Sorting

Merge Sort (Top-Down)

Split the array recursively, sort each half, then merge two sorted runs into one sorted result.

Algorithm

The checked-in replay follows the same small input and final output across all 21 DSA books, so this Python DSA implementation can be compared directly with the other languages.

divide and conquer Each recursive call solves a smaller sorted subproblem.
merge step Two sorted halves are combined by repeatedly taking the smaller front item.

Visual walkthrough

The pinned input is [5, 1, 4, 2, 8]. The diagrams show the split into recursive halves, the sorted subarrays, and the final merge choices.

Step 1 - Split the input

The first midpoint splits [5, 1, 4, 2, 8] into left [5, 1] and right [4, 2, 8].

Top-down split used by merge_sort.[5,1,4,2,8]mid = 2[5,1]left[4,2,8]right

Step 2 - Sorted halves return

Recursive calls return [1, 5] and [2, 4, 8] before the final merge begins.

Returned subarrays before the final merge.sidebefore sortafter sortleft[5, 1][1, 5]right[4, 2, 8][2, 4, 8]

Step 3 - Merge by taking smaller fronts

Take 1 from left, then 2 and 4 from right, then the remaining 5 and 8.

Final merge produces [1, 2, 4, 5, 8].choiceleft frontright frontmergedtake 112[1]take 252[1, 2]take 454[1, 2, 4]extend58[1, 2, 4, 5, 8]

Basic Implementation

basic.py 
def merge_sort(values):
    if len(values) <= 1:
        return values
    mid = len(values) // 2
    left = merge_sort(values[:mid])
    right = merge_sort(values[mid:])
    merged = []
    i = j = 0
    while i < len(left) and j < len(right):
        if left[i] <= right[j]:
            merged.append(left[i])
            i += 1
        else:
            merged.append(right[j])
            j += 1
    merged.extend(left[i:])
    merged.extend(right[j:])
    return merged

arr = [5, 1, 4, 2, 8]
print(merge_sort(arr))

Complexity

  • Time: O(n log n)
  • Space: O(n)
  • Stable: yes

Implementation notes

  • Python slicing creates new list objects for values[:mid] and values[mid:]; those lists hold references to the same immutable int objects, not copies of the integer values themselves.
  • Each merge_sort call uses a normal Python call-stack frame. merged = [] allocates the temporary output list, append adds one selected element at a time, and extend(left[i:]) / extend(right[j:]) appends the remaining sliced tail.
  • The comparison is left[i] <= right[j], so ties would take the left element first and preserve stability. The original arr is not mutated; the replay shows the newly returned left/right lists and the final merged list, with temporary lists later handled by Python GC.