# Recursion - Recursion In C/C a function may call itself...

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1 Recursion In C/C++, a function may call itself either directly or indirectly. When a function call itself recursively, each invocation gets a fresh set of all explicit parameters and automatic (local) variables, independent of the previous set. Example: Compute the greatest common divisor (GCD) of two numbers. Let M > N > 0, and M = QN + R such that 0 <= R < N if R = 0, then M = QN, and gcd(M, N) = N if R > 0, then N > R, and gcd(M, N) = gcd(N, R) int gcd(int M, int N) /* given M > 0 and N > 0, compute gcd of M & N */ { int R; R = M % N; if (R == 0) //base case return N; else return gcd(N, R); //recursion } An equivalent non-recursive implementation. int gcd(int M, int N) /* given M > 0 and N > 0, compute gcd of M & N */ { int R; while (R = M % N) //R = M % N and R != 0 { M = N; N = R; } return N; } Usually, looping (iteration) is more efficient than recursion. Recursion is preferred when the problem is recursively defined, or when the data structure that the algorithm operates on is recursively defined .

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2 Two fundamental rules of recursion: 1. Base cases You must have some base cases, which can be solved without recursion. 2. Making progress (recursion) For the cases that are to be solved recursively, the recursive call must always be to a case that makes progress toward a base case . Example: Fibonacci number f(0) = 0; f(1) = 1; f(n) = f(n 2) + f(n 1) for n 2; long long fib(long long n) { // precondition: given n >= 0 if (n == 0) // base case #1 return 0; if (n == 1) // base case #2 return 1; return fib(n-2) + fib(n-1); // recursion } // this program works but is unacceptably slow for n > 30 Computing the Fibonacci number using iteration is much more efficient.
3 Example: Ackermann’s function otherwise 1 1 0 ) 1 , 1 ( 0 1 ) , ( )) , A(M, N A(M N if M A M if N N M A long long Ack(long long m, long long n) // precondition: given m >= 0 and n >= 0 { if (m == 0) return n+1; if (n == 0) return Ack(m-1, 1); return Ack(m-1, Ack(m, n-1)); }

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4 Example: Recursive binary search
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