*UOJ#223. 【NOI2016】国王饮水记

$n \leq 8000$的数列，问不超过$m \leq 1e9$次操作后第一个数字最大是多少。操作：选一些数，把他们变成他们的平均值。需要保留$p \leq 3000$位小数，提供了一个小数高精度库。

太长懒得写了。。总之就是个斜率优化DP，然后有奇怪性质

原题解

基本能猜到定理八，十至今没理解QAQ

用前八个定理写了一份被卡精度的代码。就是DP的时候记决策点，然后最后用决策点算答案即可。但DP过程本身精度损得厉害，所以有些点过不了。

// This is an empty program with decimal lib

#include <stdio.h>
#include <cstdlib>
#include <cstring>
#include <iostream>
#include <string>
#include <algorithm>
//using namespace std;

// ---------- decimal lib start ----------

const int PREC = 3100;

class Decimal {
    public:
        Decimal();
        Decimal(const std::string &s);
        Decimal(const char *s);
        Decimal(int x);
        Decimal(long long x);
        Decimal(double x);
        
        bool is_zero() const;
        
        // p (p > 0) is the number of digits after the decimal point
        std::string to_string(int p) const;
        double to_double() const;
        
        friend Decimal operator + (const Decimal &a, const Decimal &b);
        friend Decimal operator + (const Decimal &a, int x);
        friend Decimal operator + (int x, const Decimal &a);
        friend Decimal operator + (const Decimal &a, long long x);
        friend Decimal operator + (long long x, const Decimal &a);
        friend Decimal operator + (const Decimal &a, double x);
        friend Decimal operator + (double x, const Decimal &a);
        
        friend Decimal operator - (const Decimal &a, const Decimal &b);
        friend Decimal operator - (const Decimal &a, int x);
        friend Decimal operator - (int x, const Decimal &a);
        friend Decimal operator - (const Decimal &a, long long x);
        friend Decimal operator - (long long x, const Decimal &a);
        friend Decimal operator - (const Decimal &a, double x);
        friend Decimal operator - (double x, const Decimal &a);
        
        friend Decimal operator * (const Decimal &a, int x);
        friend Decimal operator * (int x, const Decimal &a);
        
        friend Decimal operator / (const Decimal &a, int x);
        
        friend bool operator < (const Decimal &a, const Decimal &b);
        friend bool operator > (const Decimal &a, const Decimal &b);
        friend bool operator <= (const Decimal &a, const Decimal &b);
        friend bool operator >= (const Decimal &a, const Decimal &b);
        friend bool operator == (const Decimal &a, const Decimal &b);
        friend bool operator != (const Decimal &a, const Decimal &b);
        
        Decimal & operator += (int x);
        Decimal & operator += (long long x);
        Decimal & operator += (double x);
        Decimal & operator += (const Decimal &b);
        
        Decimal & operator -= (int x);
        Decimal & operator -= (long long x);
        Decimal & operator -= (double x);
        Decimal & operator -= (const Decimal &b);
        
        Decimal & operator *= (int x);
        
        Decimal & operator /= (int x);
        
        friend Decimal operator - (const Decimal &a);
        
        // These can't be called
        friend Decimal operator * (const Decimal &a, double x);
        friend Decimal operator * (double x, const Decimal &a);
        friend Decimal operator / (const Decimal &a, double x);
        Decimal & operator *= (double x);
        Decimal & operator /= (double x);
        
    private:
        static const int len = PREC / 9 + 1;
        static const int mo = 1000000000;
        
        static void append_to_string(std::string &s, long long x);
        
        bool is_neg;
        long long integer;
        int data[len];
        
        void init_zero();
        void init(const char *s);
};

