CSCI 123 INTRODUCTION TO PROGRAMMING CONCEPTS IN C++
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1 Brad Rippe CSCI 123 INTRODUCTION TO PROGRAMMING CONCEPTS IN C++ Recursion
2 Recursion CHAPTER 14
3 Overview 14.1 Recursive Functions for Tasks 14.2 Recursive Functions for Values 14.3 Thinking Recursively
4 What is recursion? When a function is defined in terms of itself A recursive function is a function that calls itself void foo(int n) { } return foo(n-1);
5 Objectives To know what is a recursive function and the benefits of using recursive To determine the base cases in a recursive function To understand how recursive function calls are handled in a call stack To solve problems using recursion To use an overloaded helper function to derive a recursive function To understand the relationship and difference between recursion and iteration
6 Factorial Function factorial(0) = 1; factorial(n) = n*factorial(n-1); /////////////////////////////////// Definition ///////////////////////////////// int factorial(int n) { if (n == 0) { // Base Case return 1; } else { return n * factorial(n - 1); // Recursive call } } Factorial.cpp
7 Computing Factorial factorial(3) factorial(0) = 1; factorial(n) = n*factorial(n-1);
8 Computing Factorial factorial(3) = 3 * factorial(2) factorial(0) = 1; factorial(n) = n*factorial(n-1);
9 Computing Factorial factorial(3) = 3 * factorial(2) = 3 * (2 * factorial(1)) factorial(0) = 1; factorial(n) = n*factorial(n-1);
10 Computing Factorial factorial(3) = 3 * factorial(2) = 3 * (2 * factorial(1)) = 3 * ( 2 * (1 * factorial(0))) factorial(0) = 1; factorial(n) = n*factorial(n-1);
11 Computing Factorial factorial(3) = 3 * factorial(2) = 3 * (2 * factorial(1)) = 3 * ( 2 * (1 * factorial(0))) = 3 * ( 2 * ( 1 * 1))) factorial(0) = 1; factorial(n) = n*factorial(n-1);
12 Computing Factorial factorial(3) = 3 * factorial(2) = 3 * (2 * factorial(1)) = 3 * ( 2 * (1 * factorial(0))) = 3 * ( 2 * ( 1 * 1))) = 3 * ( 2 * 1) factorial(0) = 1; factorial(n) = n*factorial(n-1);
13 Computing Factorial factorial(3) = 3 * factorial(2) = 3 * (2 * factorial(1)) = 3 * ( 2 * (1 * factorial(0))) = 3 * ( 2 * ( 1 * 1))) = 3 * ( 2 * 1) = 3 * 2 factorial(0) = 1; factorial(n) = n*factorial(n-1);
14 Computing Factorial factorial(3) = 3 * factorial(2) = 3 * (2 * factorial(1)) = 3 * ( 2 * (1 * factorial(0))) = 3 * ( 2 * ( 1 * 1))) = 3 * ( 2 * 1) = 3 * 2 = 6 factorial(0) = 1; factorial(n) = n*factorial(n-1);
15 animation Trace Recursive factorial Executes factorial(4) factorial(4) Step 9: return 24 Step 8: return 6 Step 0: executes factorial(4) return 4 * factorial(3) Step 1: executes factorial(3) return 3 * factorial(2) Stack Step 7: return 2 return 2 * factorial(1) Step 2: executes factorial(2) Step 3: executes factorial(1) Step 6: return 1 return 1 * factorial(0) Step 4: executes factorial(0) Step 5: return 1 return 1 Main method
16 animation Trace Recursive factorial Step 9: return 24 Step 8: return 6 Step 7: return 2 factorial(4) return 4 * factorial(3) Step 6: return 1 Step 0: executes factorial(4) Step 1: executes factorial(3) return 3 * factorial(2) return 2 * factorial(1) Step 5: return 1 Step 2: executes factorial(2) return 1 * factorial(0) Step 3: executes factorial(1) return 1 Step 4: executes factorial(0) Executes factorial(3) Function Stack Stack Frame Stack for factorial(4) Main method Note the change in the function stack. Each call to factorial pushes the function onto the stack.
