Programming Fundamental Instructor Name: Lecture-13 Today’s Lecture What is Recursion Recursive Function General Form of Recursive Methods Recursive Function Call Understanding with Example 2 Recursion What is Recursion? A recursive function is a function that calls itself. The process of solving a problem by reducing it to smaller version of itself is called recursion. Recursion is very powerful way to solve certain problems for which the solution would otherwise be very complicated. There are many problems and specific areas where you can see the repetitive behaviour (pattern) or you can find a thing, which can be modelled in such a way that it repeats itself. Once you understand recursive methods, they are often simpler to write than their iterative equivalents We will begin by studying the form of general recursive methods and comparing iterative and recursive solution. 3 Understanding Recursion Handshake Problem There are n people in a room. If each person shakes hands once with every other person. What is the total number h(n) of handshakes? h(n) = h(n-1) + n-1 h(4) = h(3) + 3 h(3) = h(2) + 2 h(2) = 1 h(n): Sum of integer from 1 to n-1 = n(n-1) / 2 4 Understanding Recursion Fund Raising : Iteration vs Recursion Problem: Collect $1,000.00 for charity Assumption: Everyone is willing to donate a penny Iterative Solution Visit 100,000 people, asking each for a penny Recursive Solution If you are asked to collect a penny, give a penny to the person who asked you for it Otherwise Visit 10 people and ask them to each raise 1/10th of the amount of money that you have been asked to raise Collect the money that they give you and combine it into one bag Give it to the person who asked you to collect the money 5 Recursion A recursive function is a function that calls itself.[ Recursive problem-solving approaches have a number of elements in common. Recursive Function has two parts (i) Base Case(s). The function actually knows how to solve only the simplest case(s), If the function is called with a base case, the function simply returns a result. (ii) Complex Case(s) If the function is called with a more complex problem, It typically divides the problem into two conceptual pieces. a) A piece that the function knows how to do and b) A piece that it does not know how to do Complex Case(s) To make recursion feasible, the latter piece must resemble the original problem But be a slightly simpler or slightly smaller version. Function launches (calls) a fresh copy of itself to work on the smaller problem this is referred to as a recursive call and is also called the recursion step. The recursion step often includes the keyword return, General Form of Recursive Methods Solve(Problem) { if (Problem is minimal/not decomposable: a base case) solve Problem directly; i.e., without recursion else { (1) Decompose Problem into one or more similar, strictly smaller subproblems: SP1, SP2, ... , SPN (2) Recursively call Solve (this method) on each subproblem: Solve(SP1), Solve(SP2),..., Solve(SPN) (3) Combine the solutions to these subproblems into a solution that solves the original Problem } } 8 General Form of Recursive Methods A recursive definition has two parts: the base case - a stopping condition the recursive step - an expression of the computation or definition in terms of itself There are many recursive definitions in mathematics. Consider the factorial function: n! = n * (n-1) * (n -2) * … * 2 * 1 The same function can be defined recursively by giving a base case and a recursive step: 0! = 1 (by definition) n! = n * (n - 1)! (the recursive step) 9 Understanding Recursive Methods In some problems, it may be natural to define the problem in terms of the problem itself. Recursion is useful for problems that can be represented by a simpler version of the same problem. Example: the factorial function 6! = 6 * 5 * 4 * 3 * 2 * 1 We could write: 6! = 6 * 5! In general, we can express the factorial function as follows: n! = n * (n-1)! Is this correct? Well… almost. The factorial function is only defined for positive integers. So we should be a bit more precise: n! = 1 (if n is equal to 1) n! = n * (n-1)! (if n is larger than 1) 10 Understanding Recursive Methods Factorial Function The C++ equivalent of this definition: int fac(int numb){ if(numb<=1) return 1; else return numb * fac(numb-1); } 11 Understanding Recursive Methods Assume the number typed is 3, that is, numb=3. fac(3) : 3 <= 1 ? fac(3) = 3 * fac(2) No. fac(2) : 2 <= 1 ? No. fac(2) = 2 * fac(1) fac(1) : 1 <= 1 ? Yes. return 1 fac(2) = 2 * 1 = 2 return fac(2) fac(3) = 3 * 2 = 6 return fac(3) fac(3) has the value 6 int fac(int numb){ if(numb<=1) return 1; else return numb * fac(numb-1); } 12 Understanding Recursive Methods Assume the number typed is 4, that is, numb=4. int fac(int numb){ if(numb<=1) return 1; else return numb * fac(numb-1); } 13 Understanding Recursive Methods Assume the number typed is 5, that is, numb=5. 