Conditions from left-most node of right sub-tree or parent... Exponential conditions!! Rapidly becomes impossible to code Non-Recursive Tree Traversal IV Cannot do non-recursively Definition of a binary tree is recursive Binary tree operations are all recursive Recursion High-level concept of the algorithm Trust recursion to execute the subproblems properly Readings and Assignments Suggested Readings from Shaffer (third edition) 2.,.2 2