湖南集训day5

难度:☆☆☆☆☆☆☆

/*
二分答案
算斜率算截距巴拉巴拉很好推的公式
貌似没这么麻烦我太弱了......
唉不重要...
*/
#include<iostream>
#include<cstdio>
#include<cmath>
#include<algorithm>

#define N 100007

using namespace std;
int n,m,cnt;
int X[N],Y[N];
double k,x,y,b,dis2;
struct segment
{
    double k,b;
}e[N<<1];

inline int read()
{
    int x=0,f=1;char c=getchar();
    while(c>'9'||c<'0'){if(c=='-')f=-1;c=getchar();}
    while(c>='0'&&c<='9'){x=x*10+c-'0';c=getchar();}
    return x*f;
}

double work(double x,double y)
{
    return sqrt(double(x)*double(x)+double(y)*double(y));
}

int main()
{
    freopen("geometry.in","r",stdin);
    freopen("geometry.out","w",stdout);
    n=read();
    for(int i=1;i<=n;i++) X[i]=read();
    for(int i=1;i<=n;i++) Y[i]=read();
    sort(X+1,X+n+1);sort(Y+1,Y+n+1);
    for(int i=1;i<=n;i++)
    {
        e[i].k=double(Y[i])/-double(X[i]);
        e[i].b=Y[i]; 
    }
    m=read();
    for(int i=1;i<=m;i++)
    {
        x=read();y=read();cnt=0;
        if(x==0)
        {
            int l=1,r=n,mid;
            while(l<=r)
            {
                mid=l+r>>1;
                if(Y[mid]>=y) cnt=mid,l=mid+1;
                else r=mid-1;
            } 
        }
        double dis1=work(double(x),double(y));
        k=double(y)/double(x);
        int l=1,r=n,mid;
        while(l<=r)
        {
            mid=l+r>>1;
            if(work(double(e[mid].b/(k-e[mid].k)),k*double(e[mid].b/(k-e[mid].k)))<=dis1)
              cnt=mid,l=mid+1;
            else r=mid-1;
        }
        printf("%d
",cnt);
    }
    fclose(stdin);fclose(stdout);
    return 0;
}

/*
貌似数据水，贪心能贪70分...
差分约束做法
记录前缀和S[i],i>=8时首先要满足条件 s[i]-s[i-8]>=a[i]
0<=s[i]-s[i-1]<=b[i]
当i<8时有s[23]-s[16+i]+s[i]>=a[i]    
因为第三个式子不满足差分约束形式，可以看出s[23]有单调性，可以二分…所以可以二分答案当常量处理 
Spfa负环则无解
还有一种网络流做法，表示很懵逼。
*/
#include<iostream>
#include<cstdio>
#include<cstring>
#include<queue>

#define N 33

using namespace std;
int n,m,ans,cnt;
int head[N<<1],dis[N],a[N],b[N];
bool inq[N];
struct edge{
    int u,v,dis,net;
}e[N<<1];

inline void add(int u,int v,int w)
{
    e[++cnt].v=v;e[cnt].net=head[u];e[cnt].dis=w;head[u]=cnt;
}

inline int read()
{
    int x=0,f=1;char c=getchar();
    while(c>'9'||c<'0'){if(c=='-')f=-1;c=getchar();}
    while(c>='0'&&c<='9'){x=x*10+c-'0';c=getchar();}
    return x*f;
}

bool spfa()
{
    queue<int>q;
    memset(dis,-0x7f7f7f,sizeof dis);
    memset(inq,0,sizeof inq);
    dis[0]=0;inq[0]=1;q.push(0);
    while(!q.empty())
    {
        int u=q.front();q.pop();inq[u]=0;
        if(dis[u]>1000) return false;
        for(int i=head[u];i;i=e[i].net)
        {
            int v=e[i].v;
            if(dis[v]<dis[u]+e[i].dis)
            {
                dis[v]=dis[u]+e[i].dis;
                if(!inq[v]) inq[v]=1,q.push(v);
            }
        }
    }return true;
}