Decimal::Decimal() {
    this->init_zero();
}

Decimal::Decimal(const char *s) {
    this->init(s);
}

Decimal::Decimal(const std::string &s) {
    this->init(s.c_str());
}

Decimal::Decimal(int x) {
    this->init_zero();
    
    if (x < 0) {
        is_neg = true;
        x = -x;
    }
    
    integer = x;
}

Decimal::Decimal(long long x) {
    this->init_zero();
    
    if (x < 0) {
        is_neg = true;
        x = -x;
    }
    
    integer = x;
}

Decimal::Decimal(double x) {
    this->init_zero();
    
    if (x < 0) {
        is_neg = true;
        x = -x;
    }
    
    integer = (long long)x;
    x -= integer;
    
    for (int i = 0; i < len; i++) {
        x *= mo;
        if (x < 0) x = 0;
        data[i] = (int)x;
        x -= data[i];
    }
}

void Decimal::init_zero() {
    is_neg = false;
    integer = 0;
    memset(data, 0, len * sizeof(int));
}

bool Decimal::is_zero() const {
    if (integer) return false;
    for (int i = 0; i < len; i++) {
        if (data[i]) return false;
    }
    return true;
}

void Decimal::init(const char *s) {
    this->init_zero();
    
    is_neg = false;
    integer = 0;
    
    // find the first digit or the negative sign
    while (*s != 0) {
        if (*s == '-') {
            is_neg = true;
            ++s;
            break;
        } else if (*s >= 48 && *s <= 57) {
            break;
        }
        ++s;
    }
    
    // read the integer part
    while (*s >= 48 && *s <= 57) {
        integer = integer * 10 + *s - 48;
        ++s;
    }
    
    // read the decimal part
    if (*s == '.') {
        int pos = 0;
        int x = mo / 10;
        
        ++s;
        while (pos < len && *s >= 48 && *s <= 57) {
            data[pos] += (*s - 48) * x;
            ++s;
            x /= 10;
            if (x == 0) {
                ++pos;
                x = mo / 10;
            }
        }
    }
}

void Decimal::append_to_string(std::string &s, long long x) {
    if (x == 0) {
        s.append(1, 48);
        return;
    }
    
    char _[30];
    int cnt = 0;
    while (x) {
        _[cnt++] = x % 10;
        x /= 10;
    }
    while (cnt--) {
        s.append(1, _[cnt] + 48);
    }
}

std::string Decimal::to_string(int p) const {
    std::string ret;
    
    if (is_neg && !this->is_zero()) {
        ret = "-";
    }
    
    append_to_string(ret, this->integer);
    
    ret.append(1, '.');
    
    for (int i = 0; i < len; i++) {
        // append data[i] as "%09d"
        int x = mo / 10;
        int tmp = data[i];
        while (x) {
            ret.append(1, 48 + tmp / x);
            tmp %= x;
            x /= 10;
            if (--p == 0) {
                break;
            }
        }
        if (p == 0) break;
    }
    
    if (p > 0) {
        ret.append(p, '0');
    }
    
    return ret;
}

double Decimal::to_double() const {
    double ret = integer;
    
    double k = 1.0;
    for (int i = 0; i < len; i++) {
        k /= mo;
        ret += k * data[i];
    }
    
    if (is_neg) {
        ret = -ret;
    }
    
    return ret;
}

bool operator < (const Decimal &a, const Decimal &b) {
    if (a.is_neg != b.is_neg) {
        return a.is_neg && (!a.is_zero() || !b.is_zero());
    } else if (!a.is_neg) {
        // a, b >= 0
        if (a.integer != b.integer) {
            return a.integer < b.integer;
        }
        for (int i = 0; i < Decimal::len; i++) {
            if (a.data[i] != b.data[i]) {
                return a.data[i] < b.data[i];
            }
        }
        return false;
    } else {
        // a, b <= 0
        if (a.integer != b.integer) {
            return a.integer > b.integer;
        }
        for (int i = 0; i < Decimal::len; i++) {
            if (a.data[i] != b.data[i]) {
                return a.data[i] > b.data[i];
            }
        }
        return false;
    }
}

