uoj233/BZOJ4654/洛谷P1721 [Noi2016]国王饮水记 【dp + 斜率优化】
题目链接
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
- 最优解中多个一起联通不超过\(14\)次
除了最后一个外还是很显然的
\(K\)足够大肯定逐个联通，否则有一些还是需要一起联通的
我们可以设\(f[i][j]\)表示前\(i\)个城市，联通\(j\)次\(h[1]\)的最高高度
那么，去掉小于\(h[1]\)的城市，将\(h[i]\)排序，记\(s[i]\)为\(h[i]\)前缀和
\[f[i][j] = max\{\frac{f[k][j - 1] + s[i] - s[k]}{i - k + 1}\}
\]
可以斜率优化且具有决策单调性
又由于最多不超过\(14\)次多个联通，所以找到\(14\)次中最优决策之后剩余的就可以逐个联通了
复杂度\(O(np)\)
#include <iostream>
#include <cstdlib>
#include <cstring>
#include <cstdio>
#include <cmath>
#include <algorithm>
using namespace std;
// ---------- 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;
}
// ---------- decimal lib end ----------
inline int read(){
int out = 0,flag = 1; char c = getchar();
while (c < 48 || c > 57){if (c == '-') flag = 0; c = getchar();}
while (c >= 48 && c <= 57){out = (out << 1) + (out << 3) + c - 48; c = getchar();}
return flag ? out : -out;
}
const int maxn = 8005;
Decimal ans;
int n,K,P,tot,h[maxn],pre[maxn][15],S[maxn];
int q[maxn],head,tail;
double f[maxn][15];
struct point{double x,y;}p[maxn];
double slope(const point& a,const point& b){
return (a.y - b.y) /(a.x - b.x);
}
Decimal calc(int i,int j){
if (!j) return h[1];
return (calc(pre[i][j],j - 1) + S[i] - S[pre[i][j]]) / (i - pre[i][j] + 1);
}
int main(){
//freopen("1.in","r",stdin);
//freopen("1.out","w",stdout);
n = read(); K = read(); P = read(); h[tot = 1] = read();
for (int i = 2; i <= n; i++){
h[i] = read();
if (h[i] > h[1]) h[++tot] = h[i];
}
n = tot; sort(h + 1,h + 1 + n);
for (int i = 1; i <= n; i++) S[i] = S[i - 1] + h[i];
K = min(n,K);
int E = min(K,14);
for (int i = 1; i <= n; i++) f[i][0] = h[1];
for (int j = 1; j <= E; j++){
f[1][j] = h[1];
q[head = tail = 1] = 1;
for (int i = 1; i <= n; i++) p[i] = (point){i - 1,S[i] - f[i][j - 1]};
for (int i = 2; i <= n; i++){
point u = (point){i,S[i]};
while (head < tail && slope(u,p[q[head]]) < slope(u,p[q[head + 1]])) head++;
pre[i][j] = q[head];
int v = q[head];
f[i][j] = (f[v][j - 1] + S[i] - S[v]) / (i - v + 1);
while (head < tail && slope(p[i],p[q[tail]]) < slope(p[q[tail - 1]],p[q[tail]])) tail--;
q[++tail] = i;
}
}
int m = n - K + E;
double mx = 0; int pos;
for (int i = 0; i <= E; i++)
if (f[m][i] > mx) mx = f[m][i],pos = m;
ans = calc(pos,E);
for (int i = pos + 1; i <= n; i++)
ans = (ans + h[i]) / 2;
cout << ans.to_string(P << 1) << endl;
return 0;
}
