HDU 6274 Master of Sequence (暴力+下整除)

题意

两个1e5的数组a,b，定义\(S(t)=\left \lfloor \frac{t-b_i}{a_i} \right \rfloor\)，有三个操作

1 x y：将\(a[x]\)变为\(y\)

2 x y：将\(b[x]\)变为\(y\)

3 x：求使得\(S(t)\geq k\)的最小\(k\)

其中\(a_i\leq 1000\),\(b_i,k\leq 1e9\)

思路

这题主要突破口在于\(a_i\leq 1000\)

我们先从下整除下手，

\(\left \lfloor \frac{t-b_i}{a_i} \right \rfloor=\left \lfloor \frac{k_1a_i+c_1-k_2a_i-c_2}{a_i} \right \rfloor=k_1-k_2+\left \lfloor \frac{c_1-c_2}{a_i} \right \rfloor\)

其中

\(c_1=t\bmod a_i\)

\(c_2=b_i \bmod a_i\)

\(k_1=t/ a_i\)

\(k_2=b_i/ a_i\)

\(\left \lfloor \frac{c_1-c_2}{a_i} \right \rfloor=\left\{\begin{matrix}
-1,c_1<c_2\\ 0,c_1\geq c_2
\end{matrix}\right.\)

那么

\(S(t)=ret+\sum_{i=1}^{1000}(\frac{t}{i}cnt[i]-f[i][t\bmod i+1])\)

其中

\(f[x][y]\)表示\(a_i=x\)时，\(y \leq c_2 \leq 1000\)的个数

\(cnt[x]\)表示\(a_i=x\)的个数，即\(f[x][0]\)

\(ret\)表示所有数\(k_2\)的和

这题就做完了。。

代码

#include<iostream>
#include<cstdio>
#include<algorithm>
#include<cmath>
#include<cstring>
#include<string>
#include<stack>
#include<queue>
#include<deque>
#include<set>
#include<vector>
#include<map>
#include<functional>

#define fst first
#define sc second
#define pb push_back
#define mem(a,b) memset(a,b,sizeof(a))
#define lson l,mid,root<<1
#define rson mid+1,r,root<<1|1
#define lc root<<1
#define rc root<<1|1
#define lowbit(x) ((x)&(-x)) 

using namespace std;

typedef double db;
typedef long double ldb;
typedef long long ll;
typedef unsigned long long ull;
typedef pair<int,int> PI;
typedef pair<ll,ll> PLL;

const db eps = 1e-6;
const int mod = 998244353;
const int maxn = 2e6+100;
const int maxm = 2e6+100;
const int inf = 0x3f3f3f3f;
const db pi = acos(-1.0);

int n,m;
ll a[maxn],b[maxn];
ll f[1111][1111];
ll ret;
ll S(ll t){
	ll ans = 0;
	for(int i = 1; i <= 1000; i++){
		ans+=t/i*f[i][0]-f[i][t%i+1];
	}
	return ans-ret;
}

int main() {
    int T;
    scanf("%d", &T);
    while(T--){
    	scanf("%d %d", &n, &m);
    	for(int i = 1; i <= n; i++){
    		scanf("%lld",&a[i]);
    	}
    	for(int i = 1; i <= n; i++){
    		scanf("%lld", &b[i]);
    	}
    	mem(f,  0);
    	ret = 0;
    	for(int i = 1; i <= n; i++){
    		ret+=b[i]/a[i];
    		f[a[i]][b[i]%a[i]]++;
    	}
    	for(int i = 1; i <= 1000; i++){
    		for(int j = i-1; j >= 0; j--){
    			f[i][j]+=f[i][j+1];
    		}
    	}
    	while(m--){
    		int op,x;
    		ll y;
    		scanf("%d",&op);
    		if(op<=2){
    			scanf("%d %lld", &x ,&y);
    			if(op==1){
	    			ret-=b[x]/a[x];
	    			ret+=b[x]/y;
	    			for(int i = b[x]%a[x]; i >= 0; i--)f[a[x]][i]--;
	    			for(int i = b[x]%y; i >= 0; i--)f[y][i]++;
	    			a[x]=y;
    			}
    			else if(op==2){
    				ret-=b[x]/a[x];
    				ret+=y/a[x];
    				for(int i = b[x]%a[x]; i >= 0; i--)f[a[x]][i]--;
    				for(int i = y%a[x]; i >= 0; i--)f[a[x]][i]++;
    				b[x]=y;
    			}
    		}
    		else{
    			ll ans = 0;
    			scanf("%lld", &y);
    			ll l = 0,r = 1e13;
    			while(l<=r){
    				ll mid = l+r>>1;
    				if(S(mid)>=y){
    					ans=mid;r=mid-1;
    				}
    				else l=mid+1;
    			}
    			printf("%lld\n",ans);
    		}

    	}

    }
    return 0;
}
/*
2
4 6
2 4 6 8
1 3 5 7
1 2 3
2 3 3
3 15
1 3 8
3 90
3 66
8 5
2 4 8 3 1 3 6 24
2 2 39 28 85 25 98 35
3 67
3 28
3 73
3 724
3 7775
 */