P3132 [USACO16JAN]愤怒的奶牛Angry Cows
题目背景
【题目描述】
奶牛Bessie设计了一个游戏:“愤怒的奶牛”。游戏的原型是:有一些可爆炸的草堆分布在一条数轴的某些坐标上,玩家用弹弓把一头奶牛发射到数轴上。奶牛砸到数轴上的冲击波会引发附近的草堆爆炸,而被引爆的草堆可能会引爆其他草堆。游戏的目标是玩家用一只奶牛炸掉所有的草堆。
有N个草堆在数轴的不同位置,坐标为x1,x2,….,xn。如果玩家以能量R把奶牛发射到坐标x,就会引爆半径R及以内的的草堆,即坐标范围[x−R,x+R]的草堆都会燃爆,每个被奶牛引爆的草堆又会2次引爆半径R-1及以内的的草堆,2次引爆的草堆又会3次引爆半径R-2及以内的的草堆...直到一次引爆后没有其他草堆被波及或半径为0。
现在只有1头奶牛,能量为R,请计算如果要引爆所有的草堆,最小的R是多少?
【输入格式】
第一行:1个整数N(2≤N≤50,000)。
下面有N行,每行一个整数:x1,x2,…,xn,范围在[0…1,000,000,000]。
【输出格式】
输出最小的能量R,保留1位小数。
感谢wenyu0909和世界第一弱的翻译。
题目描述
Bessie the cow has designed what she thinks will be the next big hit video game: "Angry Cows". The premise, which she believes is completely original, is that the player shoots a cow with a slingshot into a one-dimensional scene consisting of a set of hay bales located at various points on a number line; the cow lands with sufficient force to detonate the hay bales in close proximity to her landing site, which in turn might set of a chain reaction that causes additional hay bales to explode. The goal is to use a single cow to start a chain reaction that detonates all the hay bales.
There are NN hay bales located at distinct integer positions x_1, x_2, \ldots, x_Nx1,x2,…,xN on the number line. If a cow is launched with power RR landing at position xx, this will causes a blast of "radius RR", engulfing all hay bales within the range x-R \ldo s x+Rx−R\ldosx+R. These hay bales then themselves explode (all simultaneously), each with a blast radius of R-1R−1. Any not-yet-exploded bales caught in these blasts then all explode (all simultaneously) with blast radius R-2R−2, and so on.
Please determine the minimum amount of power RR with which a single cow may be launched so that, if it lands at an appropriate location, it will cause subsequent detonation of every single hay bale in the scene.
输入输出格式
输入格式:
The first line of input contains NN (2 \leq N \leq 50,0002≤N≤50,000). The remaining
NN lines all contain integers x_1 \ldots x_Nx1…xN (each in the range
0 \ldots 1,000,000,0000…1,000,000,000).
输出格式:
Please output the minimum power RR with which a cow must be launched in order
to detonate all the hay bales. Answers should be rounded and printed to exactly
1 decimal point.
输入输出样例
说明
In this example, a cow launched with power 3 at, say, location 5, will cause
immediate detonation of hay bales at positions 3 and 8. These then explode
(simultaneously) each with blast radius 2, engulfing bales at positions 1 and
10, which next explode (simultaneously) with blast radius 1, engulfing the final
bale at position 11, which finally explodes with blast radius 0.
#include<iostream> #include<cstdio> #include<algorithm> using namespace std; inline int read() { int x=0; char c=getchar(); while(c<'0'||c>'9') c=getchar(); while(c<='9'&&c>='0') { x=x*10+c-'0'; c=getchar(); } return x; } int n; double a[50005]; double ans; double eps=0.001; double f[50005],g[50005]; void pre() { f[1]=g[n]=0; for(int i=2;i<=n;i++) { f[i]=999999999; int l=1,r=i; while(l+1<r) { int mid=(l+r)/2; if(f[mid-1]+1<a[i]-a[mid-1]) l=mid; else r=mid; } f[i]=min(max(a[i]-a[l-1],f[l-1]+1),max(a[i]-a[r-1],f[r-1]+1)); } for(int i=n-1;i>=1;i--) { g[i]=9999999999; int l=i,r=n-1; while(l+1<r) { int mid=(l+r)/2; if(g[mid+1]+1<a[mid+1]-a[i]) r=mid; else l=mid; } g[i]=min(max(a[l+1]-a[i],g[l+1]+1),max(a[r+1]-a[i],g[r+1]+1)); } } bool check(double x) { for(int i=n;i>=1;i--) { if(f[i]+1<=x) { for(int j=i;j<=n&&a[j]<=a[i]+2.0*x;j++) if(g[j]+1<=x) return true; break; } } return false; } void solve() { double l=1.0,r=a[n]*1.0; while(eps<r-l) { double mid=(l+r)/2.0; if(check(mid)) r=mid; else l=mid; } ans=l; } int main() { freopen("angrycow.txt","r",stdin); n=read(); for(int i=1;i<=n;i++) a[i]=read(); sort(a+1,a+1+n); pre(); solve(); printf("%.1f",ans); return 0; }