Sorting

Quick Sort (Lomuto)

Choose the last item as a pivot, partition smaller values to its left, then recurse on the two sides.

Algorithm

Basic Implementation

basic.py 
def partition(arr, low, high):
    pivot = arr[high]
    i = low - 1
    for j in range(low, high):
        if arr[j] <= pivot:
            i += 1
            arr[i], arr[j] = arr[j], arr[i]
    arr[i + 1], arr[high] = arr[high], arr[i + 1]
    return i + 1

def quick_sort(arr, low, high):
    if low < high:
        pivot_index = partition(arr, low, high)
        quick_sort(arr, low, pivot_index - 1)
        quick_sort(arr, pivot_index + 1, high)

arr = [4, 1, 5, 2, 3]
quick_sort(arr, 0, len(arr) - 1)
print(arr)

The pinned first partition uses [4, 1, 5, 2, 3] with pivot 3. The diagrams track the boundary, swaps, and recursive ranges.

Step 1 - Choose the last value as pivot

The pivot is arr[4] = 3, and i starts just before the current range.

Initial partition state for [4, 1, 5, 2, 3].i0i1i2i3i441523j startspivot

Step 2 - Swap small values left

1 and 2 are <= pivot, so they move into the left partition.

After scanning values before the pivot: [1, 2, 5, 4, 3].i0i1i2i3i412543<= 3<= 3> 3> 3pivot

Step 3 - Place pivot, then recurse

Swapping pivot 3 into index 2 gives [1, 2, 3, 4, 5]; recurse on [1, 2] and [4, 5].

Pivot lands at index 2 and splits the remaining work.left rangepivotright range[1, 2]3 at i2[4, 5]quick_sort(0,1)fixedquick_sort(3,4)

Complexity

  • Time: O(n^2) worst, O(n log n) average
  • Space: O(log n) average call stack
  • Stable: no

Implementation notes

  • Python stores arr as one list object, and quicksort mutates that list's element slots in place. pivot = arr[high] binds the pivot integer locally; it does not remove the pivot from the list.
  • The swaps use tuple-assignment syntax, arr[i], arr[j] = arr[j], arr[i] and arr[i + 1], arr[high] = arr[high], arr[i + 1], so the right-hand references are read before either slot is overwritten. No replacement list is allocated, and swap temporaries are not part of the replayed state.
  • range(low, high) scans up to but not including the pivot slot, and low < high is the recursion guard for call-stack frames. The trace exposes the first partition with pivot 3, then summarizes the left and right recursive ranges after the pivot lands at index 2.
pivot The final element is moved to the boundary between smaller and larger values.
partition One scan rearranges the current range before the recursive calls.