14 Factorial Function Iterative Vs Recursive For certain problems (such as the factorial function), a recursive solution often leads to short and elegant code. Compare the recursive solution with the iterative solution: Recursive solution int fac(int numb){ if(numb<=1) return 1; else return numb*fac(numb-1); } Iterative solution int fac(int numb){ int product=1; while(numb>1){ product *= numb; numb--; } return product; } 15 Understanding Recursive Methods Fibonacci Sequence The Fibonacci Sequence is the series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... The next number is found by adding up the two numbers before it. The 2 is found by adding the two numbers before it (1+1) Similarly, the 3 is found by adding the two numbers before it (1+2), And the 5 is (2+3), and so on! Example: the next number in the sequence above is 21+34 = 55 16 Understanding Recursive Methods Fibonacci Sequence – Recursive Definition F(0) = 0; F(1) = 1; F(number) = F(number-1)+ F(number-2); C++ Recursive function for Fibonacci sequence int fib(int number) { if (number == 0) return 0; if (number == 1) return 1; return (fib(number-1) + fib(number-2)); } 17 Fibonacci Sequence - Recursive Methods 18 Fibonacci Sequence - Recursive Methods • Assume the input number is 4, that is, num=4: fib(4): 4 == 0 ? No; 4 == 1? No. fib(4) = fib(3) + fib(2) fib(3): 3 == 0 ? No; 3 == 1? No. fib(3) = fib(2) + fib(1) fib(2): 2 == 0? No; 2==1? No. fib(2) = fib(1)+fib(0) fib(1): 1== 0 ? No; 1 == 1? Yes. fib(1) = 1; return fib(1); int fib(int num) { if (num == 0) return 0; if (num == 1) return 1; return (fib(num-1)+fib(num-2)); } 19 Fibonacci Sequence - Recursive Methods Trace a Fibonacci Number fib(0): 0 == 0 ? Yes. fib(0) = 0; return fib(0); fib(2) = 1 + 0 = 1; return fib(2); fib(3) = 1 + fib(1) fib(1): 1 == 0 ? No; 1 == 1? Yes fib(1) = 1; return fib(1); fib(3) = 1 + 1 = 2; return fib(3) 20 Fibonacci Sequence - Recursive Methods Trace a Fibonacci Number fib(2): 2 == 0 ? No; 2 == 1? No. fib(2) = fib(1) + fib(0) fib(1): 1== 0 ? No; 1 == 1? Yes. fib(1) = 1; return fib(1); fib(0): 0 == 0 ? Yes. fib(0) = 0; return fib(0); fib(2) = 1 + 0 = 1; return fib(2); fib(4) = fib(3) + fib(2) = 2 + 1 = 3; return fib(4); 21 Example The Fibonacci Series long fab (long); main() { int i; Cin>>I; cout<<"fabonaci"<<i<<"="<<fab(i)<<endl; getch(); } long fab(long x) { if (x==1 || x==0) return x; else return (fab(x-1)+fab(x-2)); } Important (Iteration & Recursion) If we use iteration, we must be careful not to create an infinite loop by accident: for(int incr=1; incr!=10;incr+=2) ... Oops! int result = 1; while(result >0){ ... result++; } Oops! Similarly, if we use recursion we must be careful not to create an infinite chain of function calls: int fac(int numb){ return numb * fac(numb-1); } Oops! int fac(int numb){ if (numb<=1) return 1; else return numb * fac(numb+1); } Oops! 23 Important (Iteration & Recursion) Recursion We must always make sure that recursion bottoms out A recursive function must contains at least one non recursive branch The recursive call must eventually leads to non recursive branch Recursion is one way to decompose a task into smaller subtasks. At least one of the subtasks is a smaller example of the same task. The smallest example of the same task has a non-recursive solution. Example: The factorial function n! = n * (n-1)! and 1! = 1 24 Home Practice Home Practice - Recursion Write a recursive solution to compute sum of first n integers Write a recursive solution to computer sum of first n odd and even integers Write a recursive solution that counts the numbers of zero digits in an integer for example zeros(10200) returns 3 Write a exponential Functional that recursively computes np An important function in probability is the binomial coefficient, or choose, function, written B(n,k), and defined as: B(n,k) = n! / k!(n - k)! n≥0, 0<=k≤n Write recursive solution based on following recursive formula B(n, k) = B(n − 1, k − 1) + B(n − 1, k). 25 26

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