Score : $300$ points

问题描述

在 $xy$ 平面上有 $N$ 个人。第 $i$ 个人位于坐标 $(X_i, Y_i)$，所有人的位置各不相同。

我们有一个长度为 $N$ 的字符串 $S$，由字符 L 和 R 组成。
若 $S_i =$ R，表示第 $i$ 个人面向右方；若 $S_i =$ L，则表示第 $i$ 个人面向左方。所有人同时开始朝他们面对的方向行走。其中，右和左分别对应正和负的 $x$ 轴方向。

例如，下图展示了当 $(X_1, Y_1) = (2, 3)$，$(X_2, Y_2) = (1, 1)$，$(X_3, Y_3) = (4, 1)$，且 $S =$ RRL 时人们的移动情况。

我们称，当两个朝相反方向行走的人到达同一个位置时发生碰撞。如果所有人都无限期地继续行走，会发生碰撞吗？

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

There are $N$ people in an $xy$-plane. Person $i$ is at $(X_i, Y_i)$. The positions of all people are different.

We have a string $S$ of length $N$ consisting of L and R.
If $S_i =$ R, Person $i$ is facing right; if $S_i =$ L, Person $i$ is facing left. All people simultaneously start walking in the direction they are facing. Here, right and left correspond to the positive and negative $x$-direction, respectively.

For example, the figure below shows the movement of people when $(X_1, Y_1) = (2, 3), (X_2, Y_2) = (1, 1), (X_3, Y_3) =(4, 1), S =$ RRL.

We say that there is a collision when two people walking in opposite directions come to the same position. Will there be a collision if all people continue walking indefinitely?

Constraints

• $2 \leq N \leq 2 \times 10^5$
• $0 \leq X_i \leq 10^9$
• $0 \leq Y_i \leq 10^9$
• $(X_i, Y_i) \neq (X_j, Y_j)$ if $i \neq j$.
• All $X_i$ and $Y_i$ are integers.
• $S$ is a string of length $N$ consisting of L and R.

Input

Input is given from Standard Input in the following format:

$N$

$X_1$ $Y_1$

$X_2$ $Y_2$

$\vdots$

$X_N$ $Y_N$

$S$

Output

If there will be a collision, print Yes; otherwise, print No.

Sample Input 1

3
2 3
1 1
4 1
RRL

Sample Output 1

Yes

This input corresponds to the example in the Problem Statement.
If all people continue walking, Person $2$ and Person $3$ will collide. Thus, Yes should be printed.

Sample Input 2

2
1 1
2 1
RR

Sample Output 2

No

Since Person $1$ and Person $2$ walk in the same direction, they never collide.

Sample Input 3

10
1 3
1 4
0 0
0 2
0 4
3 1
2 4
4 2
4 4
3 3
RLRRRLRLRR

Sample Output 3

Yes

update @ 2024/3/10 10:27:01