google code jam 2008 Number Sets(模拟,数论)

google的题目总是看上去很直白，但却藏有很多优化方式。

给个官方的分析。

The process described is fairly slow:

 

• Create singleton sets of integers.
• Take each pair of integers;
• factor the two integers and see if they have a prime factor in common that is greater than or equal to P;
• if so, merge the sets that have these two integers.

We need to find the result of this process using a faster method. A few observations help here.

Firstly, we only need to find prime factors less than the size of the interval. If a prime is larger than or equal to the size of the interval, then at most one integer in the interval can have that prime as a factor, so it will never be used to merge sets. （这个条件看似简单，但却有很明显的优化效果）

Secondly, we can take a faster approach than finding the prime factors of each integer separately. Instead we consider each prime in turn and find all the integers in the interval which have this prime as a factor, a technique generally called a sieve. （逆着往里代，而不要顺着往下找）

Thirdly, joining sets can be implemented efficiently with a data structure called union-find (or the disjoint sets data structure). As we consider each prime and find all of the integers with that prime as a factor, we join all of the sets containing those integers. We could also build an undirected graph with nodes for the integers in the interval and the primes, and edges representing which integers are divisible by which primes; then the final sets are the connected components of this graph. （find-union，并查集，大家应该都很熟悉）

#include "stdafx.h" #include "string.h" #include "math.h" #include <queue> #include <algorithm> #include <iostream> #define int64 __int64 #define _clr(a,b) memset(a,b,sizeof(a)) template<class T> T _abs(T a) {if(a<0) a=-a;return a;} template<class T> void Get_Min(T& a,T b) { if(a>b) a=b;} template<class T> void Get_Max(T& a,T b) { if(a<b) a=b;} using namespace std; int64 A,B,P; int belong[1000000+5]; int rank[1000000+5]; int is_primer[1000000+5]; int index; void cal_primer()//求质数表 { _clr(is_primer,true); int k=sqrt((double)1000001); for(int i=2;i<=k;i++) { if(is_primer[i]) { for(int j=i*i;j<1000001;j+=i) is_primer[j]=false; } } } int Find(int a) { if(belong[a]==a) return a; else { belong[a]=Find(belong[a]); return belong[a]; } } void Union(int a,int b) { if(rank[a]>rank[b]) belong[b]=a; else { belong[a]=b; if(rank[a]==rank[b]) rank[b]++; } } int main() { freopen("e://1.in","r",stdin); freopen("e://1.out","w",stdout); int t; scanf("%d",&t); index=0; cal_primer(); for(int i=1;i<=t;i++) { scanf("%I64d%I64d%I64d",&A,&B,&P); for(int j=0;j<B-A+1;j++) belong[j]=j; _clr(rank,0); int ans=B-A+1; for(int j=P;j<B-A+1;j++) { if(is_primer[j]==false) continue; int64 start=A/j*j; if(start<A) start+=j; for(int64 temp=start+j;temp<=B;temp+=j) { int m=Find(start-A); int n=Find(temp-A); if(m!=n) { Union(m,n); ans--; } } } printf("Case #%d: %d/n",i,ans); } return 0; }