Air Conditioners
Contest: Codeforces Round 731 (Div. 3)
Difficulty rating: 1500
Tags: data structures, dp, implementation, shortest paths, sortings, two pointers

On a strip of land of length $$$n$$$ there are $$$k$$$ air conditioners: the $$$i$$$-th air conditioner is placed in cell $$$a_i$$$ ($$$1 \le a_i \le n$$$). Two or more air conditioners cannot be placed in the same cell (i.e. all $$$a_i$$$ are distinct).

Each air conditioner is characterized by one parameter: temperature. The $$$i$$$-th air conditioner is set to the temperature $$$t_i$$$.

Example of strip of length $$$n=6$$$, where $$$k=2$$$, $$$a=[2,5]$$$ and $$$t=[14,16]$$$.

For each cell $$$i$$$ ($$$1 \le i \le n$$$) find it's temperature, that can be calculated by the formula $$$$$$\min_{1 \le j \le k}(t_j + |a_j - i|),$$$$$$

where $$$|a_j - i|$$$ denotes absolute value of the difference $$$a_j - i$$$.

In other words, the temperature in cell $$$i$$$ is equal to the minimum among the temperatures of air conditioners, increased by the distance from it to the cell $$$i$$$.

Let's look at an example. Consider that $$$n=6, k=2$$$, the first air conditioner is placed in cell $$$a_1=2$$$ and is set to the temperature $$$t_1=14$$$ and the second air conditioner is placed in cell $$$a_2=5$$$ and is set to the temperature $$$t_2=16$$$. In that case temperatures in cells are:

1. temperature in cell $$$1$$$ is: $$$\min(14 + |2 - 1|, 16 + |5 - 1|)=\min(14 + 1, 16 + 4)=\min(15, 20)=15$$$;
2. temperature in cell $$$2$$$ is: $$$\min(14 + |2 - 2|, 16 + |5 - 2|)=\min(14 + 0, 16 + 3)=\min(14, 19)=14$$$;
3. temperature in cell $$$3$$$ is: $$$\min(14 + |2 - 3|, 16 + |5 - 3|)=\min(14 + 1, 16 + 2)=\min(15, 18)=15$$$;
4. temperature in cell $$$4$$$ is: $$$\min(14 + |2 - 4|, 16 + |5 - 4|)=\min(14 + 2, 16 + 1)=\min(16, 17)=16$$$;
5. temperature in cell $$$5$$$ is: $$$\min(14 + |2 - 5|, 16 + |5 - 5|)=\min(14 + 3, 16 + 0)=\min(17, 16)=16$$$;
6. temperature in cell $$$6$$$ is: $$$\min(14 + |2 - 6|, 16 + |5 - 6|)=\min(14 + 4, 16 + 1)=\min(18, 17)=17$$$.

For each cell from $$$1$$$ to $$$n$$$ find the temperature in it.

Input format:
The first line contains one integer $$$q$$$ ($$$1 \le q \le 10^4$$$) — the number of test cases in the input. Then test cases follow. Before each test case, there is an empty line.

Each test case contains three lines. The first line contains two integers $$$n$$$ ($$$1 \le n \le 3 \cdot 10^5$$$) and $$$k$$$ ($$$1 \le k \le n$$$) — the length of the strip of land and the number of air conditioners respectively.

The second line contains $$$k$$$ integers $$$a_1, a_2, \ldots, a_k$$$ ($$$1 \le a_i \le n$$$) — positions of air conditioners on the strip of land.

The third line contains $$$k$$$ integers $$$t_1, t_2, \ldots, t_k$$$ ($$$1 \le t_i \le 10^9$$$) — temperatures of air conditioners.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$3 \cdot 10^5$$$.

Output format:
For each test case output $$$n$$$ integers separated by space: temperatures of air in cells.

Constraints:
Time limit: 2.0 s
Memory limit: 512.0 MB

Example input:
5

6 2
2 5
14 16

10 1
7
30

5 5
3 1 4 2 5
3 1 4 2 5

7 1
1
1000000000

6 3
6 1 3
5 5 5

Example output:
15 14 15 16 16 17 
36 35 34 33 32 31 30 31 32 33 
1 2 3 4 5 
1000000000 1000000001 1000000002 1000000003 1000000004 1000000005 1000000006 
5 6 5 6 6 5

Editorial:
Let's calculate two arrays $$$L$$$ and $$$R$$$, where:
• $$$L_i$$$ is the temperature in cell $$$i$$$ if we take only air conditioners with numbers less than or equal to $$$i$$$;
• $$$R_i$$$ is the temperature in cell $$$i$$$ if we take only air conditioners with numbers greater than or equal to $$$i$$$;
Let's show how to calculate array $$$L$$$. We will calculate values from left to right using DP and next formula: $$$L_i = \min(L_{i+1}+1, c_i)$$$, where $$$c_i$$$ is the temperature of air conditioner in cell $$$i$$$ (or infinity if there is no air conditioner in this cell). Indeed, the value of $$$L_i$$$ is either determined by the air conditioner in this cell (i.e. equals $$$c_i$$$) or by some air conditioner to the left, but this means that we should take the answer from the previous cell and increase it by $$$1$$$. The full code for calculating $$$L$$$ looks like this:
In exactly the same way (but from right to left) we will calculate $$$R$$$:
The answer for cell $$$i$$$ is $$$\min(L[i], R[i])$$$.