LINK：国王饮水记
看起来很不可做的样子.
但实际上还是需要先考虑贪心.
当k==1的时候 只有一次操作机会。显然可以把那些比第一个位置小的都给扔掉.
然后可以得知剩下序列中的最大值一定会被选择.
考虑是否选出其他数字 容易想到如果选择了必然要比拿到最大的之后的平均数要大.
这样贪心下去即可.
考虑当k不等于1的时候.
设\(f_i\)表示第i次使用过后的最大值.显然有\(f_i>f_{i-1}\)
那么由此我们得到某个点至多被选择一次 再选择没有任何收益.
所以k可以对n取min.
此时观察数据范围 可以暴力状压 再对k==1单独处理 期望得分40.
然后我们思考如果先选最大的刚开始可能增幅最大 可是反过来削弱了一些原本可以有贡献的点.
先让小的先增幅 大的也再增幅这无疑是更优的.
那么我们考虑从小到大选 然后我们可以发现每次选择应该是连续的一段.
如果不连续 那么我们可以发现两端中间的元素 会被左边元素削弱 理由同上还是存在先选大的再选小的了.
所以我们可以设一个状态了 \(f_{i,j}\)表示到第i个位置选了j次的最大值.
显然有 \(f_{i,j}=(f_{k,j-1}+S_i-S_k)/(i-k+1)\)
这样复杂度为 \(n^2\cdot k\cdot p\) 期望得分60.
不过这个需要使用题目中给出的高精浮点型 观察一下调用即可.
code
//#include<bits/stdc++.h>
#include<iostream>
#include<cstdio>
#include<ctime>
#include<cctype>
#include<queue>
#include<deque>
#include<stack>
#include<iostream>
#include<iomanip>
#include<cstdio>
#include<cstring>
#include<string>
#include<ctime>
#include<cmath>
#include<cctype>
#include<cstdlib>
#include<queue>
#include<deque>
#include<stack>
#include<vector>
#include<algorithm>
#include<utility>
#include<bitset>
#include<set>
#include<map>
#define ll long long
#define db double
#define INF 1000000000000000000ll
#define inf 100000000000000000ll
#define ldb long double
#define pb push_back
#define put_(x) printf("%d ",x);
#define get(x) x=read()
#define gt(x) scanf("%d",&x)
#define gi(x) scanf("%lf",&x)
#define put(x) printf("%d\n",x)
#define putl(x) printf("%lld\n",x)
#define rep(p,n,i) for(RE int i=p;i<=n;++i)
#define go(x) for(int i=lin[x],tn=ver[i];i;tn=ver[i=nex[i]])
#define fep(n,p,i) for(RE int i=n;i>=p;--i)
#define vep(p,n,i) for(RE int i=p;i<n;++i)
#define pii pair<int,int>
#define mk make_pair
#define RE register
#define P 1000000007ll
#define gf(x) scanf("%lf",&x)
#define pf(x) ((x)*(x))
#define uint unsigned long long
#define ui unsigned
#define EPS 1e-10
#define sq sqrt
#define S second
#define F first
#define mod 1000000007
#define r(x) t[x].r
#define l(x) t[x].l
#define mx1(x) t[x].mx1
#define mx2(x) t[x].mx2
#define max(x,y) ((x)<(y)?y:x)
using namespace std;
char *fs,*ft,buf[1<<15];
inline char gc()
{
return (fs==ft&&(ft=(fs=buf)+fread(buf,1,1<<15,stdin),fs==ft))?0:*fs++;
}
inline int read()
{
RE int x=0,f=1;RE char ch=gc();
while(ch<'0'||ch>'9'){if(ch=='-')f=-1;ch=gc();}
while(ch>='0'&&ch<='9'){x=x*10+ch-'0';ch=gc();}
return x*f;
}
const int MAXN=1010;
int n,k,p,cnt,W;
int a[MAXN];
db f[MAXN][MAXN];
int main()
{
//freopen("1.in","r",stdin);
get(n);get(k);get(p);get(W);
rep(2,n,i)
{
int get(x);
if(x>W)a[++cnt]=x;
}
n=cnt;
if(p<=5)
{
sort(a+1,a+1+n);
rep(1,n,i)a[i]+=a[i-1];
k=min(k,n);
rep(0,n,j)f[j][0]=W;
rep(0,n,i)
{
rep(1,k,j)
{
f[i][j]=f[i][j-1];
rep(0,i-1,l)
{
f[i][j]=max(f[i][j],(f[l][j-1]+a[i]-a[l])/(i-l+1));
}
}
}
printf("%.8lf",f[n][k]);
return 0;
}
return 0;
}
感觉这种式子会存在决策单调性 先枚举j再枚举i打表 发现决策确实具有单调性.
证明可以自己观察式子吧 我没仔细分析 但是感觉存在单调性...
分治法\(n\cdot k\cdot p\cdot logn\)
一个以前从没注意过的东西 这个式子可以斜率优化.
转移式的形式可以看成两点的斜率大小 这个东西也可以进行斜率优化.
维护下凸壳和当前点横纵左边都递增维护队列列两点斜率的大小关系即可.
