[Noi2016]国王饮水记

传送门

Description

有(n)个数，最多(k)次操作，每次可选择一些数，使得它们全都变成它们的平均数

最大化第一个数的值

保留(p)位小数

(nleq 8000,pleq3000)

Solution 

因为对精度的要求，套用高精度小数类的话，单次计算为(O(p))

因此我们动规的过程中，采用记录决策点，并用long double记录(F)数组的值，最后在进行计算答案

根据结论，最后的方案一定是按照从小到大的顺序依次合并比(h_1)大的数，每次操作的包含(h_1)

将原数组(只考虑比(h_1)大的那些数字)

设(a_i=sum_{j=1}^{i}h_i)

考虑(dp)转移

[f_{k,i}=Max[frac{a_i-(a_j-f_{k-1,j})}{i-(j-1)}] ]

所以相当于求((i,a_i))与所有((j-1,a_j-f_{k-1,j}))的最大斜率

显然这样的点一定会在((j-1,a_j-f_{k-1,j}))的下凸壳上

然后这个显然满足决策单调性

因为设(A(i+1,a_i+h_{i+1}))，(B(i,a_i))，(B)的决策点是(P(j-1,a_j-f_j))，根据(h_i)按照从小到大的顺序排列，显然有(k_{AP}>k_{BP})，那么(AP)就与这个下凸壳相交，需要将决策点右移

最后，发现每次选的数的数量都不大于上一次，否则，将最小的改为上一次合并一定更加优秀

然后也不知道怎么证明的，大于(1)的段不超过(14)段，所以只要(dp)出前(14)层就可以了

Code 

#include<bits/stdc++.h>
#define ll long long
#define ld long double
#define reg register
#define max(a,b) ((a)>(b)?(a):(b))
#define min(a,b) ((a)<(b)?(a):(b))

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

const int PREC = 3005;

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 ----------

inline 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<<3)+(x<<1)+ch-'0';ch=getchar();}
    return x*f;
}

int n,k,p;
const int MN=10005;
ll a[MN],_1;
int from[20][MN];
ld f[20][MN];
Decimal ans;
struct Point{
	ld x,y;
	Point(ld _a=0,ld _b=0):x(_a),y(_b){}
	ld sl(const Point&o){return (ld)(o.y-y)/(ld)(o.x-x);}
};

class xxx{
	Point st[MN];
	int hd,tl;
	public:
	void init(){hd=tl=1;st[1]=Point(-1,(ld)(-_1));}
	void get(int K,int i)
	{
		Point cur=Point((ld)i,(ld)a[i]);
		while(hd<tl&&st[hd+1].sl(cur)>st[hd].sl(cur))++hd;
		from[K][i]=(int)st[hd].x+1.;
		f[K][i]=st[hd].sl(cur);
	}
	void ins(int K,int j)
	{
		Point cur=Point((ld)j-1,(ld)(a[j]-f[K][j]));
		while(hd<tl&&st[tl-1].sl(st[tl])>st[tl].sl(cur))--tl;
		st[++tl]=cur;
	}
}que;

void cal(int x,int step)
{
	if(!x||!step)return;
	cal(from[step][x],step-1);
	ans=(ans+(a[x]-a[from[step][x]]))/(x-from[step][x]+1);
}

int main()
{
	n=read()-1;k=read();p=read();_1=read();
	k=min(k,n);
	reg int i,K,tot=0;
	for(i=1;i<=n;++i)
	{
		a[++tot]=read();
		if(a[tot]<=_1) --tot;
	}
	n=tot;if(n==0)
    {
        ans=Decimal(_1);
        std::cout<<ans.to_string(p+3)<<std::endl;
        return 0;
    }
	std::sort(a+1,a+n+1);
	f[0][0]=_1;int lim=min(14,k);
	for(i=1;i<=n;++i) a[i]+=a[i-1],f[0][i]=_1;
	for(K=1;K<=lim;++K)
	{
		f[K][0]=_1;que.init();
		for(i=1;i<=n;++i)
			que.get(K,i),que.ins(K-1,i);
	}
	
	ans=Decimal(_1);cal(n-(k-lim),lim);
	for(i=n-(k-lim)+1;i<=n;++i) ans=(ans+a[i]-a[i-1])/2;
	std::cout<<ans.to_string(p+2)<<std::endl;
	return 0;
}

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