bool solve(int ans)
{
    memset(head,0,sizeof head);
    cnt=0;
    for(int i=9;i<=24;i++) add(i-8,i,a[i]);
    for(int i=1;i<=8;i++)  add(i+16,i,a[i]- ans);
    for(int i=1;i<=24;i++) add(i-1,i,0);
    for(int i=1;i<=24;i++) add(i,i-1,-b[i]);
    add(0,24,ans);add(24,0,-ans);
    return spfa();
}

int main()
{
    freopen("cashier.in","r",stdin);
    freopen("cashier.out","w",stdout);
    n=read();
    while(n--)
    {
        for(int i=1;i<=24;i++) a[i]=read();
        for(int i=1;i<=24;i++) b[i]=read();
        ans=0;
        while(1)
        {
            if(++ans>1000)
            {
                ans=-1;break;
            }
            if(solve(ans)) break;
        }
        printf("%d
",ans);
    }
    return 0;
}

线段树

部分分可以dp f[i][j]表示前i个数选了j个的答案

f[i][j]=f[i-1][j]+f[i-1][j-1]*a[i] (i选不选)

k=1时线段树区间求和区间修改，我只会这个....

线段树区间合并操作。

k比较小，所以线段树每个节点维护一个区间答案记为f[i]

考虑一段区间i,左边取j个右边就取i-j个 答案是每个方案的左边乘右边的和。

就是i左儿子f[j]和右边的f[i-j] 所以f[i]=Σ(j=0~i) lc f[j]*rc f[i-j]

考虑取反操作，i是奇数就取反，偶数无影响(因为是相乘)

考虑区间加， 开始f[i] 是 a1*a2……an  后来是(a1+c)*(a2+c)……(an+c)

考虑类似二项式定理,当上述a1~an  n个方案如果取了j个了，分别为al1,al2……alj

那考虑最后答案，如果已经选了j个方案是(al1+c)(al2+c)……(alj+c)再还能选i-j个 最后答案是C(len-i,i-j)*f[j]*c^(i-j)

复杂度 O(k^2*nlogn)

只懂思路，代码....果断粘std

#include <cstdio>
#include <cstdlib>
#define MOD 1000000007
#define N 100005
typedef long long LL;
using namespace std;
struct Node
{
	LL f[11];
} node[N * 4];
LL a[N], lazy1[N * 4];
bool lazy2[N * 4];
LL C[N][11];

Node merge(Node lc, Node rc)
{
	Node o;
	o.f[0] = 1;
	for (int i = 1; i <= 10; i++)
	{
		o.f[i] = 0;
		for (int j = 0; j <= i; j++)
			o.f[i] = (o.f[i] + lc.f[j] * rc.f[i - j] % MOD) % MOD;
	}
	return o;
}

void build(int o, int l, int r)
{
	if (l == r)
	{
		for (int i = 0; i <= 10; i++) node[o].f[i] = 0;
		node[o].f[0] = 1;
		node[o].f[1] = (a[l] % MOD + MOD) % MOD;
		return ;
	}
	int mid = (l + r) >> 1;
	build(o * 2, l, mid);
	build(o * 2 + 1, mid + 1, r);
	node[o] = merge(node[o * 2], node[o * 2 + 1]);
	return ;
}

void update1(int o, int l, int r, int c)
{
	int len = r - l + 1;
	LL ff[11];
	for (int i = 0; i <= 10; i++) ff[i] = node[o].f[i];
	for (int i = 1; i <= 10; i++)
	{
		node[o].f[i] = 0;
		LL t = 1;
		for (int j = 0; j <= i; j++)
		{
			LL tmp = ff[i - j] * C[len - (i - j)][j] % MOD * t % MOD;
			node[o].f[i] = (node[o].f[i] + tmp) % MOD;
			t = t * c % MOD;
		}
	}
	return ;
}

