[2021.8集训Day10/JZOJ.3410]【GDOI2014模拟】Tree
题目
思路
我们枚举一个参考平均数(注意不是真正的平均数),把每条边按边权与参考平均数的差的绝对值从小到大排序,跑一遍Kruskal,即可求出对应的标准差:
double now_average;//参考平均数
struct EDGE {
int u , v , c;
}ed[M];
bool cmp(EDGE a , EDGE b) {
return fabs((double)a.c - now_average) < fabs((double)b.c - now_average);
}
bool choosed[M];
double GetDeviation() {
UF::init();
std::sort(ed + 1 , ed + m + 1 , cmp);
std::memset(choosed , 0 , sizeof(choosed));
double average;
for(int i = 1 ; i <= m ; i++) {
int u = ed[i].u , v = ed[i].v;
if(UF::findroot(u) == UF::findroot(v))
continue;
UF::uni(ed[i].u , ed[i].v);
choosed[i] = true;
average += ed[i].c;
}
double deviation = 0;
average /= (double)(n - 1);//真正的平均数
for(int i = 1 ; i <= m ; i++)
if(choosed[i])
deviation += fabs(average - (double)ed[i].c) * fabs(average - (double)ed[i].c);
return sqrt(deviation / (double)(n - 1));
}
不难想到, 若\(x=\text{参考平均数}\),\(f(x)=参考平均数对应的标准差\),\(f(x)\)应该是一条比较丝滑的曲线,因此,我用了模拟退火.其实直接一个一个枚举也行,不过模拟退火跑得巨快300ms内就可以出结果.
代码
这不是GMOJ的AC代码
同样的数据,本地A,洛谷云IDE A , GMOJ WA(雾)
#include <iostream>
#include <cstdio>
#include <cmath>
#include <cstdlib>
#include <algorithm>
#include <cstring>
unsigned seed;
int read() {
int re = 0;
char c = getchar();
bool negt = false;
while(c < '0' || c > '9')
negt |= (c == '-') , c = getchar();
while(c >= '0' && c <= '9')
re = (re << 1) + (re << 3) + c - '0' , c = getchar();
seed *= re;
return negt ? -re : re;
}
const int N = 110 , M = 2010;
struct EDGE {
int u , v , c;
}ed[M];
const double delta_t = 0.993;
const double originT = 100;//初始温度没必要太大,因为c的范围小
int n , m;
double ans_average , ans_deviation = 1e10;
namespace UF {
int fa[N];
void init() {
for(int i = 1 ; i <= n ; i++)
fa[i] = i;
}
int findroot(int x) {
return fa[x] == x ? x : (fa[x] = findroot(fa[x]));
}
inline void uni(int u , int v) {
if(findroot(u) != findroot(v))
fa[findroot(u)] = fa[v];
}
}
double now_average = 50;
bool cmp(EDGE a , EDGE b) {
return fabs((double)a.c - now_average) < fabs((double)b.c - now_average);
}
bool choosed[M];
int minc , maxc;
double GetDeviation() {
UF::init();
std::sort(ed + 1 , ed + m + 1 , cmp);
std::memset(choosed , 0 , sizeof(choosed));
double average;
for(int i = 1 ; i <= m ; i++) {
int u = ed[i].u , v = ed[i].v;
if(UF::findroot(u) == UF::findroot(v))
continue;
UF::uni(ed[i].u , ed[i].v);
choosed[i] = true;
average += ed[i].c;
}
double deviation = 0;
average /= (double)(n - 1);
for(int i = 1 ; i <= m ; i++)
if(choosed[i])
deviation += fabs(average - (double)ed[i].c) * fabs(average - (double)ed[i].c);
return sqrt(deviation / (double)(n - 1));
}
void simulate_anneal() {
double average = ans_average;
double t = originT;
while(t > 1e-5) {//这个也没必要太小,否则答案没影响
now_average = average + (rand() * 2 - RAND_MAX) * t;
if(now_average < 0 || now_average > 100) {//优化下,温度的范围变小后,模拟退火可以跑得巨快.
t *= delta_t;
continue;
}
double now_deviation = GetDeviation();
double DE = now_deviation - ans_deviation;
if(DE < 0) {
ans_average = average = now_average;
ans_deviation = now_deviation;
}
else if(exp(-DE / t) * RAND_MAX > rand()) {
average = now_average;
}
t *= delta_t;
}
}
int main() {
n = read() , m = read();
for(int i = 1 ; i <= m ; i++) {
ed[i].u = read() , ed[i].v = read() , ed[i].c = read();
}
std::srand(seed);
for(int i = 1 ; i <= (m <= 20 ? 10 : 3) ; i++)
simulate_anneal();
std::printf("%.4f" , ans_deviation);
return 0;
}