Score : $300$ points

问题描述

给定$N$个区间，编号为从1到$N$，它们分别为：

• 如果$t_i=1$，则第$i$个区间为$[l_i,r_i]$；
• 如果$t_i=2$，则第$i$个区间为$[l_i,r_i)$；
• 如果$t_i=3$，则第$i$个区间为$(l_i,r_i]$；
• 如果$t_i=4$，则第$i$个区间为$(l_i,r_i)$。

请问有多少对整数$(i,j)$满足$1 \leq i < j \leq N$，使得第$i$个区间和第$j$个区间相交？

什么是$[X,Y],[X,Y),(X,Y],(X,Y)$？

• 闭区间$[X,Y]$是一个包含所有实数$x$的区间，满足$X \leq x \leq Y$。
• 半开区间$[X,Y)$是一个包含所有实数$x$的区间，满足$X \leq x < Y$。
• 半开区间$(X,Y]$是一个包含所有实数$x$的区间，满足$X < x \leq Y$。
• 开区间$(X,Y)$是一个包含所有实数$x$的区间，满足$X < x < Y$。

简单来说，方括号[]表示端点被包含在内，圆括号()表示端点被排除在外。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

You are given $N$ intervals numbered $1$ through $N$, that are as follows:

• if $t_i=1$, Interval $i$ is $[l_i,r_i]$;
• if $t_i=2$, Interval $i$ is $[l_i,r_i)$;
• if $t_i=3$, Interval $i$ is $(l_i,r_i]$;
• if $t_i=4$, Interval $i$ is $(l_i,r_i)$.

How many pairs of integers $(i,j)$ satisfying $1 \leq i \lt j \leq N$ are there such that Interval $i$ and Interval $j$ intersect?

What are $[X,Y],[X,Y),(X,Y],(X,Y)$?

• A closed interval $[X,Y]$ is an interval consisting of all real numbers $x$ such that $X \leq x \leq Y$.
• A half-open interval $[X,Y)$ is an interval consisting of all real numbers $x$ such that $X \leq x < Y$.
• A half-open interval $(X,Y]$ is an interval consisting of all real numbers $x$ such that $X < x \leq Y$.
• A open interval $(X,Y)$ is an interval consisting of all real numbers $x$ such that $X < x < Y$.

Roughly speaking, square brackets $[]$ mean the endpoint is included, and curly brackets $()$ mean the endpoint is excluded.

Constraints

• $2 \leq N \leq 2000$
• $1 \leq t_i \leq 4$
• $1 \leq l_i \lt r_i \leq 10^9$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$

$t_1$ $l_1$ $r_1$

$t_2$ $l_2$ $r_2$

$\hspace{1cm}\vdots$

$t_N$ $l_N$ $r_N$

Output

Print the number of pairs of integers $(i,j)$ such that Interval $i$ and Interval $j$ intersect.

Sample Input 1

3
1 1 2
2 2 3
3 2 4

Sample Output 1

2

As defined in the Problem Statement, Interval $1$ is $[1,2]$, Interval $2$ is $[2,3)$, and Interval $3$ is $(2,4]$.

There are two pairs of integers $(i,j)$ such that Interval $i$ and Interval $j$ intersect: $(1,2)$ and $(2,3)$. For the first pair, the intersection is $[2,2]$, and for the second pair, the intersection is $(2,3)$.

Sample Input 2

19
4 210068409 221208102
4 16698200 910945203
4 76268400 259148323
4 370943597 566244098
1 428897569 509621647
4 250946752 823720939
1 642505376 868415584
2 619091266 868230936
2 306543999 654038915
4 486033777 715789416
1 527225177 583184546
2 885292456 900938599
3 264004185 486613484
2 345310564 818091848
1 152544274 521564293
4 13819154 555218434
3 507364086 545932412
4 797872271 935850549
2 415488246 685203817

Sample Output 2

102

update @ 2024/3/10 09:20:37