Classic merge sort (part 2)

(Initial discussion from Classic merge sort, since it is new code, I start a new thread)

Post my code below, my major question is, I have to create another array result to hold sub-parts merge sort result. Is there a way I can just use original number to save additional space in result?

Any other comments on code bugs, performance (in terms of algorithm time complexity), code style, etc. are appreciated.

Code written in Python 2.7.

def merge_sort(numbers, start, end):
    if start == end:
        return
    pivot_index = start + (end-start)//2
    merge_sort(numbers, start, pivot_index)
    merge_sort(numbers, pivot_index+1, end)
    i = start
    j = pivot_index+1
    result = []
    while i <= pivot_index and j <= end:
        if numbers[i] <= numbers[j]:
            result.append(numbers[i])
            i+=1
        else:
            result.append(numbers[j])
            j+=1
    if i <= pivot_index:
        result.extend(numbers[i:pivot_index+1])
    if j <= end:
        result.extend(numbers[j:end+1])
    k=0
    for i in range(start, end+1):
        numbers[i] = result[k]
        k+=1

if __name__ == "__main__":
    numbers = [1,4,2,5,6,8,3,4,0]
    merge_sort(numbers, 0, len(numbers)-1)
    print numbers

You can make your merge sort run 35% faster by allocating the auxiliary memory only once and reusing it throughout the algorithm:

def coderodde_merge(source,
                    target,
                    source_offset,
                    target_offset,
                    left_run_length,
                    right_run_length):
    left_run_index = source_offset
    right_run_index = source_offset + left_run_length

    left_run_index_bound = right_run_index
    right_run_index_bound = right_run_index + right_run_length

    while left_run_index != left_run_index_bound and right_run_index != right_run_index_bound:
        if source[right_run_index] < source[left_run_index]:
            target[target_offset] = source[right_run_index]
            right_run_index += 1
        else:
            target[target_offset] = source[left_run_index]
            left_run_index += 1
        target_offset += 1

    while left_run_index != left_run_index_bound:
        target[target_offset] = source[left_run_index]
        target_offset += 1
        left_run_index += 1

    while right_run_index != right_run_index_bound:
        target[target_offset] = source[right_run_index]
        target_offset += 1
        right_run_index += 1

def coderodde_mergesort_impl(source,
                             target,
                             source_offset,
                             target_offset,
                             range_length):
    if range_length < 2:
        return

    left_run_length = range_length // 2
    right_run_length = range_length - left_run_length

    coderodde_mergesort_impl(target,
                             source,
                             target_offset,
                             source_offset,
                             left_run_length)

    coderodde_mergesort_impl(target,
                             source,
                             target_offset + left_run_length,
                             source_offset + left_run_length,
                             right_run_length)

    coderodde_merge(source,
                    target,
                    source_offset,
                    target_offset,
                    left_run_length,
                    right_run_length)

def coderodde_mergesort(array, start, end):
    range_length = end - start

    if range_length < 2:
        return

    aux = [array[index] for index in range(start, end)]
    coderodde_mergesort_impl(aux, array, 0, start, range_length)

def coderodde_mergesort_all(array):
    coderodde_mergesort(array, 0, len(array))

When running this demonstration, I get the following results:

OP mergesort in 17628 milliseconds.
coderodde mergesort in 11411 milliseconds.
Algorithms agree: True

Also, as an additional nitpick, by PEP 8 you should separate binary operators by a space before and after, i.e., not i+=1, but i += 1.

Check you range

If I do something as crazy as

ar = [1, 2, 3]
merge_sort(ar, 0, -1)

you will get a stack overflow.

StackExchange Code Review Q#149722, answer score: 2