The Painter’s Partition Problem

In this post, we are going to discuss a famous Binary Search problem that is The Painter’s Partition Problem. Also, I am going to discuss a general way to tackle some similar types of problems on Binary Search.

So, let’s start..!!!

Problem Statement

There are paint n boards of length {l1, l2…ln} and there are k painters available. Each painter takes 1 unit of time to paint 1 unit of the board.

The problem is to find the minimum time to get this job done under the constraints that any painter will only paint continuous sections of boards.

Example 1 :

Input : k = 10, Board[] = [1, 8, 11, 3] 
Output : 11

Since there are 10 painters and each painter take 1 sec time to paint 1 unit of board, we will give task of painting Board[] = {1, 8} to painter1,Board[] = {11} to painter2 and Board[] = {3} to painter3.

If all painters will start there task at the same time the maximum of 11 unit time will be required to paint all boards.

Example 2

Input : k = 2, Board[] = [10, 20, 30, 40]
Output : 60

Here painter1 will paint Board[] = {10, 20, 30} and painter2 will paint Board[] = {40} and this is minimum way we can divide the task.

Approach for the painter’s partition problem

This problem can be solved by Binary Search and it is one of the problems of Binary Search patterns.

Let’s discuss how we can apply a binary search on this problem.

The main idea is to apply binary search on the search space and according to the problem we have to reduce the search space which will finally give us the final result.

What is Search Space ?

Search space is the maximum range where the answer contains. For example, in [10, 20, 30, 40] what is maximum and minimum time will be required to paint all the board.

The maximum time will be (10+20+30+40) and the minimum will be 40. Forget about the number of painters, just think of the best and worst situations.

So, basically our search space will be [40 – 100]. Isn’t it ?

Binary Search on Search space

Let’s take an example,

k = 2, Board[] = [10, 20, 30, 40]

Search Space will be,

painter's partition problem

Now, let divide the search space, mid = 70 {(40+100)/2}.

Now let’s assume that no painter will paint more than 70 units of the board.

Board[] = [10, 20, 30, 40]

1st painter will paint (10+20+30) unit board, 2nd painter will paint (40) unit board which concludes that a minimum of 2 painters will be required where each painter can paint at most 70 units of the board.

Since K = 2, we can have a maximum of two painters. Now we know for 70 also two painters will be required.

So, we can cancel the second half and now our search space will change to [40, 70] to check whether any less possible answer is there or not.

NOTE our search space got reduced to half by binary search.

painter's partition problem

For, this search space we will see find mid = 55 and we will continue the same process.

Again check for the case where no painter will paint more than 55 units of the board and for that minimum 3 painters will be required.

1st painter = [10, 20], 2nd painter = [30], 3rd painter = [40]

Now, 55 will surely not be our answer so, we will neglect the left part, and surely answer lies on Search Space [55, 70].

We will continue this process by binary search and finally, we will get our answer which is 60 in this case.

Code in C++ | The Painter’s Partition Problem

Since we have discussed the idea, now let’s discuss the code.

int find(vector<int> &board, int at_most) 
    int n = board.size();
    int s = 0, painters = 1; 
    for (int i = 0; i < n; i++) 
        s += board[i]; 
        if (s > at_most) 
            s = board[i]; 
    return painters; 

int partition(vector<int> &board, int k) 
    int n = board.size(), s = 0, m = 0;
    for(int i = 0; i < n; i++)
        m = max(m, board[i]);
        s += board[i];
    int low = m, high = s;
    while (low < high) 
        int mid = low + (high - low) / 2; 
        int painters = find(board, mid); 
        if (painters <= k) high = mid; 
        else low = mid + 1; 
    return low; 

The code is simple, I have written the same logic which we have discussed earlier. I hope, you understood the painter’s partition problem very well.

Some more Problems,

Book Allocation Problem
Minimum Number of Days to Make m Bouquets

That’s all folks..!!!

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