Chapter 6 Lafores Book Recursion Hwajung Lee ITEC

Chapter 6 (Lafore’s Book) Recursion Hwajung Lee ITEC 324 Principle of CS III

Divide and Conquer (algorithmic technique) �Definition: An algorithmic technique. To solve a problem on an instance of size n, the instance is: divided into two or more smaller instances. Each of these smaller instances is recursively solved, and the solutions are combined to produce a solution for the original instance.

Recursion (algorithmic technique) �Definition: An algorithmic technique where a function, in order to accomplish a task, calls itself with some part of the task.

Structure of Recursive Solution �Every recursive solution involves two major parts or cases, the second part having three components. base case(s), in which the problem is simple enough to be solved directly, and recursive case(s). A recursive case has three components: ▪ divide the problem into one or more simpler or smaller parts of the problem, ▪ call the function (recursively) on each part, and ▪ combine the solutions of the parts into a solution for the problem.

Example of Recursion Fibonacci number �Definition: A member of the sequence of numbers such that each number is the sum of the preceding two. The first seven numbers are 1, 1, 2, 3, 5, 8, and 13. �Formal Definition: The nth Fibonacci number is F(n) = F(n-1) + F(n-2), where F(1)=1 and F(2)=1

Example of Recursion Factorials (1) �Definition: The factorial is defined for a positive integer n as n! = n*(n-1)*…*3*2*1 So, for example, 4! = 4*3*2*1. �The special case 0! is defined to have value 1.

Example of Recursion Factorials (2) �Write a recursive method to calculate Factorials:

Example of Recursion Triangular Numbers (1) Definition: A member of the sequence of numbers such that the nth term in the series is obtained by adding n to the previous term. The first seven numbers are 1, 3, 6, 10, 15, 21, and 28.

Example of Recursion Triangular Numbers (2) �Non-recursive program: Program with loop int tri(int n) { int total = 0; while (n > 0) { total = total + n; n = n - 1; } return total; }

Example of Recursion Triangular Numbers (3) Complete Recursive Program int tri(int n) { if (n == 1) return 1; else return n + tri(n - 1); }

Example of Recursion Triangular Numbers (4) �Triangular Trace Each call gets a new, separate copy of the parameter Example: Trace – int t = tri(4) V 1 - tri(4): n is 4 V 2 - tri(3): n is 3 V 3 - tri(2): n is 2 V 4 - tri(1): n is 1 return 1 add n to 1 = 2 + 1 = 3 return 3 add n to 3 = 3 + 3 = 6 return 6 add n to 6 = 4 + 6 = 10 return 10 as value of tri(4)

Characteristics of a Recursive Method Handles some version of the problem directly (ie without calling itself) - this version is called the base case 2. Handles some version of the problem recursively (ie by calling itself) - this version is called the recursive case 3. Recursive call solves a simpler version of the problem (what does simpler mean in this context? ) 4. Uses solution from recursive call to solve original problem 1. Example: Triangular Numbers, Factorial, Anagrams, Recursive Binary Search, The Towers of Hanoi, Mergesort

Efficiency of Recursive Methods �Arguments and return addresses must be stacked �Local variables must be allocated �Consider trace of int tri(int n) { int answer; if (n == 1) answer = 1; else answer = n + tri(n - 1); return answer; }

Helper Functions �Sometimes a recursive routine uses a helper function �Triangle. java

Why We Use Recursive Methods (cont. ) �Particularly common for algorithms involving recursively defined data structures: Example: list is an empty list or a node followed by a list Example: binary tree is an empty tree or a node with a left and right (sub)tree

Power of Recursion �Although recursion makes programming easier, does it make a language more powerful? In other words, are there problems that require recursion to be solved? �No - recursion and while loops provide equivalent power �languages with if and loops = languages with if and recursion Anything that can be computed with recursion can be computed with loops Anything that can be computed with loops can be computed with recursion

Example of Recursion Anagrams (1) �Objective: To find all possible permutations of a specified word. Definition: (Math) A permutation, also called an "arrangement number" or "order, " is a rearrangement of the elements of an ordered list S into a one-to-one correspondence with S itself. The number of permutations on a set of n elements is given by n!

Example of Recursion Anagrams (2) �[One approach] Assume the word has n letters. Anagram the rightmost n-1 letters Rotate all n letters Repeat these steps n times

Example of Recursion Anagrams (3) Word Display Word? First Letter Remaining Letters Action cat Yes c at Rotate at cta Yes c ta Rotate ta cat No c at Rotate cat atc Yes a tc Rotate tc act Yes a ct Rotate ct atc No a tc Rotate atc tca Yes t ca Rotate ca tac Yes t ac Rotate ac tca No t ca Rotate tca cat No c at Done

Example of Recursion Anagrams (4) �Anagram. App. java

Example of Recursion Recursive Binary Search �Objective: To find a search. Key in an array a[] Non-Recursive Binary Search Recursive Binary. Search. java [Q 1] What is the performance of the binary search in the big. Oh notation w. r. t. an input size n? [Q 2] Prove your answer.

Example of Recursion Merge sort �Objective: Given a series of numbers (or objects), decide its sequence. Merge Sort Applet merge. java merge. Sort. java [Q 1] What is the performance of the merge sort in the big. Oh notation w. r. t. an input size n? [Q 2] Prove your answer.