102757: [AtCoder]ABC275 Ex - Monster

Problem Statement

There are $N$ monsters along a number line. At the coordinate $i$ $(1\leq i\leq N)$ is a monster with a stamina of $A_i$.
Additionally, at the coordinate $i$, there is a permanent shield of a strength $B_i$.
This shield persists even when the monster at the same coordinate has a health of $0$ or below.

Takahashi, a magician, can perform the following operation any number of times.

• Choose integers $l$ and $r$ satisfying $1\leq l\leq r\leq N$.
• Then, consume $\max(B_l, B_{l+1}, \ldots, B_r)$ MP (magic points) to decrease by $1$ the stamina of each of the monsters at the coordinates $l,l+1,\ldots,r$.

When choosing $l$ and $r$, it is fine if some of the monsters at the coordinates $l,l+1,\ldots,r$ already have a stamina of $0$ or below.
Note, however, that the shields at all those coordinates still exist.

Takahashi wants to make the stamina of every monster $0$ or below.
Find the minimum total MP needed to achieve his objective.

Constraints

• $1 \leq N \leq 10^5$
• $1 \leq A_i,B_i \leq 10^9$
• All values in the input are integers.

Input

The input is given from Standard Input in the following format:

$N$
$A_1$ $A_2$ $\ldots$ $A_N$
$B_1$ $B_2$ $\ldots$ $B_N$

Output

Print the minimum total MP needed to achieve his objective.

Sample Input 1

5
4 3 5 1 2
10 40 20 60 50

Sample Output 1

210

Takahashi can achieve his objective as follows.

• Choose $(l,r)=(1,5)$. Consume $\max(10,40,20,60,50)=60$ MP, and the staminas of the monsters are $(3,2,4,0,1)$.
• Choose $(l,r)=(1,5)$. Consume $\max(10,40,20,60,50)=60$ MP, and the staminas of the monsters are $(2,1,3,-1,0)$.
• Choose $(l,r)=(1,3)$. Consume $\max(10,40,20)=40$ MP, and the staminas of the monsters are $(1,0,2,-1,0)$.
• Choose $(l,r)=(1,1)$. Consume $\max(10)=10$ MP, and the staminas of the monsters are $(0,0,2,-1,0)$.
• Choose $(l,r)=(3,3)$. Consume $\max(20)=20$ MP, and the staminas of the monsters are $(0,0,1,-1,0)$.
• Choose $(l,r)=(3,3)$. Consume $\max(20)=20$ MP, and the staminas of the monsters are $(0,0,0,-1,0)$.

Here, he consumes a total of $60+60+40+10+20+20=210$ MP, which is the minimum possible.

Sample Input 2

1
1000000000
1000000000

Sample Output 2

1000000000000000000

Sample Input 3

10
522 4575 6426 9445 8772 81 3447 629 3497 7202
7775 4325 3982 4784 8417 2156 1932 5902 5728 8537

Sample Output 3

77917796

问题描述

在数轴上有$N$个怪物。在坐标$i$ $(1\leq i\leq N)$处有一个体力为$A_i$的怪物。
此外，在坐标$i$处有一个永久性的护盾，强度为$B_i$。
即使同坐标的怪物体力为$0$或以下，这个护盾仍然存在。

魔法师高桥可以无数次地执行以下操作。

• 选择满足$1\leq l\leq r\leq N$的整数$l$和$r$。
• 然后，消耗$\max(B_l, B_{l+1}, \ldots, B_r)$ MP（魔法点数），使得坐标为$l,l+1,\ldots,r$的怪物体力各减少$1$。

选择$l$和$r$时，如果坐标$l,l+1,\ldots,r$的怪物中有一些已经体力为$0$或以下也是可以的。
但是请注意，所有这些坐标的护盾仍然存在。

高桥想要使所有怪物的体力都为$0$或以下。
找出实现他的目标所需的最低总MP。

限制条件

• $1 \leq N \leq 10^5$
• $1 \leq A_i,B_i \leq 10^9$
• 输入中的所有值都是整数。

输入

输入将从标准输入以以下格式给出：

$N$
$A_1$ $A_2$ $\ldots$ $A_N$
$B_1$ $B_2$ $\ldots$ $B_N$

输出

打印实现他的目标所需的最低总MP。

样例输入1

5
4 3 5 1 2
10 40 20 60 50

样例输出1

210

高桥可以按照以下方式实现他的目标。

• 选择$(l,r)=(1,5)$。消耗$\max(10,40,20,60,50)=60$ MP，怪物的体力分别为$(3,2,4,0,1)$。
• 选择$(l,r)=(1,5)$。消耗$\max(10,40,20,60,50)=60$ MP，怪物的体力分别为$(2,1,3,-1,0)$。
• 选择$(l,r)=(1,3)$。消耗$\max(10,40,20)=40$ MP，怪物的体力分别为$(1,0,2,-1,0)$。
• 选择$(l,r)=(1,1)$。消耗$\max(10)=10$ MP，怪物的体力分别为$(0,0,2,-1,0)$。
• 选择$(l,r)=(3,3)$。消耗$\max(20)=20$ MP，怪物的体力分别为$(0,0,1,-1,0)$。
• 选择$(l,r)=(3,3)$。消耗$\max(20)=20$ MP，怪物的体力分别为$(0,0,0,-1,0)$。

在这里，他总共消耗了$60+60+40+10+20+20=210$ MP，这是最低可能的。

样例输入2

1
1000000000
1000000000

样例输出2

1000000000000000000

样例输入3

10
522 4575 6426 9445 8772 81 3447 629 3497 7202
7775 4325 3982 4784 8417 2156 1932 5902 5728 8537

样例输出3

77917796