Question

Please discuss numerical algorithms and briefly talk about their time complexities. Some examples are randomizing values, breaking numbers into their prime factors, finding greatest common divisors, computing geometric areas, etc.

Note: Question is from advanced algorithms. I just need one paragraph of information on this.

Answer 1

The algorithms I have taken are finding GCD, Merge sort and randomizing values.

1. Finding GCD

Algorithm

int gcd(int a, int b)

{

    if (a == 0)

        return b; ---> Executed once

    return gcd(b % a, a); ---> Gets executed log min(a,b) times

}

Time complexity = O(log min(a,b)) if a and b are two numbers for which we need to find the GCD.

2. Merge sort

MERGE-SORT(A, p,r) 1 if p <r q= [(p +r)/2] MERGE-SORT(A, p, q) MERGE-SORT(A,9 +1,r) MERGE(A, p, q,r)

p and r represent the start and end index of the array to be sorted. Line 3 splits the array into halves (This represents first half). Line 4 sorts the second half. This process happens recursively. Once, the split is complete, the merge happens.

So, the recurrence relation T(n) = 2T(n/2) + n

2T(n/2) represents the time complexity of splitting the array into two halves recursively.

n represents the complexity of merging the array

On solving this recurrence relation, Time complexity T(n) = O(n log n) where n is the size of the array

3. Randomizing values

Consider the following piece of code.

int main()
{
    int no = 8;
    while(no!=rand()) {
       printf("Hello world!\n");
    }
    return 0;
}

In this case, the time complexity cannot be determined. If rand() generates number '8' in the first attempt, the time complexity would be O(1). On the other hand, if the rand() function generates numbers other than '8', the time complexity goes on increasing.