102525: [AtCoder]ABC252 F - Bread

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 $A_i$.

问题描述

我们有一个长度为$L$的面包，要将其切开并分发给$N$个孩子。

第$i$个孩子（$1\leq i\leq N$）想要一个长度为$A_i$的面包。

现在，高桥将重复以下操作，将面包切成长度为$A_1, A_2, \ldots, A_N$的面包分发给孩子们。

选择长度为$k$的面包和一个介于$1$和$k-1$（包括）之间的整数$x$。将面包切成长度为$x$和$k-x$的两个面包。

无论$x$的值如何，此操作的成本为$k$。

每个孩子$i$必须收到长度恰好为$A_i$的面包，但允许留下一些面包不分发。

找出将面包分发给所有孩子的最小成本。

约束

• $2 \leq N \leq 2\times 10^5$
• $1\leq A_i\leq 10^9$
• $A_1+A_2+\cdots+A_N\leq L\leq 10^{15}$
• 输入中的所有值都是整数。

输入

输入从标准输入以以下格式给出：

$N$ $L$
$A_1$ $A_2$ $\ldots$ $A_N$

输出

打印将面包分发给所有孩子的最小成本。

样例输入1

5 7
1 2 1 2 1

样例输出1

16

高桥可以按照以下方式将面包分发给孩子们。

• 选择长度为$7$的面包和$x=3$，将其切成长度为$3$和$4$的面包，成本为$7$。
• 选择长度为$3$的面包和$x=1$，将其切成长度为$1$和$2$的面包，成本为$3$。将前者分发给第$1$个孩子。
• 选择长度为$2$的面包和$x=1$，将其切成两个长度为$1$的面包，成本为$2$。将它们分发给第$3$个和第$5$个孩子。
• 选择长度为$4$的面包和$x=2$，将其切成两个长度为$2$的面包，成本为$4$。将它们分发给第$2$个和第$4$个孩子。

这将导致成本为$7+3+2+4=16$，这是可能的最小成本。不会有剩下的面包。

样例输入2

3 1000000000000000
1000000000 1000000000 1000000000

样例输出2

1000005000000000

请注意，每个孩子$i$必须收到长度恰好为$A_i$的面包。