*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
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 = ; 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 / + ;
static const int mo = ; 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 < ) {
is_neg = true;
x = -x;
} integer = x;
} Decimal::Decimal(long long x) {
this->init_zero(); if (x < ) {
is_neg = true;
x = -x;
} integer = x;
} Decimal::Decimal(double x) {
this->init_zero(); if (x < ) {
is_neg = true;
x = -x;
} integer = (long long)x;
x -= integer; for (int i = ; i < len; i++) {
x *= mo;
if (x < ) x = ;
data[i] = (int)x;
x -= data[i];
}
} void Decimal::init_zero() {
is_neg = false;
integer = ;
memset(data, , len * sizeof(int));
} bool Decimal::is_zero() const {
if (integer) return false;
for (int i = ; i < len; i++) {
if (data[i]) return false;
}
return true;
} void Decimal::init(const char *s) {
this->init_zero(); is_neg = false;
integer = ; // find the first digit or the negative sign
while (*s != ) {
if (*s == '-') {
is_neg = true;
++s;
break;
} else if (*s >= && *s <= ) {
break;
}
++s;
} // read the integer part
while (*s >= && *s <= ) {
integer = integer * + *s - ;
++s;
} // read the decimal part
if (*s == '.') {
int pos = ;
int x = mo / ; ++s;
while (pos < len && *s >= && *s <= ) {
data[pos] += (*s - ) * x;
++s;
x /= ;
if (x == ) {
++pos;
x = mo / ;
}
}
}
} void Decimal::append_to_string(std::string &s, long long x) {
if (x == ) {
s.append(, );
return;
} char _[];
int cnt = ;
while (x) {
_[cnt++] = x % ;
x /= ;
}
while (cnt--) {
s.append(, _[cnt] + );
}
} 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(, '.'); for (int i = ; i < len; i++) {
// append data[i] as "%09d"
int x = mo / ;
int tmp = data[i];
while (x) {
ret.append(, + tmp / x);
tmp %= x;
x /= ;
if (--p == ) {
break;
}
}
if (p == ) break;
} if (p > ) {
ret.append(p, '');
} return ret;
} double Decimal::to_double() const {
double ret = integer; double k = 1.0;
for (int i = ; 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 = ; 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 = ; 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 = ; 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 = ; 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 = ; 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 = ; 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 = ; 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 = ; 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 = ; 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 >= ) {
integer += x;
} else {
bool last = false;
for (int i = len - ; i >= ; 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 >= ) {
integer -= x;
} else {
bool last = false;
for (int i = len - ; i >= ; 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 < ) {
is_neg ^= ;
x = -x;
} int last = integer % x;
integer /= x; for (int i = ; 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 == ) {
int i;
for (i = ; i < len; i++) {
if (data[i] != ) {
break;
}
}
if (i == len) {
is_neg = false;
}
} return *this;
} Decimal & Decimal::operator *= (int x) {
if (x < ) {
is_neg ^= ;
x = -x;
} else if (x == ) {
init_zero();
return *this;
} int last = ;
for (int i = len - ; i >= ; 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 == ) {
int i;
for (i = ; i < Decimal::len; i++) {
if (ret.data[i] != ) break;
}
if (i < Decimal::len) {
ret.is_neg = true;
}
} else {
ret.is_neg ^= ;
}
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 - ; i >= ; 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 - ; i >= ; i--) {
ret.data[i] -= b.data[i] + last;
if (ret.data[i] < ) {
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 - ; i >= ; i--) {
ret.data[i] -= a.data[i] + last;
if (ret.data[i] < ) {
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=; while ((c=getchar())<'' || c>'');
do s=s*+c-''; while ((c=getchar())>='' && c<=''); return s;
}
#define lld Decimal
#define ld long double
int n,m,P;
#define maxn 8011
int a[maxn],sum[maxn];
ld f[][maxn]; short cur=,p[maxn][maxn];
short sta[maxn],top=; 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+);} int main()
{
n=qread(); m=qread(); P=qread();
int a0=qread(); for (int i=;i<n;i++) a[i]=qread();
n--; {int m=; for (int i=;i<=n;i++) if (a[i]>a0) a[++m]=a[i]; n=m;}
std::sort(a+,a++n); for (int i=;i<=n;i++) sum[i]=sum[i-]+a[i]; for (int i=;i<=n;i++) f[][i]=a0;
ld ans=a0; short xx=,yy=n;
for (int i=,to=std::min(n,m);i<=to;i++)
{
memset(f[cur^],,sizeof(f[cur^]));
f[cur^][i]=(f[cur][i-]+a[i])/; p[i][i]=i-;
if (ans<f[cur^][i]) ans=f[cur^][i],xx=i,yy=i;
head=tail=; que[tail++]=(Queue){i-,sum[i-]-f[cur][i-]};
for (int j=i+;j<=n;j++)
{
Queue u=(Queue){j-,sum[j-]-f[cur][j-]};
while (head<tail- && que[tail-]*que[tail-]>u*que[tail-]) tail--;
que[tail++]=u;
while (head<tail- && calc(j,que[head].x+)<calc(j,que[head+].x+)) head++;
f[cur^][j]=calc(j,que[head].x+); p[i][j]=que[head].x+;
if (ans<f[cur^][j]) ans=f[cur^][j],xx=i,yy=j;
}
cur^=;
} lld Ans=lld(a0);
for (;xx;yy=p[xx][yy],xx--) sta[++top]=yy; sta[++top]=yy;
for (int i=top;i>;i--) Ans=(Ans+(sum[sta[i-]]-sum[sta[i]]))/(sta[i-]-sta[i]+); std::string s=Ans.to_string(P+);
std::cout<<s<<std::endl;
return ;
}
*UOJ#223. 【NOI2016】国王饮水记的更多相关文章
- [UOJ#223][BZOJ4654][Noi2016]国王饮水记
[UOJ#223][BZOJ4654][Noi2016]国王饮水记 试题描述 跳蚤国有 n 个城市,伟大的跳蚤国王居住在跳蚤国首都中,即 1 号城市中.跳蚤国最大的问题就是饮水问题,由于首都中居住的跳 ...
