Sorting

Quick Sort (Lomuto)

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

Algorithm

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

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.

Visual walkthrough

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)

Basic Implementation

basic.lua 
local function partition(arr, low, high)
	local pivot = arr[high]
	local i = low - 1
	for j = low, high - 1 do
		if arr[j] <= pivot then
			i = i + 1
			arr[i], arr[j] = arr[j], arr[i]
		end
	end
	arr[i + 1], arr[high] = arr[high], arr[i + 1]
	return i + 1
end

local function quick_sort(arr, low, high)
	if low < high then
		local pivot_index = partition(arr, low, high)
		quick_sort(arr, low, pivot_index - 1)
		quick_sort(arr, pivot_index + 1, high)
	end
end

local arr = {4, 1, 5, 2, 3}
quick_sort(arr, 1, #arr)
io.write("[")
for k = 1, #arr do
	if k > 1 then io.write(", ") end
	io.write(tostring(arr[k]))
end
io.write("]\n")

Complexity

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

Implementation notes

  • local arr = {4, 1, 5, 2, 3} is sorted in place by quick_sort(arr, 1, #arr).
  • partition(arr, low, high) chooses the pivot with local pivot = arr[high]; in the first call, Lua slot 5 holds pivot 3.
  • Lua source indexes are 1-based: local i = low - 1 starts at 0, and for j = low, high - 1 do scans slots 1 through 4.
  • The partition comparison is if arr[j] <= pivot then, so only values at or below the pivot move to the left side.
  • Swaps use Lua multiple assignment: arr[i], arr[j] = arr[j], arr[i].
  • The replay labels positions in zero-based cross-language form: it keeps 4 right of pivot 3, swaps 1 left, keeps 5 right, then swaps 2 left.
  • Those partition steps change the table from [4, 1, 5, 2, 3] to [1, 4, 5, 2, 3], then [1, 2, 5, 4, 3].
  • The final pivot swap arr[i + 1], arr[high] = arr[high], arr[i + 1] puts 3 in Lua slot 3; the trace reports that as pivot index 2.
  • quick_sort recurses on low..pivot_index - 1 and pivot_index + 1..high; the replay summarizes the left side [1, 2] and right side [4, 5].
  • The final io.write loop prints [1, 2, 3, 4, 5].