void push_down(int o, int l, int r)
{
	int mid = (l + r) >> 1;
	if (lazy1[o])
	{
		if (lazy2[o * 2])
			lazy1[o * 2] = (lazy1[o * 2] + MOD - lazy1[o]) % MOD;
		else
			lazy1[o * 2] = (lazy1[o * 2] + lazy1[o]) % MOD;
		if (lazy2[o * 2 + 1])
			lazy1[o * 2 + 1] = (lazy1[o * 2 + 1] + MOD - lazy1[o]) % MOD;
		else
			lazy1[o * 2 + 1] = (lazy1[o * 2 + 1] + lazy1[o]) % MOD;
		update1(o * 2, l, mid, lazy1[o]);
		update1(o * 2 + 1, mid + 1, r, lazy1[o]);
		lazy1[o] = 0;
	}
	if (lazy2[o])
	{
		lazy2[o * 2] ^= 1;
		lazy2[o * 2 + 1] ^= 1;
		for (int j = 1; j <= 10; j += 2)
		{
			node[o * 2].f[j] = MOD - node[o * 2].f[j];
			node[o * 2 + 1].f[j] = MOD - node[o * 2 + 1].f[j];
		}
		lazy2[o] = 0;
	}
}

void modify1(int o, int l, int r, int ll, int rr, int c)
{
	if (ll <= l && rr >= r)
	{
		if (lazy2[o]) lazy1[o] = (lazy1[o] + MOD - c) % MOD;
		else lazy1[o] = (lazy1[o] + c) % MOD;
		update1(o, l, r, c);
		return ;
	}
	int mid = (l + r) >> 1;
	push_down(o, l, r);
	if (ll <= mid) modify1(o * 2, l, mid, ll, rr, c);
	if (rr > mid) modify1(o * 2 + 1, mid + 1, r, ll, rr, c);
	node[o] = merge(node[o * 2], node[o * 2 + 1]);
	return ;
}

void modify2(int o, int l, int r, int ll, int rr)
{
	if (ll <= l && rr >= r)
	{
		for (int i = 1; i <= 10; i += 2) node[o].f[i] = MOD - node[o].f[i];
		lazy2[o] ^= 1;
		return ;
	}
	int mid = (l + r) >> 1;
	push_down(o, l, r);
	if (ll <= mid) modify2(o * 2, l, mid, ll, rr);
	if (rr > mid) modify2(o * 2 + 1, mid + 1, r, ll, rr);
	node[o] = merge(node[o * 2], node[o * 2 + 1]);
	return ;
}

Node query(int o, int l, int r, int ll, int rr)
{
	if (ll <= l && rr >= r)
		return node[o];
	int mid = (l + r) >> 1;
	push_down(o, l, r);
	if (rr <= mid) return query(o * 2, l, mid, ll, rr);
	if (ll > mid) return query(o * 2 + 1, mid + 1, r, ll, rr);
	Node lc = query(o * 2, l, mid, ll, rr);
	Node rc = query(o * 2 + 1, mid + 1, r, ll, rr);
	return merge(lc, rc);
}

int main(int argc, char ** argv)
{
	freopen("game.in", "r", stdin);
	freopen("game.out", "w", stdout);
	int n, m;
	scanf("%d %d", &n, &m);
	C[0][0] = 1;
	for (int i = 1; i <= n; i++)
	{
		C[i][0] = 1;
		for (int j = 1; j <= 10; j++)
			C[i][j] = (C[i - 1][j - 1] + C[i - 1][j]) % MOD;
	}
	for (int i = 1; i <= n; i++)
		scanf("%d", &a[i]);
	build(1, 1, n);
	for (int i = 1; i <= m; i++)
	{

		int l, r, opt;
		scanf("%d%d%d",&opt, &l, &r);
		if (opt == 1)
		{
			int c;
			scanf("%d", &c);
			c = (c % MOD + MOD) % MOD;
			modify1(1, 1, n, l, r, c);
		}
		else if (opt == 2)
		{
			modify2(1, 1, n, l, r);
		}
		else
		{
			int k;
			scanf("%d", &k);
			Node o = query(1, 1, n, l, r);
			printf("%d
", o.f[k] % MOD);
		}
	}
	return 0;
}