Question

Lets say that I have sets that I know are already sorted such as {0,2,10,23,65} and {3,5,8..}. What would be the best sorting algorithm that could combine any number of pre-sorted sets into one sorted set? For how effecient would this type of sorting be?

Answer 1

You do not need to sort them, you need to merge. This is done in O(M+N) using a simple loop that keeps two indexes looking at the current element of the two parts, adding the smaller of the two to the final sequence, and advancing the index by one.

Here is pseudocode:

int[] parts1, parts2 // Sorted parts
int i = 0, j = 0;
while i != parts1.Length || j != parts2.Length
    if i != parts1.Length || j != parts2.Length
        if parts1[i] < parts2[j]
            res.Add(parts1[i++])
        else
            res.Add(parts2[j++])
    else if i != parts1.Length
        res.Add(parts1[i++])
    else
        res.Add(parts2[j++])

At each step the loop advances either i or j, executing parts1.Lenght + part2.Length times.