【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)1in1jmgcd(i AND j,i OR j)。其中\gcd(a,b)gcd(a,b)表示a,ba,b的最大公约数,当a\neq 0a0时规定\gcd(0,a)=agcd(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tt,表示有tt组数据。
接下来tt组数据。每组数据包含一行两个整数n,mn,m。

(1\le t \le 3,1\le n,m \le 150001t3,1n,m15000,至多有22组数据使得\max\{n,m\}>2000max{n,m}>2000,至多有11组数据使得\max\{n,m\}>8000max{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】

l=\log_2 \max{n,m}l=log2max{n,m}

枚举i \text{ AND } ji AND ji\text{ OR }ji 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^l3l的,用枚举子集的方法。

然后要计算有多少组i,ji,j满足i \text{ AND } j=ai AND j=a, i\text{ OR }j=bi OR j=b, 且1\le i \le n, 1\le j\le m1in,1jmiijj肯定都包含了aa,剩下b-aba的那些bit要分配给iijj,你可以算出最大分配多少才能使ii不超过nnjj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×3l)=O(n1.59logn)

claris老司机提供了一个O(3^l)O(3l)的做法。考虑数位dp,状态是f[i][and][or][2][2]f[i][and][or][2][2],这样的状态数是1+3+\cdots + 3^l=O(3^l)1+3++3l=O(3l)的,然后转移也可以O(1)O(1)。求gcd也不需要log,可以用一个O(n)O(n)预处理支持O(1)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)

*/


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