标签：set text leq version Easy integer 100 Balance mex

D1. Balance (Easy version)

This is the easy version of the problem. The only difference is that in this version there are no "remove" queries.

Initially you have a set containing one element — $0$. You need to handle $q$ queries of the following types:

+  $x$ — add the integer $x$ to the set. It is guaranteed that this integer is not contained in the set;
?  $k$ — find the $k\text{-mex}$ of the set.
In our problem, we define the $k\text{-mex}$ of a set of integers as the smallest non-negative integer $x$ that is divisible by $k$ and which is not contained in the set.

Input

The first line contains an integer $q$ $(1 \leq q \leq 2 \cdot {10}^{5})$ — the number of queries.

The following $q$ lines describe the queries.

An addition query of integer $x$ is given in the format + $x$ $(1 \leq x \leq {10}^{18})$. It is guaranteed that $x$ was not contained in the set.

A search query of $k\text{-mex}$ is given in the format ? $k$ $(1 \leq k \leq {10}^{18})$.

It is guaranteed that there will be at least one query of type ? .

Output

For each query of type ? output a single integer — the $k\text{-mex}$ of the set.

Examples

input

15
+ 1
+ 2
? 1
+ 4
? 2
+ 6
? 3
+ 7
+ 8
? 1
? 2
+ 5
? 1
+ 1000000000000000000
? 1000000000000000000

output

3
6
3
3
10
3
2000000000000000000

input

6
+ 100
? 100
+ 200
? 100
+ 50
? 50

output

200
300
150

Note

In the first example:

After the first and second queries, the set will contain elements $\{0,1,2\}$. The smallest non-negative number that is divisible by $1$ and is not contained in the set is $3$.

After the fourth query, the set will contain the elements $\{0,1,2,4\}$. The smallest non-negative number that is divisible by $2$ and is not contained in the set is $6$.

In the second example:

• Initially, the set contains only the element $\{0\}$.
• After adding an integer $100$ the set contains elements $\{0,100\}$.
• $100\text{-mex}$ of the set is $200$.
• After adding an integer $200$ the set contains elements $\{0,100,200\}$.
• $100\text{-mex}$ of the set is $300$.
• After adding an integer $50$ the set contains elements $\{0,50,100,200\}$.
• $50\text{-mex}$ of the set is $150$.

 

解题思路

　　先想一个最暴力的思路，就是我们用一个哈希表维护集合中存在的数，对于每个询问$k$，我们依次枚举$k$的倍数看看是否出现在哈希表中，直到枚举到某个$k$的倍数在哈希表中不存在。

　　但这种做法必定会超时，如果依次插入$k$的倍数，然后每次询问都是查询$k$，那么计算量就是$O(n^2)$级别的。因此可以开个哈希表来记录对于某个$k$上一次枚举到哪个数，如果后面再次询问$k$，那么直接从上次枚举的到的数开始往后枚举$k$的倍数就可以了。这是因为对于$\text{mex}$而言，当插入的数越多，$\text{mex}$值只会变大而不会变小。

　　这种做法的时间复杂度是多少呢？考虑最糟糕的情况，就是依次插入$1 \sim n$，然后依次询问$1 \sim n$，那么计算量大概就是$n + \frac{n}{2} + \frac{n}{3} + \dots + \frac{n}{n} \approx n \log{n}$。

　　这题卡std::unordered_map和std::unordered_set，因此要么人为手动初始化大小，要么改用std::map和std::set。　　

　　AC代码如下：

 1 #include <bits/stdc++.h>
 2 using namespace std;
 3 
 4 typedef long long LL;
 5 
 6 const int N = 1e6 + 10;
 7 
 8 void solve() {
 9     int n;
10     scanf("%d", &n);
11     unordered_set<LL> st(N);
12     unordered_map<LL, LL> mp(N);
13     while (n--) {
14         char op[2];
15         LL x;
16         scanf("%s %lld", op, &x);
17         if (op[0] == '+') {
18             st.insert(x);
19         }
20         else {
21             if (!mp.count(x)) mp[x] = x;
22             while (st.count(mp[x])) {
23                 mp[x] += x;
24             } 
25             printf("%lld\n", mp[x]);
26         }
27     }
28 }
29 
30 int main() {
31     int t = 1;
32     // scanf("%d", &t);
33     while (t--) {
34         solve();
35     }
36     
37     return 0;
38 }

 

参考资料

　　Codeforces Round #830 (Div. 2) D1/D2(mex问题)：https://zhuanlan.zhihu.com/p/576620849

标签：set,text,leq,version,Easy,integer,100,Balance,mex
From： https://www.cnblogs.com/onlyblues/p/16821998.html