Quick Sort Partition Lomuto and Hoare in DSA C++

Mental Model

Partition rearranges elements around a pivot so smaller go left and bigger go right. Lomuto uses one pointer scanning forward, Hoare uses two pointers scanning inward.

Analogy: Imagine sorting books on a shelf by height. Lomuto picks a book and moves through the shelf placing shorter books to the left and taller to the right. Hoare uses two people starting at each end swapping books that are out of place until they meet in the middle.

Array: [5, 3, 8, 4, 2]
Pivot chosen (Lomuto): 2 (last element)
Pointers:
Lomuto: i and j scanning right
Hoare: i from left, j from right

Initial:
[5, 3, 8, 4, 2]
 i,j -> start

After partition:
[2, 3, 8, 4, 5]
Pivot at correct place

Dry Run Walkthrough

Input: array: [5, 3, 8, 4, 2]

Goal: Partition the array around a pivot using Lomuto and Hoare methods

Step 1: Lomuto: choose pivot=2 (last element), set i=-1

[5, 3, 8, 4, 2]
i=-1, j=0

Why: Pivot chosen at end; i tracks smaller elements boundary

Step 2: Lomuto: j=0, 5 > 2 no swap, i stays -1

[5, 3, 8, 4, 2]
i=-1, j=0

Why: 5 is bigger than pivot, no change

Step 3: Lomuto: j=1, 3 > 2 no swap, i stays -1

[5, 3, 8, 4, 2]
i=-1, j=1

Why: 3 is bigger than pivot, no change

Step 4: Lomuto: j=2, 8 > 2 no swap, i stays -1

[5, 3, 8, 4, 2]
i=-1, j=2

Why: 8 is bigger than pivot, no change

Step 5: Lomuto: j=3, 4 > 2 no swap, i stays -1

[5, 3, 8, 4, 2]
i=-1, j=3

Why: 4 is bigger than pivot, no change

Step 6: Lomuto: swap pivot with element at i+1 (index 0)

[2, 3, 8, 4, 5]
i=0, pivot index=0

Why: Place pivot in correct position

Step 7: Hoare: choose pivot=5 (first element), i=0, j=4

[5, 3, 8, 4, 2]
i=0, j=4

Why: Pivot chosen at start; i and j scan inward

Step 8: Hoare: i moves right while arr[i]<pivot: i=1 (3<5), i=2 (8>5 stop)

[5, 3, 8, 4, 2]
i=2, j=4

Why: Find element on left bigger than pivot

Step 9: Hoare: j moves left while arr[j]>pivot: j=4 (2<5 stop)

[5, 3, 8, 4, 2]
i=2, j=4

Why: Find element on right smaller than pivot

Step 10: Hoare: swap arr[i] and arr[j]: swap 8 and 2

[5, 3, 2, 4, 8]
i=2, j=4

Why: Put smaller element left, bigger right

Step 11: Hoare: i moves right: i=3 (4<5), i=4 (8>5 stop)

[5, 3, 2, 4, 8]
i=4, j=4

Why: Find next bigger element on left

Step 12: Hoare: j moves left: j=3 (4<5 stop)

[5, 3, 2, 4, 8]
i=4, j=3

Why: Find next smaller element on right

Step 13: Hoare: i >= j stop, partition index = j=3

[5, 3, 2, 4, 8]
i=4, j=3

Why: Pointers crossed, partition done

Result:

Lomuto partitioned array: [2, 3, 8, 4, 5]
Pivot index: 0
Hoare partitioned array: [5, 3, 2, 4, 8]
Partition index: 3

Annotated Code

DSA C++

#include <iostream>
#include <vector>
using namespace std;

// Lomuto partition scheme
int lomuto_partition(vector<int>& arr, int low, int high) {
    int pivot = arr[high];
    int i = low - 1; // boundary for smaller elements
    for (int j = low; j < high; j++) {
        if (arr[j] <= pivot) {
            i++;
            swap(arr[i], arr[j]);
        }
    }
    swap(arr[i + 1], arr[high]);
    return i + 1; // pivot index
}

// Hoare partition scheme
int hoare_partition(vector<int>& arr, int low, int high) {
    int pivot = arr[low];
    int i = low - 1;
    int j = high + 1;
    while (true) {
        do {
            i++;
        } while (arr[i] < pivot);
        do {
            j--;
        } while (arr[j] > pivot);
        if (i >= j) return j;
        swap(arr[i], arr[j]);
    }
}

void print_array(const vector<int>& arr) {
    for (int v : arr) cout << v << " ";
    cout << endl;
}

int main() {
    vector<int> arr1 = {5, 3, 8, 4, 2};
    vector<int> arr2 = arr1; // copy for hoare

    int lomuto_pivot = lomuto_partition(arr1, 0, arr1.size() - 1);
    cout << "Lomuto partitioned array: ";
    print_array(arr1);
    cout << "Pivot index: " << lomuto_pivot << endl;

    int hoare_pivot = hoare_partition(arr2, 0, arr2.size() - 1);
    cout << "Hoare partitioned array: ";
    print_array(arr2);
    cout << "Partition index: " << hoare_pivot << endl;

    return 0;
}

int pivot = arr[high];

select pivot as last element for Lomuto

int i = low - 1;

initialize boundary for smaller elements

for (int j = low; j < high; j++) { if (arr[j] <= pivot) { i++; swap(arr[i], arr[j]); } }

scan and swap elements smaller or equal to pivot to left

swap(arr[i + 1], arr[high]);

place pivot after smaller elements

int pivot = arr[low];

select pivot as first element for Hoare

int i = low - 1; int j = high + 1;

initialize two pointers outside array bounds

do { i++; } while (arr[i] < pivot);

move i right until element >= pivot

do { j--; } while (arr[j] > pivot);

move j left until element <= pivot

if (i >= j) return j;

stop when pointers cross

swap(arr[i], arr[j]);

swap out-of-place elements

Output

Lomuto partitioned array: 2 3 8 4 5 

Pivot index: 0

Hoare partitioned array: 5 3 2 4 8 

Partition index: 3

Complexity Analysis

Time: O(n) because partition scans the array once comparing and swapping elements

Space: O(1) because partitioning is done in-place without extra arrays

vs Alternative: Both Lomuto and Hoare partition run in linear time, but Hoare often does fewer swaps and can be faster in practice

Edge Cases

empty array

partition returns immediately without changes

DSA C++

for (int j = low; j < high; j++) { ... } in lomuto_partition

array with one element

partition returns the single index as pivot position

DSA C++

int pivot = arr[high]; in lomuto_partition

all elements equal

Lomuto places pivot at end; Hoare pointers cross quickly with minimal swaps

DSA C++

if (arr[j] <= pivot) in lomuto_partition and do-while loops in hoare_partition

When to Use This Pattern

When you see a problem asking to sort or rearrange elements efficiently, recognize the partition step of quicksort and choose Lomuto or Hoare partition schemes to split the array around a pivot.

Common Mistakes

Mistake: Using Lomuto partition but forgetting to swap the pivot at the end
Fix: Always swap the pivot with element at i+1 after scanning to place it correctly

Mistake: In Hoare partition, returning i instead of j when pointers cross
Fix: Return j because it marks the correct partition boundary

Summary

Partitions an array around a pivot so smaller elements go left and larger go right.

Use when implementing quicksort to efficiently divide the array for recursive sorting.

Lomuto uses one pointer scanning forward; Hoare uses two pointers scanning inward and often swaps fewer times.