CS101: Fundamentals of Computer Programming. Dr. Tejada www-bcf.usc.edu/~stejada Week 14: Recursion

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1 CS101: Fundamentals of Computer Programming Dr. Tejada www-bcf.usc.edu/~stejada Week 14: Recursion

2 Recursion in Nature 2

3 The Art of Problem Solving It has been proven that the process of problem solving (or algorithm development) is not an algorithmic process In 1945, George Polya (a mathematician), wrote a famous book, How to Solve It. Identified four principles for solving problems 1. Understand the Problem 2. Devise a Plan 3. Carry out the Plan 4. Look Back

4 Understanding the Problem To ensure that you understand a problem, you can ask yourself the following questions: Do you understand all the words used in stating the problem? What are you asked to find or show? Can you restate the problem in your own words? Can you think of a picture or a diagram that might help you understand the problem? Is there enough information to enable you to find a solution?

5 Devise a Plan Polya mentions that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. A partial list of strategies is below: Guess and check Use symmetry Make an orderly list Consider special cases Eliminate possibilities Use direct reasoning Solve an equation Look for a pattern Draw a picture Solve a simpler problem Use a formula Use a model Be ingenious Work backward Look for similar problems

6 Carry Out the Plan This step is usually easier than devising the plan In general, all you need is care and patience, given that you have the necessary skills. Keep trying with the plan that you have chosen. If it continues not to work discard it and choose another.

7 Look Back Much can be gained by taking the time to reflect and look back at what you have done, what worked and what didn't Doing this will enable you to predict what strategy to use to solve future problems It may also help you optimize your solution

8 Preconceptions:

9 9 Algorithm Development: Fibonacci Number Consider the following sequence of numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34,... Called the Fibonacci sequence Given the input n The n th number a n, n >= 3, of this sequence is given by: a n = a n- 1 + a n- 2

10 Algorithm Development Design an algorithm using pseudocode that does the following: Input: a positive integer, n (>=1) Output: the n th Fibonacci number

11 11 Problem Analysis and Algorithm Design Algorithm: 1. IniNalize the first two Fibonacci numbers 2. Get n, the posinon of the desired Fibonacci number in the sequence 3. Calculate the next Fibonacci number Add the previous two elements of the sequence 4. Repeat Step 3 unnl the n th Fibonacci number is found 5. Output the n th Fibonacci number

12 12 Programming Example: Variables int previous1; //first number int previous2; //second number int current; //current Fibonacci number int counter; //counts iterations int nthfibonacci; //nth desired position

13 13 Programming Example: IniNalize previous1 and previous2 to 1 Store the posinon of the desired Fibonacci number into nthfibonacci if (nthfibonacci == 1) The desired Fibonacci number is the first Fibonacci number; copy the value of previous1 into current else if (nthfibonacci == 2) The desired Fibonacci number is the second Fibonacci number; copy the value of previous2 into current

14 14 Programming Example: else calculate the desired Fibonacci number as follows: Start by determining the third Fibonacci number IniNalize counter to 3 to keep track of the calculated Fibonacci numbers. Calculate the next Fibonacci number, as follows: current = previous2 + previous1;

15 15 Programming Example: Assign the value of previous2 to previous1 Assign the value of current to previous2 Increment counter Repeat unnl Fibonacci number is calculated: while (counter <= nthfibonacci) { } current = previous2 + previous1; previous1 = previous2; previous2 = current; counter++; Return current

16 16 Problem Solving: Recursion Recursion: solving a problem by reducing it to smaller versions of itself Provides a powerful way to solve certain problems which would be complicated otherwise

17 17 Recursive DefiniNons Recursive defininon: defining a problem in terms of a smaller version of itself Base case: the case for which the solunon is obtained directly Every recursive defininon must have one (or more) base case(s) The base case stops the recursion General case: must eventually reduce to a base case

18 18 Recursive Functions Consider a function for solving the count-down problem from some number num down to 0: The base case is when num is already 0: the problem is solved and we blast off! If num is greater than 0, we count off num and then recursively count down from num-1

19 19 Recursive Functions A recursive function for counting down to 0: void countdown(int num) { if (num == 0) cout << "Blastoff!"; else { cout << num << "..."; countdown(num-1); // recursive } // call }

20 What Happens When Called? first call to countdown num is 2 countdown(1); second call to countdown num is 1 OUTPUT: 2... countdown(0); 1... third call to countdown num is 0 // no // recursive // call Blastoff!

21 21 Recursive DefiniNons Example: Fibonacci Fib(0) = 0 (1) Fib(1) = 1 (2) Fib(n) = Fib(n- 1) + Fib(n- 2) (3) EquaNon (1) and (2) are called the base cases EquaNon (3) is called the general or recursive case

22 22 Recursive funcnon To design a recursive funcnon: IdenNfy base cases and provide a direct solunon to each base case IdenNfy general cases and provide a solunon to each general case in terms of smaller versions of itself

23 23 Recursive Fibonacci solution int rfibnum(int n) { } if (n <= 1) return n; else return rfibnum(n - 1) + rfibnum(n - 2);

24 24 Recursive DefiniNons: factorials Example: factorials 0! = 1 (1) n! = n x (n- 1)! if n > 0 (2) EquaNon (1) is called the base case EquaNon (2) is called the general case

25 25 Recursive Factorial solution int fact(int num) { if (num == 0) return 1; else return num * fact(num - 1); }

26 Recursive Binary Search lo m hi If a[m] == X, we found X, so return m If a[m] > X, recursively search a[lo..m-1] If a[m] < X, recursively search a[m+1..hi]

27 Recursive Binary Search int bsearch(int a[],int lo,int hi,int X) { int m = (lo + hi) /2; if(lo > hi) return -1; // base if(a[m] == X) return m; // base if(a[m] > X) return bsearch(a,lo,m-1,x); else return bsearch(a,m+1,hi,x); }

28 28 Recursion or IteraNon? Overhead associated with execunng a (recursive) funcnon in terms of: Memory space Computer Nme A recursive funcnon executes more slowly than its iteranve counterpart Today s computers are fast Overhead of a recursion funcnon is not nonceable SomeNmes iteranve solunon is more obvious and easier to understand If the defininon of a problem is inherently recursive, consider a recursive solunon

29 29 Lecture Exercise Use recursion to program your robot to draw the Fibonacci Spiral Determine the time complexity of your algorithm in Big O-notation Make it as efficient as you can Take a picture of the spiral and post to your website along with your code and links to your teammate s page

Standard Version of Starting Out with C++, 4th Edition. Chapter 19 Recursion. Copyright 2003 Scott/Jones Publishing

Standard Version of Starting Out with C++, 4th Edition Chapter 19 Recursion Copyright 2003 Scott/Jones Publishing Topics 19.1 Introduction to Recursion 19.2 The Recursive Factorial Function 19.3 The Recursive