17 animation Trace Recursive factorial factorial(4) Step 0: executes factorial(4) Step 9: return 24 return 4 * factorial(3) Step 1: executes factorial(3) Step 8: return 6 return 3 * factorial(2) Executes factorial(2) Stack Step 7: return 2 return 2 * factorial(1) Step 2: executes factorial(2) Step 3: executes factorial(1) Step 6: return 1 return 1 * factorial(0) Step 4: executes factorial(0) Step 5: return 1 return 1 for factorial(3) for factorial(4) Main method
18 animation Trace Recursive factorial factorial(4) Step 0: executes factorial(4) Step 9: return 24 return 4 * factorial(3) Step 1: executes factorial(3) Step 8: return 6 return 3 * factorial(2) Executes factorial(1) Stack Step 2: executes factorial(2) Step 7: return 2 return 2 * factorial(1) Step 3: executes factorial(1) Step 6: return 1 return 1 * factorial(0) Step 4: executes factorial(0) Step 5: return 1 return 1 for factorial(2) for factorial(3) for factorial(4) Main method
19 animation Trace Recursive factorial factorial(4) Step 0: executes factorial(4) Step 9: return 24 return 4 * factorial(3) Step 1: executes factorial(3) Step 8: return 6 Executes factorial(0) Stack return 3 * factorial(2) Step 2: executes factorial(2) Step 7: return 2 return 2 * factorial(1) Step 3: executes factorial(1) Step 6: return 1 return 1 * factorial(0) Step 4: executes factorial(0) Step 5: return 1 return 1 for factorial(1) for factorial(2) for factorial(3) for factorial(4) Main method
20 animation Trace Recursive factorial Step 9: return 24 Step 8: return 6 Step 7: return 2 factorial(4) return 4 * factorial(3) Step 6: return 1 Step 0: executes factorial(4) Step 1: executes factorial(3) return 3 * factorial(2) return 2 * factorial(1) Step 5: return 1 Step 2: executes factorial(2) return 1 * factorial(0) Step 3: executes factorial(1) return 1 Step 4: executes factorial(0) returns 1 Stack for factorial(0) for factorial(1) for factorial(2) for factorial(3) for factorial(4) Main method
21 animation Trace Recursive factorial factorial(4) Step 0: executes factorial(4) Step 9: return 24 return 4 * factorial(3) Step 1: executes factorial(3) Step 8: return 6 returns factorial(0) Stack return 3 * factorial(2) Step 2: executes factorial(2) Step 7: return 2 return 2 * factorial(1) Step 3: executes factorial(1) Step 6: return 1 return 1 * factorial(0) Step 4: executes factorial(0) Step 5: return 1 return 1 for factorial(1) for factorial(2) for factorial(3) for factorial(4) Main method
22 animation Trace Recursive factorial factorial(4) Step 0: executes factorial(4) Step 9: return 24 return 4 * factorial(3) Step 1: executes factorial(3) Step 8: return 6 return 3 * factorial(2) returns factorial(1) Stack Step 2: executes factorial(2) Step 7: return 2 return 2 * factorial(1) Step 3: executes factorial(1) Step 6: return 1 return 1 * factorial(0) Step 4: executes factorial(0) Step 5: return 1 return 1 for factorial(2) for factorial(3) for factorial(4) Main method
23 animation Trace Recursive factorial factorial(4) Step 0: executes factorial(4) Step 9: return 24 return 4 * factorial(3) Step 1: executes factorial(3) Step 8: return 6 return 3 * factorial(2) returns factorial(2) Stack Step 7: return 2 return 2 * factorial(1) Step 2: executes factorial(2) Step 3: executes factorial(1) Step 6: return 1 return 1 * factorial(0) Step 4: executes factorial(0) Step 5: return 1 return 1 for factorial(3) for factorial(4) Main method
24 animation Trace Recursive factorial factorial(4) Step 0: executes factorial(4) Step 9: return 24 return 4 * factorial(3) Step 1: executes factorial(3) Step 8: return 6 return 3 * factorial(2) returns factorial(3) Stack Step 7: return 2 return 2 * factorial(1) Step 2: executes factorial(2) Step 3: executes factorial(1) Step 6: return 1 return 1 * factorial(0) Step 4: executes factorial(0) Step 5: return 1 return 1 for factorial(4) Main method
