RecursionRecursion

Recursion You should be able to identify the base case(s) and the general case in a recursive definition To be able to write a recursive algorithm for a problem involving only simple variables.

A Recursive call A function call in which the function being called is the same as the one making the call. Put differently it is a function call to itself. When a function calls itself this is said to be a recursive call. The word recursive means: having the characteristics of coming up again or repeating. Can be used instead of iteration or looping

Efficiency considerations Recursive solutions are generally less efficient than iterative solutions. Some problems that we will see lend themselves to recursion Some languages do not allow recursion e.g. Fortran, Basic and Cobol Lisp on the other hand is especially suited for recursive algorithms Initially we will look at recursive algorithms on simple variables we will then look at recursion on structured data

What is recursion? The simplest use of the recursive algorithm is to compute a number raised to a given power, e.g. 2^3 is 2x2x2, 4^5 is 4x4x4x4x4 and so on. y^n, yxyxyxyxyxyx…. (n times) y^n is yxy^(n-1) y^n is yxyxy^(n-2) etc

The recursive definition This is the important bit. We can write x^n as x * x^(n-1). Hence something is defined in terms of a smaller version of itself Base case: the case for which the solution can be stated non recursively General case: the case for which the solution is expressed in terms of a smaller version of itself

See program power.cpp demo Run power.cpp See also power*.cpp in the cpteach directory included in the week 8 directory See all Visual Basic illustrations in this directory go t examples directory

Some comments on the program Each recursive call to Power can be thought of as creating its own copy of the parameters x and n. x remains the same for each call but the value of n decreases each time. Lets consider the case with number =2 and exponent = 3, so we are computing 2^3 which is 8

Tracing through the Power program Call 1. Power is called by main, with number equal to 2 and exponent equal to 3. Because n does not equal 1 Power is called recursively with x and n-1as arguments. Call 2: x is equal to 2 and n is equal to 2. Because n does not equal 1 the function Power is called recursively. Call 3: x = 2 and n = 1. As n=1, the value of x is returned. This call to the function has terminated

Infinite recursion If the function Power was called with n negative then the condition n==1 would never be satisfied since starting at a value less than zero and continually subtracting 1 would mean that a value of 1 is never obtained. I have altered this now so that negative powers are calculated

Execution of Power(2,3) X=2, n=1 X=2, n=2 X=2, n=3 Call 3 Call 2 Call 1 Power(2,3)

Recursive algorithms with Simple Variables Lets look at a special mathematical function important in so many areas of mathematics. It is the factorial function defined as n! = nx(n-1)x(n-2)x……1 E.g. 4! = 4*3*2*1 = 24

Run Factorial demo Factorial.cpp

Tracing through the Factorial function n=0 n=1 n=2 n=3 n=4 Call 5 Call 4 Call 3 Call 2 Call 1

Iterative solution Writing the code without using recursion gives: int Factorial(int n) { int factor; int count; Factor = 1; for (count=2;count<=n;count++) Factor=Factor*count; return Factor; }

The Towers of Hanoi Very old problem demonstrating recursion. Only one disc can be moved at a time After each move, all discs must be on the poles No disc may be placed on top of a disc which is smaller than itself

Hanoi.cpp Discuss Hanoi.cpp and demonstrate