$\forall i,j \in \{1, \dots, n-1\}$ such that $i > j$, $(A[i + 1] - A[i]) > (A[j + 1] - A[j])$
Which one of the following gives the worst case time complexity of the fastest algorithm that can be designed for the problem?
Algorithm Condition Analysis
The problem requires checking a specific condition on an array $A$ of size $n$. The condition is stated as:
$ \forall i,j \in \{1, \dots, n-1\} \text{ such that } i > j, \quad (A[i + 1] - A[i]) > (A[j + 1] - A[j]) $
Let's define the difference between adjacent elements as $D[k] = A[k+1] - A[k]$ for $k \in \{1, \dots, n-1\}$.
The condition simplifies to ensuring that the sequence of differences, $D[1], D[2], \dots, D[n-1]$, is strictly increasing. That is:
$ D[1] < D[2] < \dots < D[n-1] $
Fastest Algorithm Verification
To verify this condition, the fastest algorithm must perform the following steps:
- Compute all adjacent differences $D[k] = A[k+1] - A[k]$ for $k$ from 1 to $n-1$. This requires accessing elements $A[1]$ through $A[n]$.
- Check if these computed differences are strictly increasing, i.e., $D[k] < D[k+1]$ for all $k$ from 1 to $n-2$.
Time Complexity Calculation
The process involves:
- Computing Differences: Calculating $n-1$ differences requires $n-1$ subtraction operations. Accessing the necessary array elements ($A[1]$ to $A[n]$) takes time proportional to $n$. This step is $\Theta(n)$.
- Checking Order: Comparing adjacent differences requires $n-2$ comparison operations. This step is also $\Theta(n)$.
The total time complexity is the sum of the time taken for these steps: $\Theta(n) + \Theta(n) = \Theta(n)$.
Since the algorithm must inspect all adjacent differences to guarantee the condition holds, it needs to access a significant portion of the array, establishing $\Omega(n)$ as a lower bound. As we have an algorithm that achieves $\Theta(n)$, this is the tightest bound for the fastest algorithm.
Conclusion
The worst-case time complexity of the fastest algorithm to check the given array condition is $\Theta(n)$.
Important Questions from Array
Which of the following initialization statement store six integer values in array?
Which of the following expression will delete the entire array pointed to by q?
Consider the following C program segment.
#include <stdio.h>
int main()
{
char s1[7] = "1234";
char *p;p = s1 + 2;
*p = '0';printf("%s", s1);
return 0;
What will be printed by the program?
}#include <stdio.h> void foo(int *p, int x){ *p = x; } int main(){ int *z; int a = 20, b = 25; z = &a; foo(z, b); printf("%d", a); return 0; }The output of the given C program is __________. (Answer in integer)What is the output the given C language code snippet?
#include<stdio.h>
int main()
{
int x[] = {10,20,30};
int *p = x;
p++;
printf("%d",*p);
return 0;
}