bool operator > (const Decimal &a, const Decimal &b) {
    if (a.is_neg != b.is_neg) {
        return !a.is_neg && (!a.is_zero() || !b.is_zero());
    } else if (!a.is_neg) {
        // a, b >= 0
        if (a.integer != b.integer) {
            return a.integer > b.integer;
        }
        for (int i = 0; i < Decimal::len; i++) {
            if (a.data[i] != b.data[i]) {
                return a.data[i] > b.data[i];
            }
        }
        return false;
    } else {
        // a, b <= 0
        if (a.integer != b.integer) {
            return a.integer < b.integer;
        }
        for (int i = 0; i < Decimal::len; i++) {
            if (a.data[i] != b.data[i]) {
                return a.data[i] < b.data[i];
            }
        }
        return false;
    }
}

bool operator <= (const Decimal &a, const Decimal &b) {
    if (a.is_neg != b.is_neg) {
        return a.is_neg || (a.is_zero() && b.is_zero());
    } else if (!a.is_neg) {
        // a, b >= 0
        if (a.integer != b.integer) {
            return a.integer < b.integer;
        }
        for (int i = 0; i < Decimal::len; i++) {
            if (a.data[i] != b.data[i]) {
                return a.data[i] < b.data[i];
            }
        }
        return true;
    } else {
        // a, b <= 0
        if (a.integer != b.integer) {
            return a.integer > b.integer;
        }
        for (int i = 0; i < Decimal::len; i++) {
            if (a.data[i] != b.data[i]) {
                return a.data[i] > b.data[i];
            }
        }
        return true;
    }
}

bool operator >= (const Decimal &a, const Decimal &b) {
    if (a.is_neg != b.is_neg) {
        return !a.is_neg || (a.is_zero() && b.is_zero());
    } else if (!a.is_neg) {
        // a, b >= 0
        if (a.integer != b.integer) {
            return a.integer > b.integer;
        }
        for (int i = 0; i < Decimal::len; i++) {
            if (a.data[i] != b.data[i]) {
                return a.data[i] > b.data[i];
            }
        }
        return true;
    } else {
        // a, b <= 0
        if (a.integer != b.integer) {
            return a.integer < b.integer;
        }
        for (int i = 0; i < Decimal::len; i++) {
            if (a.data[i] != b.data[i]) {
                return a.data[i] < b.data[i];
            }
        }
        return true;
    }
}

bool operator == (const Decimal &a, const Decimal &b) {
    if (a.is_zero() && b.is_zero()) return true;
    if (a.is_neg != b.is_neg) return false;
    if (a.integer != b.integer) return false;
    for (int i = 0; i < Decimal::len; i++) {
        if (a.data[i] != b.data[i]) return false;
    }
    return true;
}

bool operator != (const Decimal &a, const Decimal &b) {
    return !(a == b);
}

Decimal & Decimal::operator += (long long x) {
    if (!is_neg) {
        if (integer + x >= 0) {
            integer += x;
        } else {
            bool last = false;
            for (int i = len - 1; i >= 0; i--) {
                if (last || data[i]) {
                    data[i] = mo - data[i] - last;
                    last = true;
                } else {
                    last = false;
                }
            }
            integer = -x - integer - last;
            is_neg = true;
        }
    } else {
        if (integer - x >= 0) {
            integer -= x;
        } else {
            bool last = false;
            for (int i = len - 1; i >= 0; i--) {
                if (last || data[i]) {
                    data[i] = mo - data[i] - last;
                    last = true;
                } else {
                    last = false;
                }
            }
            integer = x - integer - last;
            is_neg = false;
        }
    }
    return *this;
}