25 animation Trace Recursive factorial returns factorial(4) factorial(4) Step 9: return 24 Step 8: return 6 Step 0: executes factorial(4) return 4 * factorial(3) Step 1: executes factorial(3) return 3 * factorial(2) Stack Step 7: return 2 return 2 * factorial(1) Step 2: executes factorial(2) Step 3: executes factorial(1) Step 6: return 1 return 1 * factorial(0) Step 4: executes factorial(0) Step 5: return 1 return 1 Main method
26 factorial(4) Stack Trace 5 for factorial(0) 4 for factorial(1) for factorial(1) 3 for factorial(2) for factorial(2) for factorial(2) 2 for factorial(3) for factorial(3) for factorial(3) for factorial(3) 1 for factorial(4) for factorial(4) for factorial(4) for factorial(4) for factorial(4) 6 for factorial(1) for factorial(2) 7 for factorial(2) for factorial(3) for factorial(3) 8 for factorial(3) for factorial(4) for factorial(4) for factorial(4) 9 for factorial(4)
27 Fibonacci Numbers Finonacci series: indices: fib(0) = 0; fib(1) = 1; fib(index) = fib(index -1) + fib(index -2); assume index >=2 fib(3) = fib(2) + fib(1) = (fib(1) + fib(0)) + fib(1) = (1 + 0) +fib(1) = 1 + fib(1) = = 2 Fibonacci.cpp
28 Fibonnaci Numbers, cont.
29 Characteristics of Recursion All recursive methods have the following characteristics: One or more base cases (the simplest case) are used to stop recursion. We know how to solve this case. Every recursive call reduces the original problem, bringing it increasingly closer to a base case until it becomes that case. In general, to solve a problem using recursion, you break it into subproblems. If a subproblem resembles the original problem, you can apply the same approach to solve the subproblem recursively. This subproblem is almost the same as the original problem in nature with a smaller size.
30 Problem Solving Using Recursion Let us consider a simple problem of printing a message for n times. You can break the problem into two subproblems: 1. print the message one time 2. print the message for n-1 times. The second problem is the same as the original problem with a smaller size. The base case for the problem is n==0.
31 Print Message Recursively void printmsg(char *amessage, int atimes) { if(atimes >= 1) { cout << amessage << atimes << endl; printmsg(amessage, atimes-1); } }
32 Think Recursively Many of the problems presented in the early chapters can be solved using recursion if you think recursively. For example, the palindrome problem can be solved recursively as follows: bool ispalindrome(const char* const astring) { if(strlen(astring) <= 1) { return true; } else if(astring[0]!= astring[strlen(astring)-1]) { return false; } else { return ispalindrome(substring(astring, 1, strlen(astring)-2)); } }
33 Recursive Selection Sort 1. Find the largest number in the list and swaps it with the last number. 2. Ignore the last number and sort the remaining ints in the list recursively. recursiveselectionsort.cpp
34 Towers of Hanoi There are n disks labeled 1, 2, 3,..., n, and three towers labeled A, B, and C. No disk can be on top of a smaller disk at any time. All the disks are initially placed on tower A. Only one disk can be moved at a time, and it must be the top disk on the tower.
35 Towers of Hanoi, cont. A B C A B C Original position Step 4: Move disk 3 from A to B A B C A B C Step 1: Move disk 1 from A to B Step 5: Move disk 1 from C to A A B C A B C Step 2: Move disk 2 from A to C Ste p 6: Move disk 2 from C to B A B C A B C Step 3: Move disk 1 from B to C Step 7: Mve disk 1 from A to B
36 Solution to Towers of Hanoi The Towers of Hanoi problem can be decomposed into three subproblems. n-1 disks n-1 disks A B C A B C Original position Step2: Move disk n from A to C n-1 disks n-1 disks A B C A B C Step 1: Move the first n -1 disks from A to C recursively Step3: Move n -1 disks from C to B recursively
37 Solution to Towers of Hanoi Move the first n - 1 disks from A to C with the assistance of tower B. Move disk n from A to B. Move n - 1 disks from C to B with the assistance of tower A.
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