**Loop Invariant Condition:**

Loop invariant condition is a condition about the relationship between the variables of our program which is definitely true immediately before and immediately after each iteration of the loop.

For example: Consider an array A{7, 5, 3, 10, 2, 6} with 6 elements and we have to find maximum element max in the array.

max = -INF (minus infinite) for (i = 0 to n-1) if (A[i] > max) max = A[i]

In the above example after 3rd iteration of the loop max value is 7, which holds true for first 3 elements of array A. Here, the loop invariant condition is that max is always maximum among the first i elements of array A.

**Loop Invariant condition of various algorithms:**

Prerequisite: insertion sort, selection sort, quick sort, bubblesort,

**Selection Sort:**

In selection sort algorithm we find the minimum element from the unsorted part and put it at the beginning.

min_idx = 0 for (i = 0; i < n-1; i++) { min_idx = i; for (j = i+1 to n-1) if (arr[j] < arr[min_idx]) min_idx = j; swap(&arr[min_idx], &arr[i]); }

In the above pseudo code there are two loop invariant condition:

1. In the outer loop, array is sorted for first i elements.

2. In the inner loop, min is always the minimum value in A[i to j].

**Insertion Sort:**

In insertion sort, loop invariant condition is that the subarray A[0 to i-1] is always sorted.

for (i = 1 to n-1) { key = arr[i]; j = i-1; while (j >= 0 and arr[j] > key) { arr[j+1] = arr[j]; j = j-1; } arr[j+1] = key; }

**Quicksort:**

In quicksort algorithm, after every partition call array is divided into 3 regions:

1. Pivot element is placed at its correct position.

2. Elements less than pivot element lie on the left side of pivot element.

3. Elements greater than pivot element lie on the right side of pivot element.

quickSort(arr[], low, high) { if (low < high) { pi = partition(arr, low, high); quickSort(arr, low, pi - 1); // Before pi quickSort(arr, pi + 1, high); // After pi } } partition (arr[], low, high) { pivot = arr[high]; i = (low - 1) for (j = low; j <= high- 1; j++) if (arr[j] <= pivot) i++; swap arr[i] and arr[j] swap arr[i + 1] and arr[high]) return (i + 1) }

**Bubble Sort:**

In bubble sort algorithm, after each iteration of the loop largest element of the array is always placed at right most position. Therefore, the loop invariant condition is that at the end of i iteration right most i elements are sorted and in place.

for (i = 0 to n-1) for (j = 0 to j arr[j+1]) swap(&arr[j], &arr[j+1]);

## Recommended Posts:

- Time Complexity of a Loop when Loop variable “Expands or Shrinks” exponentially
- Stability in sorting algorithms
- Time taken by Loop unrolling vs Normal loop
- Time Complexities of all Sorting Algorithms
- Asymptotic Analysis and comparison of sorting algorithms
- Lower bound for comparison based sorting algorithms
- Know Your Sorting Algorithm | Set 1 (Sorting Weapons used by Programming Languages)
- Know Your Sorting Algorithm | Set 2 (Introsort- C++’s Sorting Weapon)
- Sort 3 Integers without using if condition or using only max() function
- Regularity condition in the master theorem.
- Divide N segments into two non-empty groups such that given condition is satisfied
- Time Complexity of Loop with Powers
- Time Complexity where loop variable is incremented by 1, 2, 3, 4 ..
- External Sorting
- Alternative Sorting

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