Highest sum of tree heights when one recoloring is allowed

Question

You are given a sequence of n trees.  The height of the i-th tree is h[i].  Each tree could be of one of k different colors.  You are allowed to do ONE recoloring operation, in which you can recolor ALL trees with color C1, to color C2.  Your objective is to MAXIMIZE the total height of contiguous block of trees of one color.  Present an algorithm for this problem, and present its time complexity.

For example, suppose your trees are:

Tree T1, Height 17, Color: R (Red)

Tree T2, Height 20, Color: Y (Yellow)

Tree T3, Height 1, Color: G (Green)

Tree T4, Height 5, Color: G (Green)

Tree T5, Height 6, Color: Y (Yellow)

In this case, if you color YàG (or GàY), then the maximum height of G trees is 32.  However, if you recolor YàR (or RàY), then the maximum contiguous height becomes 37, and is therefore the correct answer.

Answer 1

1. Initialize Variables:

   • Create an array color_heights of size k, initialized to zeros. This array will store the total height for each color.
   • Calculate the total_height as the sum of the heights of all trees.
   • Initialize two pointers, left and right, for the sliding window.

2. Sliding Window:

   • Iterate through the trees using the right pointer.
   • For each tree, update the corresponding color's total height in the color_heights array.
   • If the number of distinct colors in the window is more than two, move the left pointer to reduce the window size until there are only two distinct colors.

3. Update Maximum Height:

   • Update the total_height if the current window's total height is greater than the current maximum.

4. Recoloring:

   • Iterate through all pairs of colors (C1, C2) where C1 is not equal to C2.
   • Recolor the trees by replacing all occurrences of C1 with C2.
   • Calculate the maximum total height of contiguous trees for the current recoloring using the sliding window approach.
   • Update the overall maximum height if the current recoloring yields a higher maximum total height.

5. Output:

   • The maximum total height obtained after considering all possible recoloring operations is the final result.

The time complexity of this algorithm is O(kn), where n is the number of trees and k is the number of colors. The sliding window approach ensures efficiency in finding the maximum total height of contiguous trees.

Answer 2

1.Initialization:
Make an array color_heights to hold the total height for each colour. Set current_height and maximum_height to zero.

2.Iteration
Using a sliding window method, begin iterating through the trees.
Create two pointers, one to the left and one to the right, both pointing to the first tree.

3.Window Expansion:
Increase the right pointer to expand the window to the right.
Add the newly inserted tree to color_heights and current_height.

4.Recoloring Attempt:
Attempt a recoloring operation for the full window from colour C1 to colour C2.
Correctly update color_heights and current_height.

5. Update Maximum Height:
Update max_height with the new value if the current contiguous height is greater than the      existing max_height.
     
6. Window Size Control:
If the window size is greater than k, adjust the left cursor to keep the window size constant. Update the outgoing tree's current_height and color_heights.
     
7. Repeat Iteration:
Repeat steps 3-6 until the right pointer reaches the end of the trees.
     
8. Result:
The final value of max_height represents the maximum contiguous height achievable after the recoloring operation.

9. Time Complexity:
Because it only iterates through the trees once and the sliding window assures that each tree is considered at most twice, the technique has a time complexity of O(n), where n is the number of trees.
  
Conclusion:
With a single recoloring operation, the algorithm uses a sliding window approach to maximize the total height of a continuous block of trees. Iterating through the trees, attempting recoloring, and adjusting the maximum height are all part of the step-by-step procedure. The temporal complexity of the algorithm is O(n), where n is the number of trees.