Algorithms An algorithm is a set of instructions used to solve a specific problem In order to be useful, an algorithm must have the following properties: Accurate specification of the input Definiteness of each instruction Termination Correctness Description of result or effect

Recursive Procedures A procedure that under certain circumstances invokes itself is called recursive Is the following a recursive procedure? void foo () { cout << “Hello World!” << endl; foo(); }

The Idea Behind a Recursive Procedure A recursive procedure should involve a conditional test that divides execution into two or more cases: A trivial version of the problem (i.e. the base case) Solved without a recursive call A more complicated version of the problem (i.e. recursive case) Solved by: Making the problem a little bit simpler, and Calling the recursive procedure to solve the simpler problem

Recursive Procedure – An Example void printUnsigned (unsigned int val) { // print the character representation of the unsigned argument if (val < 10) // base case: val is only one digit printChar(digitChar(val)); else { printUnsigned (val /10); // recursive call: // print high order part printChar(digitChar(val % 10)); // print last character }

Recursive Procedure – Illustrated main() calls printUnsigned (456) Recursive case invoked – calls printUnsigned (45) Recursive case invoked – calls printUnsigned (4) Base case invoked – prints ‘4’ Prints ‘5’ Prints ‘6’

Recursive Procedure – Illustrated (cont)

Let’s Write Our Own Recursive Procedure! The Problem: write a function, fac, that computes n! (n factorial) int fac (unsigned int n) { // computes n! …. }

The Recursive Factorial Procedure What is the base case? Let’s write it. What is the recursive case? Does it: Make the problem a little bit simpler? Call the recursive procedure to solve the simpler problem?

The Recursive Factorial Procedure Is this an algorithm? A set of instructions used to solve a specific problem Is it a useful algorithm? Accurate specification of the input Definiteness of each instruction Termination Correctness Description of result or effect

Proving Termination In order to prove termination, find a property or value that possesses all three following characteristics: Can be placed in 1-to-1 correspondence with the integers Is non-negative Decreases as algorithm executes

Loops are Usually Easy

Loops are Usually Easy double power (double base, unsigned int n) { // return the value yielded by raising base to the exponent, n. double result = 1.0; for (unsigned int i=0;i<n; i++) result *= base; return result; } The quantity n-i: Can be placed in 1-to-1 correspondence with the integers Is non-negative Decreases as algorithm executes

The Value Need Not Be in the Algorithm unsigned int gcd(unsigned int n unsigned int m) { // compute the gcd of two positive integers assert (n > 0 && m > 0); while (n != m) { if (n > m) n = n –m; else m = m –n; } return n; } // consider the quantity (n+m)

Other Important Considerations Time to execute (efficiency) How long does an algorithm take to arrive at a result In the best case? In the average case? In the worst case? Are there other algorithms that solve the same problem more quickly? Space utilization