A15453 | Location
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题目描述
You are given two arrays of integers $a_1, a_2, \ldots, a_n$ and $b_1, b_2, \ldots, b_n$ . You need to handle $q$ queries of the following two types:
- $1$ $l$ $r$ $x$ : assign $a_i := x$ for all $l \leq i \leq r$ ;
- $2$ $l$ $r$ : find the minimum value of the following expression among all $l \leq i \leq r$ : $$$$\frac{\operatorname{lcm}(a_i, b_i)}{\gcd(a_i, b_i)}. $$ </li></ul><p>In this problem $\\gcd(x, y)$ denotes the <a href="https://en.wikipedia.org/wiki/Greatest_common_divisor">greatest common divisor</a> of $x$ and $y$ , and $\\operatorname{lcm}(x, y)$ denotes the <a href="https://en.wikipedia.org/wiki/Least_common_multiple">least common multiple</a> of $x$ and $y$$$.
- $1$ $l$ $r$ $x$ : assign $a_i := x$ for all $l \leq i \leq r$ ;
- $2$ $l$ $r$ : find the minimum value of the following expression among all $l \leq i \leq r$ : $$$$\frac{\operatorname{lcm}(a_i, b_i)}{\gcd(a_i, b_i)}. $$ </li></ul><p>In this problem $\\gcd(x, y)$ denotes the <a href="https://en.wikipedia.org/wiki/Greatest_common_divisor">greatest common divisor</a> of $x$ and $y$ , and $\\operatorname{lcm}(x, y)$ denotes the <a href="https://en.wikipedia.org/wiki/Least_common_multiple">least common multiple</a> of $x$ and $y$$$.
输入格式
The first line contains two integers $n$ and $q$ ( $1 \leq n, q \leq 5 \cdot 10^4$ ) — the number of numbers in the arrays $a$ and $b$ and the number of queries.
The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ( $1 \leq a_i \leq 5 \cdot 10^4$ ) — the elements of the array $a$ .
The third line contains $n$ integers $b_1, b_2, \ldots, b_n$ ( $1 \leq b_i \leq 5 \cdot 10^4$ ) — the elements of the array $b$ .
Then $q$ lines follow, $j$ -th of which starts with an integer $t_j$ ( $1 \leq t_j \leq 2$ ) and means that the $j$ -th query has type $t_j$ .
If $t_j = 1$ , it is followed by three integers $l_j$ , $r_j$ , and $x_j$ ( $1 \leq l_j \leq r_j \leq n$ , $1 \leq x_j \leq 5 \cdot 10^4$ ).
If $t_j = 2$ , it is followed by two integers $l_j$ and $r_j$ ( $1 \leq l_j \leq r_j \leq n$ ).
It is guaranteed that there is at least one query of type $2$ .
The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ( $1 \leq a_i \leq 5 \cdot 10^4$ ) — the elements of the array $a$ .
The third line contains $n$ integers $b_1, b_2, \ldots, b_n$ ( $1 \leq b_i \leq 5 \cdot 10^4$ ) — the elements of the array $b$ .
Then $q$ lines follow, $j$ -th of which starts with an integer $t_j$ ( $1 \leq t_j \leq 2$ ) and means that the $j$ -th query has type $t_j$ .
If $t_j = 1$ , it is followed by three integers $l_j$ , $r_j$ , and $x_j$ ( $1 \leq l_j \leq r_j \leq n$ , $1 \leq x_j \leq 5 \cdot 10^4$ ).
If $t_j = 2$ , it is followed by two integers $l_j$ and $r_j$ ( $1 \leq l_j \leq r_j \leq n$ ).
It is guaranteed that there is at least one query of type $2$ .
输出格式
For each query of the second type, output the minimum value of the expression.
输入输出样例
输入 #1
10 10 6 10 15 4 9 25 2 3 5 30 1 2 3 4 6 9 12 15 18 30 2 1 10 1 7 10 9 2 5 10 1 1 6 14 2 4 7 2 3 9 1 2 9 30 2 1 4 2 3 7 2 5 10
输出 #1
1 2 12 2 10 5 2
输入 #2
4 4 10 2 12 5 1 12 16 1 2 2 4 1 2 3 18 1 2 2 10 2 2 3
输出 #2
5 30
In the first example:
- For the first query we can choose $i = 4$ . So the value is $\frac{\operatorname{lcm}(4, 4)}{\gcd(4, 4)} = \frac{4}{4} = 1$ .
- After the second query the array $a = [6, 10, 15, 4, 9, 25, 9, 9, 9, 9]$ .
- For the third query we can choose $i = 9$ . So the value is $\frac{\operatorname{lcm}(9, 18)}{\gcd(9, 18)} = \frac{18}{9} = 2$ .
In the second:
- For the first query we can choose $i = 4$ . So the value is $\frac{\operatorname{lcm}(1, 5)}{\gcd(1, 5)} = \frac{5}{1} = 5$ .
- After the second query the array $a = [10, 18, 18, 5]$ .
- After the third query the array $a = [10, 10, 18, 5]$ .
- For the fourth query we can choose $i = 2$ . So the value is $\frac{\operatorname{lcm}(10, 12)}{\gcd(10, 12)} = \frac{60}{2} = 30$ .
- For the first query we can choose $i = 4$ . So the value is $\frac{\operatorname{lcm}(4, 4)}{\gcd(4, 4)} = \frac{4}{4} = 1$ .
- After the second query the array $a = [6, 10, 15, 4, 9, 25, 9, 9, 9, 9]$ .
- For the third query we can choose $i = 9$ . So the value is $\frac{\operatorname{lcm}(9, 18)}{\gcd(9, 18)} = \frac{18}{9} = 2$ .
In the second:
- For the first query we can choose $i = 4$ . So the value is $\frac{\operatorname{lcm}(1, 5)}{\gcd(1, 5)} = \frac{5}{1} = 5$ .
- After the second query the array $a = [10, 18, 18, 5]$ .
- After the third query the array $a = [10, 10, 18, 5]$ .
- For the fourth query we can choose $i = 2$ . So the value is $\frac{\operatorname{lcm}(10, 12)}{\gcd(10, 12)} = \frac{60}{2} = 30$ .
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