Decimal & Decimal::operator += (int x) {
    return *this += (long long)x;
}

Decimal & Decimal::operator -= (int x) {
    return *this += (long long)-x;
}

Decimal & Decimal::operator -= (long long x) {
    return *this += -x;
}

Decimal & Decimal::operator /= (int x) {
    if (x < 0) {
        is_neg ^= 1;
        x = -x;
    }
    
    int last = integer % x;
    integer /= x;
    
    for (int i = 0; i < len; i++) {
        long long tmp = 1LL * last * mo + data[i];
        data[i] = tmp / x;
        last = tmp - 1LL * data[i] * x;
    }
    
    if (is_neg && integer == 0) {
        int i;
        for (i = 0; i < len; i++) {
            if (data[i] != 0) {
                break;
            }
        }
        if (i == len) {
            is_neg = false;
        }
    }
    
    return *this;
}

Decimal & Decimal::operator *= (int x) {
    if (x < 0) {
        is_neg ^= 1;
        x = -x;
    } else if (x == 0) {
        init_zero();
        return *this;
    }
    
    int last = 0;
    for (int i = len - 1; i >= 0; i--) {
        long long tmp = 1LL * data[i] * x + last;
        last = tmp / mo;
        data[i] = tmp - 1LL * last * mo;
    }
    integer = integer * x + last;
    
    return *this;
}

Decimal operator - (const Decimal &a) {
    Decimal ret = a;
    // -0 = 0
    if (!ret.is_neg && ret.integer == 0) {
        int i;
        for (i = 0; i < Decimal::len; i++) {
            if (ret.data[i] != 0) break;
        }
        if (i < Decimal::len) {
            ret.is_neg = true;
        }
    } else {
        ret.is_neg ^= 1;
    }
    return ret;
}

Decimal operator + (const Decimal &a, int x) {
    Decimal ret = a;
    return ret += x;
}

Decimal operator + (int x, const Decimal &a) {
    Decimal ret = a;
    return ret += x;
}

Decimal operator + (const Decimal &a, long long x) {
    Decimal ret = a;
    return ret += x;
}

Decimal operator + (long long x, const Decimal &a) {
    Decimal ret = a;
    return ret += x;
}

Decimal operator - (const Decimal &a, int x) {
    Decimal ret = a;
    return ret -= x;
}

Decimal operator - (int x, const Decimal &a) {
    return -(a - x);
}

Decimal operator - (const Decimal &a, long long x) {
    Decimal ret = a;
    return ret -= x;
}

Decimal operator - (long long x, const Decimal &a) {
    return -(a - x);
}

Decimal operator * (const Decimal &a, int x) {
    Decimal ret = a;
    return ret *= x;
}

Decimal operator * (int x, const Decimal &a) {
    Decimal ret = a;
    return ret *= x;
}

Decimal operator / (const Decimal &a, int x) {
    Decimal ret = a;
    return ret /= x;
}

Decimal operator + (const Decimal &a, const Decimal &b) {
    if (a.is_neg == b.is_neg) {
        Decimal ret = a;
        bool last = false;
        for (int i = Decimal::len - 1; i >= 0; i--) {
            ret.data[i] += b.data[i] + last;
            if (ret.data[i] >= Decimal::mo) {
                ret.data[i] -= Decimal::mo;
                last = true;
            } else {
                last = false;
            }
        }
        ret.integer += b.integer + last;
        return ret;
    } else if (!a.is_neg) {
        // a - |b|
        return a - -b;
    } else {
        // b - |a|
        return b - -a;
    }
}

Decimal operator - (const Decimal &a, const Decimal &b) {
    if (!a.is_neg && !b.is_neg) {
        if (a >= b) {
            Decimal ret = a;
            bool last = false;
            for (int i = Decimal::len - 1; i >= 0; i--) {
                ret.data[i] -= b.data[i] + last;
                if (ret.data[i] < 0) {
                    ret.data[i] += Decimal::mo;
                    last = true;
                } else {
                    last = false;
                }
            }
            ret.integer -= b.integer + last;
            return ret;
        } else {
            Decimal ret = b;
            bool last = false;
            for (int i = Decimal::len - 1; i >= 0; i--) {
                ret.data[i] -= a.data[i] + last;
                if (ret.data[i] < 0) {
                    ret.data[i] += Decimal::mo;
                    last = true;
                } else {
                    last = false;
                }
            }
            ret.integer -= a.integer + last;
            ret.is_neg = true;
            return ret;
        }
    } else if (a.is_neg && b.is_neg) {
        // a - b = (-b) - (-a)
        return -b - -a;
    } else if (a.is_neg) {
        // -|a| - b
        return -(-a + b);
    } else {
        // a - -|b|
        return a + -b;
    }
}

