Question

question origin

Given an unsorted array of size n containing objects with ids of 0 … n-1, sort the array in place and in linear time. Assume that the objects contain large members such as binary data, so instantiating new copies of the objects is prohibitively expensive.

void linearSort(int* input, const int n) {
    for (int i = 0; i < n; i++) {
        while (input[i] != i) {
            // swap
            int swapPoint = input[i];
            input[i] = input[swapPoint];
            input[swapPoint] = swapPoint;
        }
    }
}

Is this linear? Does this sort work with any kind of array of ints? If so, why do we need quicksort anymore?

Answer 1

Despite the while loop inside the for, this sort is linear O(n). If the while loop occurs multiple times for a given i then for the i values that match swapPoint there will not execute the while loop at all.

This implementation will only work for arrays of ints where there are no duplicates and the values are sequential from 0 to n-1, which is why Quicksort still is relevant being O(n log n) because it works with non-sequential values.

This can be easily tested by making the worst case:

input = new int[] {1, 2, 3, 4, 5, 6, 7, 8, 9, 0};

and then using the following code:

int whileCount = 0;
for (int i = 0; i < n; i++)
{
    while (input[i] != i)
    {
        whileCount++;
        // swap
        int swapPoint = input[i];
        input[i] = input[swapPoint];
        input[swapPoint] = swapPoint;
    }
    Console.WriteLine("for: {0}, while: {1}", i, whileCount);
}

The output will be as follows:

for: 0, while: 9
for: 1, while: 9
for: 2, while: 9
for: 3, while: 9
for: 4, while: 9
for: 5, while: 9
for: 6, while: 9
for: 7, while: 9
for: 8, while: 9
for: 9, while: 9

so you see even in the worst case where you have the while loop run n-1 times in the first iteration of the for loop, you still only get n-1 iterations of the while loop for the entire process.

Further examples with random data:

{7, 1, 2, 4, 3, 5, 0, 6, 8, 9} => 2 on i=0, 1 on i=3 and nothing more. (total 3 while loop runs)
{7, 8, 2, 1, 0, 3, 4, 5, 6, 9} => 7 on i=0 and nothing more (total 7 while loop runs)
{9, 8, 7, 4, 3, 1, 0, 2, 5, 6} => 2 on i=0, 2 on i=1, 1 on i=2, 1 on i=3 (total 6 while loop runs)