bonustimecomplexity

To measure the time complexity of the given Java code, we can analyze the algorithm theoretically to understand how 
the execution time scales with the size of the input. The code you provided is a dynamic programming solution for the Hackerland Radio Transmitters problem. 
Let's break down the key components and analyze their time complexity:

 1. Sorting
```java
Collections.sort(x);
```
- The `Collections.sort()` method in Java typically uses a variant of the TimSort algorithm, which has a time complexity of O(n log n), 
where `n` is the number of elements in the list.

2. Dynamic Programming Calculation
```java
int[] dp = new int[n];
Arrays.fill(dp, Integer.MAX_VALUE);
dp[0] = 1;
// ... loop to calculate dp[i] ...
```
- An array `dp` of size `n` is created and initialized.
- The `Arrays.fill()` operation has a complexity of O(n), as it iterates through the entire array once.
- The loop iterates `n` times, where `n` is the number of houses. Inside the loop, operations are performed which have constant time complexity (O(1)).

Time Complexity Analysis
- The sorting operation dominates the time complexity of this algorithm, with O(n log n).
- The dynamic programming calculation iterates through the list once, adding an additional O(n) to the complexity.
- Thus, the overall time complexity of the algorithm is the sum of these two parts: O(n log n) + O(n).

Since O(n log n) is the dominant term, the overall time complexity of the algorithm is O(n log n).

Space Complexity
- The space complexity is primarily determined by the size of the dynamic programming array `dp`, which is O(n), where `n` is the number of houses.

Measuring Time Complexity Practically
To empirically measure the time complexity, you could run the program with various input sizes and record the execution time for each. Plotting these times against 
the input sizes on a graph can give you a visual representation of how the algorithm scales, which should closely align with the theoretical O(n log n) complexity.