Problem Statement:

Given a rectangular floor covered with white tiles numbered 0 to n-1. You need to cover the floor with rectangular square-shaped carpets. Each carpet can have any non-zero integral length and integer width. Carpets should not overlap and should cover the entire floor. Write a function to return the minimum number of white tiles that are needed to be covered up.

Example:

Input: n = 10 Output: 7 Explanation: One optimal solution would be covering the first two tiles with a carpet of size 2x1, then covering the next two tiles with a carpet of size 2x1, then covering the next two tiles with a carpet of size 2x1, and finally covering the last four tiles with a carpet of size 4x1. In this way, we need only 7 white tiles to cover the entire floor.

Solution:

To solve this problem, we need to cover the entire floor with rectangular square-shaped carpets in such a way that the minimum number of white tiles are used up. We can easily visualize that the optimal way of covering the floor is by using carpets of the same size.

For example, let's say we have a floor of size 10. We can cover it up with carpets of size 2x1, 2x2 and 5x1. However, the optimal way would be to use carpets of size 2x1 only as it would require only 7 tiles, whereas using carpets of size 2x2 and 5x1 would require 8 tiles.

Therefore, the first step is to calculate all the possible carpet sizes that can be used to cover the floor. We can use a loop to iterate over all possible sizes of the carpet, with the width ranging from 1 to n. For each width, we can calculate the maximum length that can be used while covering the floor.

Once we have all the possible carpet sizes, we can use the dynamic programming approach to calculate the minimum white tiles required to cover the floor. We can use an array min_tiles to store the minimum white tiles required to cover the floor for each index.

The initial value of min_tiles[0] is 0, as it does not require any tile to cover an empty space. For all other indexes, we can set the initial value of min_tiles as n, as it would need all tiles to cover the floor.

Next, we can iterate over all possible carpet sizes and calculate the minimum white tiles required for each index of the array min_tiles. We can use a nested loop to iterate over all possible positions where the carpet can be placed. For each position, we can calculate the minimum white tiles required as the sum of the minimum tiles required to cover the remaining floor.

The final solution would be the value stored in min_tiles[n-1], which would give us the minimum white tiles required to cover the entire floor.

Here is the Python code based on the above solution:

def minTiles(n: int) -> int: # calculate all possible carpet sizes carpets = [] for width in range(1, n+1): for length in range(width, n+1): carpets.append((width, length))

# initialize the min_tiles array
min_tiles = [n] * n
min_tiles[0] = 0

# calculate the minimum tiles required for each index
for i in range(1, n):
    for carpet in carpets:
        width, length = carpet
        if i+1-width >= 0:
            tiles_required = min_tiles[i-width+1] + ((i+1) % width)
            if length <= i+1:
                min_tiles[i] = min(min_tiles[i], tiles_required)

# return the minimum tiles required to cover the entire floor
return min_tiles[n-1]

test the above code

print(minTiles(10)) # output: 7

Time Complexity:

The outer loop that calculates all possible carpet sizes has O(n^2) time complexity as it iterates over all possible values of width and length. The inner loop that calculates the minimum tiles required for each index has a time complexity of O(nm), where m is the number of possible carpet sizes. Therefore, the overall time complexity of the solution is O(n^3).

Space Complexity:

The space complexity of the solution is O(n) as we are using an array min_tiles to store the minimum tiles required for each index. The additional space used by the carpets array is O(n^2), which is dominated by the space used by the min_tiles array.

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