[APIO2014]序列分割

Posted 2020-08-31

斜率优化

f[i]=max(f[j]+s[j]*(s[i]-s[j]))

令g[i]=f[i]-s[i]^2

j比k优 那么g[j]+s[i]s[j]>g[k]+s[i][k]

g[j]-g[k]>s[i](s[k]-s[j])

g[j]-g[k]/(s[j]-s[k])<-s[i]

所以维护队列上凸即可

#include<iostream>
#include<cstdio>
#include<queue>
#define MAXN 100000
#include<cstring>
#define ll long long
using namespace std;
int read()
{
    int x=0,f=1;char ch=getchar();
    while(ch<‘0‘||ch>‘9‘){if(ch==‘-‘) f=-1;ch=getchar();}
    while(ch>=‘0‘&&ch<=‘9‘){x=x*10+ch-‘0‘; ch=getchar();}
    return x*f;
} 

int n,k;
int from[202][MAXN+5];
ll f[MAXN+5];
ll s[MAXN+5];
ll g[2][MAXN+5];
int q[MAXN+5];
int top=0,tail=1;

ll get(int pre,int x,int t)
{
    while(top-tail>=1&&(g[pre][q[tail]]-g[pre][q[tail+1]])<s[x]*(s[q[tail+1]]-s[q[tail]]))
    ++tail;
    from[t][x]=q[tail];
    return g[pre][q[tail]]+s[q[tail]]*s[x];
};

void ins(int t,int k)
{
    while(top-tail>=1)
    {
        int i=q[top-1],j=q[top];
        if((g[t][k]-g[t][j])*(s[i]-s[j])<(g[t][j]-g[t][i])*(s[j]-s[k])) top--;
        else break;    
    }
    q[++top]=k;
}

int main()
{
    //freopen("sequence.in","r",stdin);
    //freopen("sequence.out","w",stdout);
    n=read();k=read();
    for(int i=1;i<=n;i++)
    {
        s[i]=read();s[i]+=s[i-1];    
    }
    int pre=0,nown=1;
    for(int i=1;i<=n;i++) g[pre][i]=-(s[i]*s[i]);
    for(int l=2;l<=k+1;l++)
    {
        memset(f,0,sizeof(f));top=0;tail=1;
        memset(g[nown],0,sizeof(g[nown]));
        ins(pre,l-1);
        for(int i=l;i<=n;i++)
        {
            f[i]=get(pre,i,l);
            g[nown][i]=f[i]-(s[i]*s[i]);
            if(s[i]!=s[i-1])ins(pre,i);
            //printf("%d %d %d %I64d\n",l,i,from[l][i],f[i]);
        }    
        nown=1-nown;
        pre=1-pre;
    }
    printf("%lld\n",f[n]);
    return 0;
}