Decimal operator + (const Decimal &a, double x) {
    return a + Decimal(x);
}

Decimal operator + (double x, const Decimal &a) {
    return Decimal(x) + a;
}

Decimal operator - (const Decimal &a, double x) {
    return a - Decimal(x);
}

Decimal operator - (double x, const Decimal &a) {
    return Decimal(x) - a;
}

Decimal & Decimal::operator += (double x) {
    *this = *this + Decimal(x);
    return *this;
}

Decimal & Decimal::operator -= (double x) {
    *this = *this - Decimal(x);
    return *this;
}

Decimal & Decimal::operator += (const Decimal &b) {
    *this = *this + b;
    return *this;
}

Decimal & Decimal::operator -= (const Decimal &b) {
    *this = *this - b;
    return *this;
}

// ---------- decimal lib end ----------

int qread()
{
    char c; int s=0; while ((c=getchar())<'0' || c>'9');
    do s=s*10+c-'0'; while ((c=getchar())>='0' && c<='9'); return s;
}
#define lld Decimal
#define ld long double
int n,m,P;
#define maxn 8011
int a[maxn],sum[maxn];
ld f[2][maxn]; short cur=0,p[maxn][maxn];
short sta[maxn],top=0;

struct Queue
{
    int x; ld y;
    ld operator * (const Queue b) const {return (y-b.y)/(x-b.x);}
}que[maxn]; int head,tail;

ld calc(int i,int j) {return (f[cur][j]+sum[i]-sum[j])/(i-j+1);}

int main()
{
    n=qread(); m=qread(); P=qread();
    int a0=qread(); for (int i=1;i<n;i++) a[i]=qread();
    n--; {int m=0; for (int i=1;i<=n;i++) if (a[i]>a0) a[++m]=a[i]; n=m;}
    std::sort(a+1,a+1+n); for (int i=1;i<=n;i++) sum[i]=sum[i-1]+a[i];
    
    for (int i=0;i<=n;i++) f[0][i]=a0;
    ld ans=a0; short xx=0,yy=n;
    for (int i=1,to=std::min(n,m);i<=to;i++)
    {
        memset(f[cur^1],0,sizeof(f[cur^1]));
        f[cur^1][i]=(f[cur][i-1]+a[i])/2; p[i][i]=i-1;
        if (ans<f[cur^1][i]) ans=f[cur^1][i],xx=i,yy=i;
        head=tail=1; que[tail++]=(Queue){i-2,sum[i-1]-f[cur][i-1]};
        for (int j=i+1;j<=n;j++)
        {
            Queue u=(Queue){j-2,sum[j-1]-f[cur][j-1]};
            while (head<tail-1 && que[tail-1]*que[tail-2]>u*que[tail-1]) tail--;
            que[tail++]=u;
            while (head<tail-1 && calc(j,que[head].x+1)<calc(j,que[head+1].x+1)) head++;
            f[cur^1][j]=calc(j,que[head].x+1); p[i][j]=que[head].x+1;
            if (ans<f[cur^1][j]) ans=f[cur^1][j],xx=i,yy=j;
        }
        cur^=1;
    }
    
    lld Ans=lld(a0);
    for (;xx;yy=p[xx][yy],xx--) sta[++top]=yy; sta[++top]=yy;
    for (int i=top;i>1;i--) Ans=(Ans+(sum[sta[i-1]]-sum[sta[i]]))/(sta[i-1]-sta[i]+1);
    
    std::string s=Ans.to_string(P+2);
    std::cout<<s<<std::endl;
    return 0;
}