【HDU5648 BestCoder Round 76 (div1)C】【打表or数位DP】DZY Loves Math nm范围内i和j i或j的gcd之和

DZY Loves Math

 

 Accepts: 20

 

 Submissions: 78

 Time Limit: 10000/5000 MS (Java/Others)

 

 Memory Limit: 262144/262144 K (Java/Others)

问题描述

DZY从小就喜欢数学。 有一天他遇到这么一个题：

计算\sum_{1\le i\le n} \sum_{1\le j \le m} \gcd(i \text{ AND } j, i \text{ OR } j)∑​1≤i≤n​​∑​1≤j≤m​​gcd(i AND j,i OR j)。其中$\gcd (a,b)$表示$a,b$的最大公约数，当$a\ne 0$时规定$\gcd (0,a)=a$。AND和OR分别表示按位与、按位或运算。

DZY说这种题他已经做腻了，于是就把它丢给了你。

输入描述

第一行$t$，表示有$t$组数据。
接下来$t$组数据。每组数据包含一行两个整数$n,m$。

($1\le t\le 3,1\le n,m\le 15000$，至多有$2$组数据使得$\max \{n,m\}>2000$，至多有$1$组数据使得$\max \{n,m\}>8000$)

输出描述

每组数据输出一行答案。

输入样例

3
3 5
30 50
210 15000

输出样例

43
15467
2660805398

【官方题解】

记l=\log_2 \max{n,m}l=log​2​​max{n,m}。

枚举$i\text{ AND }j$和$i\text{ OR }j$的值分别是多少，因为是有包含关系的，所以枚举量是3^l3​l​​的，用枚举子集的方法。

然后要计算有多少组$i,j$满足$i\text{ AND }j=a$, $i\text{ OR }j=b$, 且$1\le i\le n,1\le j\le m$。$i$和$j$肯定都包含了$a$，剩下$b-a$的那些bit要分配给$i$和$j$，你可以算出最大分配多少才能使$i$不超过$n$，$j$也是同理，于是就确定了一个区间，区间内的都是合法的方案。

所以总的复杂度就是O(l \times 3^l) = O(n^{1.59} \log n)O(l×3​l​​)=O(n​1.59​​logn)

claris老司机提供了一个O(3^l)O(3​l​​)的做法。考虑数位dp，状态是$f[i][and][or][2][2]$，这样的状态数是1+3+\cdots + 3^l=O(3^l)1+3+⋯+3​l​​=O(3​l​​)的，然后转移也可以$O(1)$。求gcd也不需要log，可以用一个$O(n)$预处理支持$O(1)$询问的结构，见http://mimuw.edu.pl/~kociumaka/files/stacs2013_slides.pdf 。

这题用奇妙的打表或者其他更加厉害的做法也是有可能AC的。

#include<stdio.h>
#include<iostream>
#include<string.h>
#include<string>
#include<ctype.h>
#include<math.h>
#include<set>
#include<map>
#include<vector>
#include<queue>
#include<bitset>
#include<algorithm>
#include<time.h>
using namespace std;
void fre() { freopen("c://test//input.in", "r", stdin); freopen("c://test//output.out", "w", stdout); }
#define MS(x,y) memset(x,y,sizeof(x))
#define MC(x,y) memcpy(x,y,sizeof(x))
#define MP(x,y) make_pair(x,y)
#define ls o<<1
#define rs o<<1|1
typedef long long LL;
typedef unsigned long long UL;
typedef unsigned int UI;
template <class T1, class T2>inline void gmax(T1 &a, T2 b) { if (b>a)a = b; }
template <class T1, class T2>inline void gmin(T1 &a, T2 b) { if (b<a)a = b; }
const int N = 0, M = 0, Z = 1e9 + 7, ms63 = 0x3f3f3f3f;
int casenum, casei;
int n, m;
int gcd(int x, int y)
{
	return y == 0 ? x : gcd(y, x%y);
}
LL ans[61][61] =
{
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};
const int d = 250;
void test()
{
	LL sum[15005][15005];
	LL line[15005];
	freopen("c://test//output.out", "w", stdout);
	for (int i = 1; i <= 15000; ++i)
	{
		for (int j = 1; j <= 15000; ++j)
		{
			line[j] = line[j - 1] + gcd(i&j, i | j);
			sum[i][j] = sum[i - 1][j] + line[j];
			if (i % d == 0 && j % d == 0)
			{
				ans[i / d][j / d] = sum[i][j];
			}
		}
	}
	for (int i = 0; i <= 60; ++i)
	{
		for (int j = 0; j <= 60; ++j)
		{
			printf("%lldll", ans[i][j]);
		}
	}
}
void solve()
{
	for (int i = 1; i <= 61; ++i)
	{
		for (int j = 1; j <= 61; ++j)
		{
			if (i * d > n&&j * d > m)
			{
				LL sum = ans[i - 1][j - 1];
				int y = (i - 1) * d;
				int x = (j - 1) * d;
				for (int i = 1; i <= n; ++i)
				{
					for (int j = x + 1; j <= m; ++j)
					{
						sum += gcd(i&j, i | j);
					}
				}
				for (int i = y + 1; i <= n; ++i)
				{
					for (int j = 1; j <= x; ++j)
					{
						sum += gcd(i&j, i | j);
					}
				}
				printf("%lld\n", sum);
				return;
			}
		}
	}
}
int main()
{
	//test(); return 0;
	scanf("%d", &casenum);
	for (casei = 1; casei <= casenum; ++casei)
	{
		scanf("%d%d", &n, &m);
		solve();
	}
	return 0;
}
/*
【trick&&吐槽】
1，LL ans[61][61] = 不要忘记"="，只有c++11可以省略
2，末尾不要有多余","
3，打表之前表要初始化为空

【题意】
给你n（15000）m（15000），
让你求出for(i=1~n)for(j=1~n)sum+=gcd(i&j,i|j)的sum

【类型】
打表or
枚举子集or
数位DP

【分析】
我们可以分段打表！
分段打表大法好哇，棒哇，啦啦啦！
如何分段打表呢？
直接求出[1,1]~[y,x]的区间和
然后对于询问[n,m]
我们求[1,n][x+1,m]，求[y+1,n][1,x]的就可以啦
打表可以打一个60*60的表，也就是说250为区间段。
那么单个询问的复杂度不会超过15000*250，可以很轻松就AC啦。

【时间复杂度&&优化】
O(T*(n+m)*250)

*/