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zc
2019-12-21 19:47:00
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f_i表示[1,i]最多变成多少个

f_i=\max_{s_i=s_j}(f[j-1]+s_j\times (\sum_{i=j}^i[s_i=s_j])^2)

我们可以发现如果s_j=s_{j'}(j'< j),从j'转移一定比从j

这一条件满足决策单调性!

我们可以用单调队列维护单调性

#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,a[N],s[N],la[10011];
vector<int>q[10011];
ll f[N];
il ll calc(int p,int t){
    return f[p-1]+1ll*a[p]*t*t;
}
#define sz(x) q[x].size()
il int bound(int x,int y){
    int ret=n+1;
    int l=1,r=n,m;
    while(l<=r){
        m=(l+r)>>1;
        if(calc(x,m-s[x]+1)>=calc(y,m-s[y]+1))
            ret=m,r=m-1;
        else l=m+1;
    }
    return ret;
}
int main(){
    in(n);
    Fur(i,1,n){
        in(a[i]);
        s[i]=s[la[a[i]]]+1;
        la[a[i]]=i;
    }
    int h=1,t=0,x;
    Fur(i,1,n){
        x=a[i];
        while(sz(x)>1&&bound(q[x][sz(x)-2],q[x][sz(x)-1])<=bound(q[x][sz(x)-1],i))
            q[x].pop_back();
        q[x].push_back(i);
        while(sz(x)>1&&bound(q[x][sz(x)-2],q[x][sz(x)-1])/*相当于决策点,懒得开k数组了*/<=s[i])
            q[x].pop_back();
        f[i]=calc(q[x][sz(x)-1],s[i]-s[q[x][sz(x)-1]]+1);
    }
    cout<<f[n]<<endl;
}
LG 5504 [JSOI2011]柠檬
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