Complexity Analysis of Given C Code

Code

#include 
int main(void) {
    int n;
    scanf("%d", &n);
    int arr[n][n];
    for (int i = 0; i < n; i = i + 1){
        int k = 0;
        if (i % 2 == 0) k = 1;
        if (i % 2 != 0) k = n;
        for(int j = 0; j < n; j = j + 1){
            arr[i][j] = k;
            if (i % 2 == 0) k++;
            else k--;
        }
    }
    for (int i = 0; i < n; i = i + 1){
        for(int j = 0; j < n; j = j + 1){
            printf("%d ", arr[i][j]);
        }
        printf("\n");
    }
    return 0;
}

Time Complexity

Initialization of the Array (int arr[n][n])

• Operation: Allocation of an n x n 2D array.
• Complexity: This can be considered as O(1) in terms of operation, but typically ignored in complexity analysis since it doesn't involve loops.

First Nested Loop (Filling Array)

for (int i = 0; i < n; i = i + 1) {
    int k = 0;
    if (i % 2 == 0) k = 1;
    if (i % 2 != 0) k = n;
    for(int j = 0; j < n; j = j + 1) {
        arr[i][j] = k;
        if (i % 2 == 0) k++;
        else k--;
    }
}

• Here, we have a nested loop structure; the outer loop runs n times.
• The inner loop also runs n times for each iteration of the outer loop.
• Thus, the total number of iterations equals n * n, making this part of the code have a time complexity of O(n^2).

Second Nested Loop (Printing Array)

for (int i = 0; i < n; i = i + 1) {
    for(int j = 0; j < n; j = j + 1) {
        printf("%d ", arr[i][j]);
    }
    printf("\n");
}

• Similarly, the outer loop runs n times.
• The inner loop also runs n times for each iteration of the outer loop.
• Thus, the total number of iterations for this nested loop is also n * n, giving this part a time complexity of O(n^2).

Overall Time Complexity

• Combining both nested loops, the overall time complexity remains dominated by O(n^2) as they are sequential operations.

Space Complexity

Array Storage

• The primary storage requirement is for the arr[n][n] 2D array.
• This requires space proportional to n * n elements.
• Hence, the space complexity for the array storage is O(n^2).

Other Variables

• The other variables used (n, i, j, k) are all of constant space.
• Since the constant space is negligible, the additional space required does not affect the overall space complexity.

Overall Space Complexity

• The overall space complexity of the code is O(n^2) due to the 2D array storage.

Summary

• Time Complexity: O(n^2)
  • Filling Array: O(n^2)
  • Printing Array: O(n^2)
• Space Complexity: O(n^2)
  • Array Storage: O(n^2)

Potential Bottlenecks and Optimizations

• The primary bottleneck is the nested loops manipulating and printing an n x n array.
• Optimization Insight: Given the problem's need to fill and print the entire array, reducing time complexity beyond O(n^2) for these specific operations is generally infeasible.

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