Solution Explanation: Array Transformation

This problem involves transforming an array based on a set of rules applied iteratively until the array stops changing. The core idea is to simulate the day-by-day transformations until a stable state is reached.

Rules:

1. Interior Element Increase: If an element is smaller than both its immediate neighbors, it's incremented.
2. Interior Element Decrease: If an element is larger than both its immediate neighbors, it's decremented.
3. Boundary Elements: The first and last elements remain unchanged.

Algorithm:

The solution uses a simulation approach. The process continues until no changes occur in a given iteration.

1. Initialization: A flag variable (f in the code) tracks whether any changes were made in an iteration. It's initialized to true.
2. Iteration: The main loop continues as long as f is true.
3. Temporary Copy: A copy of the array (t in the code) is created before making any changes. This allows comparing the array before and after the transformations in the current iteration.
4. Element-wise Transformation: The code iterates through the interior elements (excluding the first and last) of the array. For each element, the rules are applied:
   • If an element satisfies the condition to be incremented or decremented, the corresponding change is made in the original array (arr). The f flag is set to true to indicate a change.
5. Convergence Check: After processing all interior elements, if f is still true, it means that changes occurred in this iteration, and the loop continues. If f is false, it means no changes happened, signifying that the array has reached a stable state, and the loop terminates.
6. Return: Finally, the transformed array (arr) is returned.

Time Complexity Analysis:

The worst-case time complexity is O(n*m), where 'n' is the length of the array and 'm' represents the maximum possible number of iterations needed before the array stabilizes. In the worst case, each element might need to change its value several times until the array converges. The value 'm' will depend on the initial values in the array. While m isn't directly related to n, it's bounded by the maximum possible value an element can hold (100 in this problem's constraints). This leads to a pseudo-polynomial time complexity. In most practical cases, convergence happens much faster.

Space Complexity Analysis:

The space complexity is O(n) because a copy of the array (t) is created in each iteration. In addition, some languages (like Java and C++) create a copy implicitly, which doesn't necessarily create extra space, but conceptually increases the space used for variables.

Code Examples (Python, Java, C++, Go):

The code examples provided in the original response accurately reflect the algorithm described above. Each example shows how to implement the simulation approach in different programming languages. The key features—the flag to track changes, temporary array copy, and element-wise transformations—are present across all implementations.