Line Empire

1000ms

262144K

Description:

You are an ambitious king who wants to be the Emperor of The Reals. But to do that, you must first become Emperor of The Integers.

Consider a number axis. The capital of your empire is initially at $$$0$$$. There are $$$n$$$ unconquered kingdoms at positions $$$0<x_1<x_2<\ldots<x_n$$$. You want to conquer all other kingdoms.

There are two actions available to you:

You can change the location of your capital (let its current position be $$$c_1$$$) to any other conquered kingdom (let its position be $$$c_2$$$) at a cost of $$$a\cdot |c_1-c_2|$$$.
From the current capital (let its current position be $$$c_1$$$) you can conquer an unconquered kingdom (let its position be $$$c_2$$$) at a cost of $$$b\cdot |c_1-c_2|$$$. You cannot conquer a kingdom if there is an unconquered kingdom between the target and your capital.

Note that you cannot place the capital at a point without a kingdom. In other words, at any point, your capital can only be at $$$0$$$ or one of $$$x_1,x_2,\ldots,x_n$$$. Also note that conquering a kingdom does not change the position of your capital.

Find the minimum total cost to conquer all kingdoms. Your capital can be anywhere at the end.

Input:

The first line contains a single integer $$$t$$$ ($$$1 \le t \le 1000$$$) — the number of test cases. The description of each test case follows.

The first line of each test case contains $$$3$$$ integers $$$n$$$, $$$a$$$, and $$$b$$$ ($$$1 \leq n \leq 2 \cdot 10^5$$$; $$$1 \leq a,b \leq 10^5$$$).

The second line of each test case contains $$$n$$$ integers $$$x_1, x_2, \ldots, x_n$$$ ($$$1 \leq x_1 < x_2 < \ldots < x_n \leq 10^8$$$).

The sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.

Output:

For each test case, output a single integer — the minimum cost to conquer all kingdoms.

Sample Input:

4
5 2 7
3 5 12 13 21
5 6 3
1 5 6 21 30
2 9 3
10 15
11 27182 31415
16 18 33 98 874 989 4848 20458 34365 38117 72030

Sample Output:

Note:

Here is an optimal sequence of moves for the second test case:

Conquer the kingdom at position $$$1$$$ with cost $$$3\cdot(1-0)=3$$$.
Move the capital to the kingdom at position $$$1$$$ with cost $$$6\cdot(1-0)=6$$$.
Conquer the kingdom at position $$$5$$$ with cost $$$3\cdot(5-1)=12$$$.
Move the capital to the kingdom at position $$$5$$$ with cost $$$6\cdot(5-1)=24$$$.
Conquer the kingdom at position $$$6$$$ with cost $$$3\cdot(6-5)=3$$$.
Conquer the kingdom at position $$$21$$$ with cost $$$3\cdot(21-5)=48$$$.
Conquer the kingdom at position $$$30$$$ with cost $$$3\cdot(30-5)=75$$$.

The total cost is $$$3+6+12+24+3+48+75=171$$$. You cannot get a lower cost than this.

Informação

Provedor Codeforces

Código CF1659C