1. More on Problem Solving

2. Factorial Computation
Given a number n, compute the factorial of n
Assume n >= 0
What is factorial?
0! = 1, 1! = 1, 2! = 1*2 = 2
3! = 1*2*3 = 6
4! = 1*2*3*4* = 24
n! = 1*2*...*(n-1)*n, for n>=1

3. The algorithm Observation: For n>=1, n! is (n-1)! multiplied by n
Strategy: Given n, compute n! by successively computing 1!, 2!, etc. till n!
Algorithm Fact:
Input n
Output Fact
1. initialize fact to 1 and index to 1
2. do while (index < = n) steps 2.1 and fact = fact * index index = index + 1
end Fact

4. Analysis Is the solution correct?
Loop invariant: At the beginning of each iteration,
fact holds the partial product 1 * * (index-1)
When the loop terminates, index = (n+1)
fact then holds (1 * ... * n)
Does the loop terminate?
There is a bound function: (n + 1 – index)
The bound function always >= 0
It decreases in each iteration

5. Efficiency Analysis How many operations? n multiplications
Can we do better?
Yes, there is a method using (log n) operations

6. Extracting the digits
Problem: Given a number, list its digits of the number from right to left (How about left to right?)
Example:
If given number is then the program should output
5,6,8,0,5
If the input is 0 then output is 0
If the input is 7680 the output is 0,8,6,7
Solution?

7. One Strategy
Divide the given number by 10. The remainder is the last digit
Divide the quotient by 10. The new remainder is the next last digit
Divide the new quotient by 10. The newer remainder is the next last digit
and so on.
Till no more digits

8. The Algorithm Algorithm Digit_Extract: Input N Output List
quotient = N
do while (quotient > 0) 2.1, 2.2, divide the quotient by obtain the remainder and the new quotient Add the remainder to List end Digit_Extract

9. Analysis of the Algorithm
Is the algorithm correct?
Does the algorithm terminate?
How many operations?

10. From Digits to Numbers This is the inverse problem
Given a list of digits from, left to right, of a number, obtain the number
Example:
Given 4,7,8 the required number is 478
Given 0,8,0,5,6, the required number is 8056
Aside: The alternate problem, where the digits are given from right to left order is, less easy.

11. Solution Strategy Observation: If the list of digits are 4,7,8 then
4  (102) + 7  100 = 478
the required number
This suggests the following strategy:
Take the first number
Multiply by 10 and add it to the next number
Multiply the resulting number by 10 and add it
to the next number
and so on

12. The algorithm Algorithm Form_no: Input: List Output: d
d = the first digit in List
do while (next_digit in List) step 2.1
2.1 d = 10*d + next_digit
end Form_no
Is the algorithm correct?
Does it terminate?
How many number of operations?

13. Another Problem Problem: Reversing the Digits of an integer Examples:
Input:
Output: 20985
Input: 4300
Output: 34

14. Solution Strategy Here is a high level strategy:
Extract the list of digits from the number
Obtain the new number from the list
How do you extract the digits?
Use the earlier algorithm
How do you obtain the new number?
Use the previous algorithm
Are the digits reversed?

15. Top Down Strategy The above solution follows the top-down methodology
Break the problem into two or more sub problems
Obtain the solutions for the sub problems
Combine the solutions to get the solution for the
original problem
Many Large problems can be solved using top-down approach
Obtain the algorithm for the problem using the two previous algorithm

16. The algorithm for Digit Reversal
Algorithm Digit_Reversal
Input: N, the number to be reversed
Output: M, the reversed number
q = N
M = 0
Do while (q > 0) steps 3.1,3.2,3.3
3.1 rem = remainder of q by 10
3.2 M = M*10 + rem
3.3 q = quotient of q by 10
end Digit_Reversal

17. Analysis of the algorithm
Is the algorithm correct?
Does it terminate?
How many number of iterations?
How many number of operations?