codeforces#P1799F. Halve or Subtract
Halve or Subtract
Description
You have an array of positive integers $a_1, a_2, \ldots, a_n$, of length $n$. You are also given a positive integer $b$.
You are allowed to perform the following operations (possibly several) times in any order:
- Choose some $1 \le i \le n$, and replace $a_i$ with $\lceil \frac{a_i}{2} \rceil$. Here, $\lceil x \rceil$ denotes the smallest integer not less than $x$.
- Choose some $1 \le i \le n$, and replace $a_i$ with $\max(a_i - b, 0)$.
However, you must also follow these rules:
- You can perform at most $k_1$ operations of type 1 in total.
- You can perform at most $k_2$ operations of type 2 in total.
- For all $1 \le i \le n$, you can perform at most $1$ operation of type 1 on element $a_i$.
- For all $1 \le i \le n$, you can perform at most $1$ operation of type 2 on element $a_i$.
The cost of an array is the sum of its elements. Find the minimum cost of $a$ you can achieve by performing these operations.
Input consists of multiple test cases. The first line contains a single integer $t$, the number of test cases ($1 \le t \le 5000$).
The first line of each test case contains $n$, $b$, $k_1$, and $k_2$ ($1 \le n \le 5000$, $1 \le b \le 10^9$, $0 \le k_1, k_2 \le n$).
The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ describing the array $a$ ($1 \le a_i \le 10^9$).
It is guaranteed the sum of $n$ over all test cases does not exceed $5000$.
For each test case, print the minimum cost of $a$ you can achieve by performing the operations.
Input
Input consists of multiple test cases. The first line contains a single integer $t$, the number of test cases ($1 \le t \le 5000$).
The first line of each test case contains $n$, $b$, $k_1$, and $k_2$ ($1 \le n \le 5000$, $1 \le b \le 10^9$, $0 \le k_1, k_2 \le n$).
The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ describing the array $a$ ($1 \le a_i \le 10^9$).
It is guaranteed the sum of $n$ over all test cases does not exceed $5000$.
Output
For each test case, print the minimum cost of $a$ you can achieve by performing the operations.
7
3 2 1 1
9 3 5
2 1 2 0
1000000000 1
5 3 1 1
2 8 3 19 3
6 9 4 2
1 2 3 4 5 6
3 10 3 3
1 2 3
5 1 0 0
999999999 999999999 999999999 999999999 999999999
5 5 4 3
5 9 10 7 4
11
500000001
23
6
0
4999999995
6
Note
In the first test case, you can do the following:
- Perform operation 2 on element $a_3$. It changes from $5$ to $3$.
- Perform operation 1 on element $a_1$. It changes from $9$ to $5$.
After these operations, the array is $a = [5, 3, 3]$ has a cost $5 + 3 + 3 = 11$. We can show that this is the minimum achievable cost.
In the second test case, note that we are not allowed to perform operation 1 more than once on $a_1$. So it is optimal to apply operation 1 once to each $a_1$ and $a_2$. Alternatively we could apply operation 1 only once to $a_1$, since it has no effect on $a_2$.
In the third test case, here is one way to achieve a cost of $23$:
- Apply operation 1 to $a_4$. It changes from $19$ to $10$.
- Apply operation 2 to $a_4$. It changes from $10$ to $7$.
After these operations, $a = [2, 8, 3, 7, 3]$. The cost of $a$ is $2 + 8 + 3 + 7 + 3 = 23$. We can show that this is the minimum achievable cost.