复杂度\(n\cdot k\cdot p\)期望得分\(80\)
接下来由两个trick可以优化.
一个是 dp的时候可以先使用double型数组做 然后最后再使用高精浮点获得决策来做.
这样前者是\(nk\)后者\(k\cdot p\)期望得分100.
我想正确性应该没什么问题 恰好做到了决策之间的细微差距再十几位小数的时候就被分辨.
第二个是 当k==n的时候 显然每次两个单独合并是最优的.
当k<n的时候必须有一些点是一起合并的 且每次合并的区间大小递减.
可以证明在当前数据范围最多有14次是整段合并 后面都是单个合并。
不会证明.不过第一个方法已经可以通过 加上这个可以更快.
(700多行代码警告!
\(14\cdot n+k\cdot p\)
code
// This is an empty program with decimal lib
#include <cstdlib>
#include <cstring>
#include <string>
// ---------- decimal lib start ----------
const int PREC = 3000;
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;
}
//#include<bits/stdc++.h>
#include<iostream>
#include<cstdio>
#include<ctime>
#include<cctype>
#include<queue>
#include<deque>
#include<stack>
#include<iostream>
#include<iomanip>
#include<cstdio>
#include<cstring>
#include<string>
#include<ctime>
#include<cmath>
#include<cctype>
#include<cstdlib>
#include<queue>
#include<deque>
#include<stack>
#include<vector>
#include<algorithm>
#include<utility>
#include<bitset>
#include<set>
#include<map>
#define ll long long
#define db double
#define INF 1000000000000000000ll
#define inf 100000000000000000ll
#define ldb long double
#define pb push_back
#define put_(x) printf("%d ",x);
#define get(x) x=read()
#define gt(x) scanf("%d",&x)
#define gi(x) scanf("%lf",&x)
#define put(x) printf("%d\n",x)
#define putl(x) printf("%lld\n",x)
#define rep(p,n,i) for(RE int i=p;i<=n;++i)
#define go(x) for(int i=lin[x],tn=ver[i];i;tn=ver[i=nex[i]])
#define fep(n,p,i) for(RE int i=n;i>=p;--i)
#define vep(p,n,i) for(RE int i=p;i<n;++i)
#define pii pair<int,int>
#define mk make_pair
#define RE register
#define P 1000000007ll
#define gf(x) scanf("%lf",&x)
#define pf(x) ((x)*(x))
#define uint unsigned long long
#define ui unsigned
#define EPS 1e-10
#define sq sqrt
#define S second
#define F first
#define mod 1000000007
#define r(x) t[x].r
#define l(x) t[x].l
#define mx1(x) t[x].mx1
#define mx2(x) t[x].mx2
#define max(x,y) ((x)<(y)?y:x)
using namespace std;
char *fs,*ft,buf[1<<15];
inline char gc()
{
return (fs==ft&&(ft=(fs=buf)+fread(buf,1,1<<15,stdin),fs==ft))?0:*fs++;
}
inline int read()
{
RE int x=0,f=1;RE char ch=gc();
while(ch<'0'||ch>'9'){if(ch=='-')f=-1;ch=gc();}
while(ch>='0'&&ch<='9'){x=x*10+ch-'0';ch=gc();}
return x*f;
}
const int MAXN=8010;
int n,k,p,cnt,W;
int a[MAXN],g[MAXN][15];
db f[MAXN][15];
struct wy
{
int id;db x,y;
inline wy operator -(wy a){return (wy){0,x-a.x,y-a.y};}
inline db operator %(wy a){return x*a.y-a.x*y;}
}q[MAXN];
inline Decimal calc(int i,int j)
{
if(!j)return f[i][0];
return (calc(g[i][j],j-1)+a[i]-a[g[i][j]])/(i-g[i][j]+1);
}
int main()
{
//freopen("1.in","r",stdin);
get(n);get(k);get(p);get(W);
rep(2,n,i)
{
int get(x);
if(x>W)a[++cnt]=x;
}
n=cnt;
sort(a+1,a+1+n);
rep(1,n,i)a[i]+=a[i-1];
k=min(k,n);
int lim=min(14,k);
rep(0,n,i)f[i][0]=W;
rep(1,lim,j)
{
int l=1,r=1;
q[r]=(wy){0,-1,a[0]-f[0][j-1]};
rep(1,n,i)
{
wy v=(wy){0,(db)i,(db)a[i]};
while(l<r&&(v-q[l+1])%(v-q[l])<0)++l;
f[i][j]=(a[i]-q[l].y)/(i-q[l].x);
g[i][j]=q[l].id;
v=(wy){i,(db)i-1,(db)a[i]-f[i][j-1]};
while(l<r&&(v-q[r])%(q[r]-q[r-1])>0)--r;
q[++r]=v;
}
}
int ww=n-(k-lim);
int cc;db mx=0;
rep(1,lim,j)if(mx<f[ww][j])mx=f[ww][j],cc=j;
Decimal ans=calc(ww,cc);
rep(ww+1,n,i)ans=(ans+(a[i]-a[i-1]))/2;
cout<<ans.to_string(p<<1)<<endl;
return 0;
}