Recursion Basic problemsolving technique is to divide a problem into smaller subproblems These subproblems may also be divided into smaller subproblems When the subproblems are small enough to solve directly the process stops A recursive algorithm is a problem solution that has been expressed in terms of two or more easier to solve subproblems
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What is recursion? A procedure that is defined in terms of itself In a computer language a function that calls itself
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Recursion Recursion A recursive definitionis one which is defined in terms of itself. Examples: • A phrase is a "palindrome" if the 1st and last letters are the same, and what's inside is itself a palindrome (or empty or a single letter) • Rotor • Rotator • 12344321
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N = 1 isa natural number if n is a natural number, then n+1 is a natural number • The definition of the natural numbers: Recursion Recursion
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1. Recursive datastructure: A data structure that is partially composed of smaller or simpler instances of the same data structure. For instance, a tree is composed of smaller trees (subtrees) and leaf nodes, and a list may have other lists as elements. a data structure may contain a pointer to a variable of the same type: struct Node { int data; Node *next; }; 2. Recursive procedure: a procedure that invokes itself 3. Recursive definitions: if A and B are postfix expressions, then A B + is a postfix expression. Recursion in Computer Science Recursion in Computer Science
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Recursive Data Structures RecursiveData Structures Linked lists and trees are recursive data structures: struct Node { int data; Node *next; }; struct TreeNode { int data; TreeNode *left; TreeNode * right; }; Recursive data structures suggest recursive algorithms.
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A mathematical look We are familiar with f(x) = 3x+5 How about f(x) = 3x+5 if x > 10 or f(x) = f(x+2) -3 otherwise
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Calculate f(5) f(x) =3x+5 if x > 10 or f(x) = f(x+2) -3 otherwise f(5) = f(7)-3 f(7) = f(9)-3 f(9) = f(11)-3 f(11) = 3(11)+5 = 38 But we have not determined what f(5) is yet!
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Calculate f(5) f(x) =3x+5 if x > 10 or f(x) = f(x+2) -3 otherwise f(5) = f(7)-3 = 29 f(7) = f(9)-3 = 32 f(9) = f(11)-3 = 35 f(11) = 3(11)+5 = 38 Working backwards we see that f(5)=29
Recursion occurs whena function/procedure calls itself. Many algorithms can be best described in terms of recursion. Example: Factorial function The product of the positive integers from 1 to n inclusive is called "n factorial", usually denoted by n!: n! = 1 * 2 * 3 .... (n-2) * (n-1) * n Recursion Recursion
The Fibonacci numbersare a series of numbers as follows: fib(1) = 1 fib(2) = 1 fib(3) = 2 fib(4) = 3 fib(5) = 5 ... fib(n) = 1, n <= 2 fib(n-1) + fib(n-2), n > 2 Recursive Definition Recursive Definition of the Fibonacci Numbers fib(3) = 1 + 1 = 2 fib(4) = 2 + 1 = 3 fib(5) = 2 + 3 = 5
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int BadFactorial(n){ int x= BadFactorial(n-1); if (n == 1) return 1; else return n*x; } What is the value of BadFactorial(2)? Recursive Definition Recursive Definition We must make sure that recursion eventually stops, otherwise it runs forever:
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Using Recursion Properly UsingRecursion Properly For correct recursion we need two parts: 1. One (ore more) base cases that are not recursive, i.e. we can directly give a solution: if (n==1) return 1; 2. One (or more) recursive cases that operate on smaller problems that get closer to the base case(s) return n * factorial(n-1); The base case(s) should always be checked before the recursive calls.
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Example 1 Writetwo recursive functions to display numbers from 1 to n (a positive integer) in both ascending and descending order void displayAscending(int n) { if (n==1) { cout<<n<<“,”; return; } displayAscending(n-1); cout<<n<<“,”; } 17
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void displayDescending (intn) { cout<<n<<“,”; if (n==1) return; displayDescending( n-1); } void main(void) { int num = 15; displayDescending(num); cout<<endl; displayAscending(num); } 18
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Example 2 Writea recursive function which adds first n positive integers int add_n_integers(int n) { if (n==1) return 1; else return n + add_n_integers(n-1); } void main(void) { int num = 4; cout<<“sum of first “<<num <<“ positive integers is: “ <<add_n_integers(num); } 19