Score : $500$ points

问题描述

我们有一条长度为 $L$ 的面包，打算将其切割并分配给 $N$ 个孩子。 第 $i$ 个孩子（$1\leq i\leq N$）希望得到长度为 $A_i$ 的面包片。

现在，Takahashi 将重复以下操作以将面包切割成长度分别为 $A_1, A_2, \ldots, A_N$ 的面包片分给孩子：

选择一条长度为 $k$ 的面包，并在 $1$ 和 $k-1$（包括两端点）之间选择一个整数 $x$。将这条面包切成长度分别为 $x$ 和 $k-x$ 的两条面包片。 不论 $x$ 的取值如何，这个操作的代价均为 $k$。

每个孩子 $i$ 必须精确收到长度为 $A_i$ 的面包片，但允许留下一些未分配的面包片。

求解分配所有孩子面包所需的最小成本。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

We have a loaf of bread of length $L$, which we will cut and distribute to $N$ children.
The $i$-th child $(1\leq i\leq N)$ wants a loaf of length $A_i$.

Now, Takahashi will repeat the operation below to cut the loaf into lengths $A_1, A_2, \ldots, A_N$ for the children.

Choose a loaf of length $k$ and an integer $x$ between $1$ and $k-1$ (inclusive). Cut the loaf into two loaves of length $x$ and $k-x$.
This operation incurs a cost of $k$ regardless of the value of $x$.

Each child $i$ must receive a loaf of length exactly $A_i$, but it is allowed to leave some loaves undistributed.

Find the minimum cost needed to distribute loaves to all children.

Constraints

• $2 \leq N \leq 2\times 10^5$
• $1\leq A_i\leq 10^9$
• $A_1+A_2+\cdots+A_N\leq L\leq 10^{15}$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $L$

$A_1$ $A_2$ $\ldots$ $A_N$

Output

Print the minimum cost needed to distribute loaves to all children.

Sample Input 1

5 7
1 2 1 2 1

Sample Output 1

16

Takahashi can cut the loaf for the children as follows.

• Choose the loaf of length $7$ and $x=3$, cutting it into loaves of length $3$ and $4$ for a cost of $7$.
• Choose the loaf of length $3$ and $x=1$, cutting it into loaves of length $1$ and $2$ for a cost of $3$. Give the former to the $1$-st child.
• Choose the loaf of length $2$ and $x=1$, cutting it into two loaves of length $1$ for a cost of $2$. Give them to the $3$-rd and $5$-th children.
• Choose the loaf of length $4$ and $x=2$, cutting it into two loaves of length $2$ for a cost of $4$. Give them to the $2$-nd and $4$-th children.

This incurs a cost of $7+3+2+4=16$, which is the minimum possible. There will be no leftover loaves.

Sample Input 2

3 1000000000000000
1000000000 1000000000 1000000000

Sample Output 2

1000005000000000

Note that each child $i$ must receive a loaf of length exactly $A_i$.

update @ 2024/3/10 10:46:04