- luogu P1721 [NOI2016]国王饮水记 斜率优化dp 贪心 决策单调性
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. ...
- BZOJ4654/UOJ223 [Noi2016]国王饮水记
本文版权归ljh2000和博客园共有,欢迎转载,但须保留此声明,并给出原文链接,谢谢合作. 本文作者:ljh2000 作者博客:http://www.cnblogs.com/ljh2000-jump/ ...
- [Noi2016]国王饮水记
来自FallDream的博客,未经允许,请勿转载,谢谢. 跳蚤国有 n 个城市,伟大的跳蚤国王居住在跳蚤国首都中,即 1 号城市中.跳蚤国最大的问题就是饮水问题,由于首都中居住的跳蚤实在太多,跳蚤国王 ...
- uoj233/BZOJ4654/洛谷P1721 [Noi2016]国王饮水记 【dp + 斜率优化】
题目链接 uoj233 题解 下面不加证明地给出几个性质: 小于\(h[1]\)的城市一定是没用的 任何城市联通包含\(1\)且只和\(1\)联通一次 联通顺序从小到大最优 单个联通比多个一起联通要优 ...
- BZOJ4654 NOI2016国王饮水记（动态规划+三分）
有很多比较显然的性质.首先每个城市(除1外)至多被连通一次,否则没有意义.其次将城市按水位从大到小排序后,用以连通的城市集合是一段前缀,并且不应存在比1城市还小的.然后如果确定了选取的城市集合,每次选 ...
- P1721 [NOI2016] 国王饮水记 题解
蒟蒻的第一篇黑题题解,求过. 题目链接 题意描述 这道题用简洁的话来说,就是: 给你 \(n\) 个数字,你可以让取其中任意若干个数字,每次操作,都会使所有取的数字变为取的数字的平均数,并且你最多只能 ...
- 【BZOJ4654】【NOI2016】国王饮水记（动态规划，斜率优化）
[BZOJ4654][NOI2016]国王饮水记(动态规划,斜率优化) 题面 BZOJ 洛谷 题解 首先肯定是找性质. 明确一点,比\(h_1\)小的没有任何意义. 所以我们按照\(h\)排序,那么\ ...
- [NOI 2016]国王饮水记
Description 题库链接 给出 \(n\) 个水杯,每个水杯装有不同高度的水 \(h_i\) ,每次可以指定任意多水杯用连通器连通后断开,问不超过 \(k\) 次操作之后 \(1\) 号水杯的 ...
随机推荐
- Dungeon Master POJ - 2251 (搜索）
Dungeon Master Time Limit: 1000MS Memory Limit: 65536K Total Submissions: 48605 Accepted: 18339 ...
- MySQL - EXISTS 和 NOT EXISTS
语法规则: SELECT * FROM tableName t WHERE 1 = 1 AND 2 = 2 AND EXISTS (SELECT * FROM tableName t2 WHERE ...
- 八、Linux 用户和用户组管理
Linux 用户和用户组管理 Linux系统是一个多用户多任务的分时操作系统,任何一个要使用系统资源的用户,都必须首先向系统管理员申请一个账号,然后以这个账号的身份进入系统. 用户的账号一方面可以帮助 ...
- 五 python并发编程之IO模型
一 IO模型介绍 同步(synchronous) IO和异步(asynchronous) IO,阻塞(blocking) IO和非阻塞(non-blocking)IO分别是什么,到底有什么区别?这个问 ...
- 【JavaScript】修改图片src属性切换图片
今天做项目时其中一个环节需要用到js修改图片src属性切换图片,现在来记录一下 以下是示例: html <img src="/before.jpg" id="img ...
- java的重写(Override) (2013-10-11-163 写的日志迁移
/* *说明方法的重写(又称方法的覆盖)子类并不想原封不动地继承父类的方法,而是想作一定的修改 */ package czbk.jxy.study; /** * @author Archon * @d ...
- day 85 Vue学习七之vue-cookie
Vue学习七之vue-cookie 通过vue如何操作cookie呢 参考链接:https://www.jianshu.com/p/535b53989b39 第一步:安装vue-cookies ...
- Neon Lights in Hong Kong【香港霓虹灯】
Neon Lights in Hong Kong Neon is to Hong Kong as red phone booths are to London and fog is to San Fr ...
- Debug调试文件
在debug.h中设置g_debug_switch即可控制调试级别. /* debug.c */ #include "debug.h" const char *get_log_le ...
- POJ 2311 Cutting Game（SG函数）
Cutting Game Time Limit: 1000MS Memory Limit: 65536K Total Submissions: 4806 Accepted: 1760 Desc ...