1. 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

2. 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(); }

3. 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

4. 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 }

5. 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’

6. Recursive Procedure – Illustrated (cont)

7. 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! …. }

8. 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?

9. 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

10. 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

11. Loops are Usually Easy

12. 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

13. 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)

14. 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