cf 979D Kuro and GCD and XOR and SUM

一 原题

D. Kuro and GCD and XOR and SUM

time limit per test

2 seconds

memory limit per test

512 megabytes

input

standard input

output

standard output

Kuro is currently playing an educational game about numbers. The game focuses on the greatest common divisor (GCD), the XOR value, and the sum of two numbers. Kuro loves the game so much that he solves levels by levels day by day.

Sadly, he's going on a vacation for a day, and he isn't able to continue his solving streak on his own. As Katie is a reliable person, Kuro kindly asked her to come to his house on this day to play the game for him.

Initally, there is an empty array $a$. The game consists of $q$ tasks of two types. The first type asks Katie to add a number $u_i$ to $a$. The second type asks Katie to find a number $v$ existing in $a$ such that $k_i\mid GCD(x_i,v)$, $x_i+v\le s_i$, and $x_i\oplus v$ is maximized, where $\oplus$ denotes the bitwise XOR operation, $GCD(c,d)$ denotes the greatest common divisor of integers $c$ and $d$, and $y\mid x$ means $x$ is divisible by $y$, or report -1 if no such numbers are found.

Since you are a programmer, Katie needs you to automatically and accurately perform the tasks in the game to satisfy her dear friend Kuro. Let's help her!

Input

The first line contains one integer $q$ ($2\le q\le {10}^5$) — the number of tasks the game wants you to perform.

$q$ lines follow, each line begins with an integer $t_i$ — the type of the task:

• If $t_i=1$, an integer $u_i$ follow ($1\le u_i\le {10}^5$) — you have to add $u_i$ to the array $a$.
• If $t_i=2$, three integers $x_i$, $k_i$, and $s_i$ follow ($1\le x_i,k_i,s_i\le {10}^5$) — you must find a number $v$ existing in the array $a$ such that $k_i\mid GCD(x_i,v)$, $x_i+v\le s_i$, and $x_i\oplus v$ is maximized, where $\oplus$ denotes the XOR operation, or report -1 if no such numbers are found.

It is guaranteed that the type of the first task is type $1$, and there exists at least one task of type $2$.

Output

For each task of type $2$, output on one line the desired number $v$, or -1 if no such numbers are found.

Examples

input

Copy

5
1 1
1 2
2 1 1 3
2 1 1 2
2 1 1 1

output

Copy

2
1
-1

input

Copy

10
1 9
2 9 9 22
2 3 3 18
1 25
2 9 9 20
2 25 25 14
1 20
2 26 26 3
1 14
2 20 20 9

output

Copy

9
9
9
-1
-1
-1

Note

In the first example, there are 5 tasks:

• The first task requires you to add $1$ into $a$. $a$ is now $\left\{1\right\}$.
• The second task requires you to add $2$ into $a$. $a$ is now $\{1,2\}$.
• The third task asks you a question with $x=1$, $k=1$ and $s=3$. Taking both $1$ and $2$ as $v$ satisfies $1\mid GCD(1,v)$ and $1+v\le 3$. Because $2\oplus 1=3>1\oplus 1=0$, $2$ is the answer to this task.
• The fourth task asks you a question with $x=1$, $k=1$ and $s=2$. Only $v=1$ satisfies $1\mid GCD(1,v)$ and $1+v\le 2$, so $1$ is the answer to this task.
• The fifth task asks you a question with $x=1$, $k=1$ and $s=1$. There are no elements in $a$ that satisfy the conditions, so we report -1 as the answer to this task.

二 分析

初始一个空数组，有两种类型的操作：1）在数组中增加一个数；2）接受3个参数：x,k,s，要求在数组中找到一个数v最大化x^v，并且v<=s-x，k|v.如果找不到，返回-1,数组中的数大小不超过10^5

给定一个数，照与它异或结果最大的数肯定是用trie了，但这题还有两个限制条件。根据数组中的数不超过10^5，我们可以建立10^5棵trie，第i棵树存储了所有能被i整除的数，同时每个trie的节点存储一下子树中最小的数。想到这些问题就解决了。

三 代码

#include <cstdio>
#include <vector>

// #define local

const int maxn = 1e5 + 5;
const int maxd = 20;
const int inf = 0x7fffffff;

int q, op, x, k, s;
bool flag[maxn];
std::vector<int> d[maxn];

void Init() {
	for (int i = 1; i < maxn; i++) {
		for (int j = i; j < maxn; j += i) {
			d[j].push_back(i);
		}
	}
}

struct TrieNode {
	int minVal;
	TrieNode *child[2];
	TrieNode() {
		minVal = inf;
		child[0] = child[1] = NULL;
	}
};

struct Trie {
	TrieNode *root;

	void insert(int val) {
		int bits[maxd], v = val;
		for (int i = 0; i < maxd; i++) {
			bits[maxd - 1 - i] = (v & 1);
			v >>= 1;
		}
		if (root == NULL) root = new TrieNode();
		TrieNode *fa = root, *cur = NULL;
		for (int i = 0; i < maxd; i++) {
			if (fa->child[bits[i]] == NULL) {
				fa->child[bits[i]] = new TrieNode();
			}
			cur = fa->child[bits[i]];
			cur->minVal = std::min(cur->minVal, val);
			fa = cur;
		}
	}

	void deleteTrie(TrieNode *cur) {
		if (cur == NULL) return;
		deleteTrie(cur->child[0]);
		deleteTrie(cur->child[1]);
		delete cur;
	}

} tries[maxn];

void Update(int idx) {
	// printf("begin insert x:%d to trie:%d\n", x, idx);
	tries[idx].insert(x);
}

int xBits[maxd];

int Search(TrieNode *cur, int dep, int ret) {
	if (dep == maxd) return ret;
	if (cur == NULL) return -1;
	int b = 1 - xBits[dep];
	if (cur->child[b] != NULL && cur->child[b]->minVal <= s - x) {
		return Search(cur->child[b], dep + 1, ret * 2 + b);
	}
	else if (cur->child[1 - b] != NULL && cur->child[1 - b]->minVal <= s - x) {
		return Search(cur->child[1 - b], dep + 1, ret * 2 + 1 - b);
	}
	else return -1;
}

int main() {
#ifdef local
	freopen("input.txt", "r", stdin);
#endif
	Init();
	scanf("%d", &q);
	while (q--) {
		scanf("%d", &op);
		if (op == 1) {
			scanf("%d", &x);
			if (flag[x]) continue;
			flag[x] = true;
			for (int item: d[x]) {
				Update(item);
			}
		}
		else {
			scanf("%d%d%d", &x, &k, &s);
			if (x % k != 0) { puts("-1"); continue; }
			int xx = x;
			for (int i = 0; i < maxd; i++) {
				xBits[maxd - 1 - i] = (xx & 1);
				xx >>= 1;
			}
			printf("%d\n", Search(tries[k].root, 0, 0));
		}
	}
	
	for (int i = 0; i < maxn; i++) {
		tries[i].deleteTrie(tries[i].root);
	}
	return 0;
}