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zc
2019-12-21 19:47:00
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f(i,k)表示将前i个分k次的分数

我们可以dpk次就可以了

f_i为前i个,g_i为上一次dp的结果

f_i=\max_{j=1}^{i-1}(g_j+s_j\times(s_i-s_j))

取任意0\le k<j <i且从j转移比从k更优,那么

g_j+s_j\times(s_i-s_j)\ge g_k+s_k\times(s_i-s_k) 展开得

g_j+s_is_j-{s_j}^2\ge g_k+s_is_k-{s_k}^2 移项得 s_i(s_j-s_k) \ge {s_j}^2-{s_k}^2+g_k-g_j 整理得 s_i\ge \frac{({s_j}^2-g_j)-({s_k}^2-g_k)}{s_j-s_k}

注意s_j-s_k有可能为0,要特判

因为要求的是最大值,所以我们维护一个上凸壳就可以了

#include<bits/stdc++.h>
namespace ZDY{
    #pragma GCC optimize(3)
    #define il __inline__ __attribute__ ((always_inline))
    #define register
    #define ll long long
    #define ull unsigned long long
    #define db double
    #define sht short
    #define MB template <class T>il
    #define Fur(i,x,y) for(int i(x);i<=y;++i)
    #define Fdr(i,x,y) for(int i(x);i>=y;--i)
    #define fl(i,x) for(int i(head[x]),to;to=e[i].to,i;i=e[i].nxt)
    #define clr(x,y) memset(x,y,sizeof(x))
    #define cpy(x,y) memcpy(x,y,sizeof(x))
    #define fin(s) freopen(s".in","r",stdin)
    #define fout(s) freopen(s".out","w",stdout)
    #define fcin ios::sync_with_stdio(false)
    #define l2(n) ((int)(log2(n)))
    #define inf 2122219134
    MB T ABS(T x){return x>0?x:-x;}
    MB T MAX(T x,T y){return x>y?x:y;}
    MB T MIN(T x,T y){return x<y?x:y;}
    MB T GCD(T x,T y){return y?GCD(y,x%y):x;}
    MB void SWAP(T &x,T &y){T t=x;x=y;y=t;}
}using namespace ZDY;using namespace std;
namespace IO{const int str=1<<20;static char in_buf[str],*in_s,*in_t;bool __=0;il char gc(){return (in_s==in_t)&&(in_t=(in_s=in_buf)+fread(in_buf,1,str,stdin)),in_s==in_t?EOF:*in_s++;}il void in(string &ch){ch.clear();if(__)return;char c;while((c=gc())!=EOF&&isspace(c));if(c==EOF){__=1;return;}ch+=c;while((c=gc())!=EOF&&!isspace(c))ch+=c;if(c==EOF)__=1;}il void in(char &ch){if(__)return;char c;while((c=gc())!=EOF&&isspace(c));if(c==EOF)__=1;else ch=c;}il void in(char *ch){*ch='\0';if(__)return;char c;while((c=gc())!=EOF&&isspace(c));if(c==EOF){__=1;return;}*ch=c;ch++;while((c=gc())!=EOF&&!isspace(c))*ch=c,ch++;if(c==EOF)__=1;*ch='\0';}template<typename T>il void in(T &x){if(__)return;char c=gc();bool f=0;while(c!=EOF&&(c<'0'||c>'9'))f^=(c=='-'),c=gc();if(c==EOF){__=1;return;}x=0;while(c!=EOF&&'0'<=c&&c<='9')x=x*10+c-48,c=gc();if(c==EOF)__=1;if(f)x=-x;}template<typename T,typename ... arr>il void in(T &x,arr & ... y){in(x),in(y...);}const char ln='\n';static char out_buf[str],*out_s=out_buf,*out_t=out_buf+str;il void flush(){fwrite(out_buf,1,out_s-out_buf,stdout);out_s=out_buf;}il void pt(char c){(out_s==out_t)?(fwrite(out_s=out_buf,1,str,stdout),*out_s++=c):(*out_s++=c);}il void out(const char* s){while(*s)pt(*s++);}il void out(char* s){while(*s)pt(*s++);}il void out(char c){pt(c);}il void out(string s){for(int i=0;s[i];i++)pt(s[i]);}template<typename T>il void out(T x){if(!x){pt('0');return;}if(x<0)pt('-'),x=-x;char a[50],t=0;while(x)a[t++]=x%10,x/= 10;while(t--)pt(a[t]+'0');}template<typename T,typename ... arr>il void out(T x,arr & ... y){out(x),out(y...);}}using namespace IO;
#define N 100011
int n,k,q[N],pre[N][201];
ll s[N],f[N],g[N];
db S(int i,int j){
    if(s[i]==s[j])return -1e18;
    return (db)(s[i]*s[i]-g[i]-s[j]*s[j]+g[j])/(db)(s[i]-s[j]);
}
void op(int n,int t){
    if(pre[n][t])op(pre[n][t],t-1);
    out(n," ");
}
int main(){
    in(n,k);
    Fur(i,1,n)in(s[i]),s[i]+=s[i-1];
    Fur(tt,1,k){
        int h=1,t=1;
        q[1]=0;
        Fur(i,1,n){
            while(h<t&&S(q[h],q[h+1])<=s[i])++h;
            f[i]=g[q[h]]+s[q[h]]*(s[i]-s[q[h]]);
            pre[i][tt]=q[h];
            while(h<t&&S(q[t-1],q[t])>=S(q[t-1],i))--t;
            q[++t]=i;
        }
        std::swap(f,g);
    }
    out(g[n],ln);
    op(pre[n][k],k-1);
    flush();
}
LG 3648 [APIO2014]序列分割
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