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  • sgu106 求直线在某一矩形范围内经过的整点,扩展gcd

    sgu106  求直线在某一矩形范围内经过的整点,扩展gcd

    X - The equation
    Time Limit:250MS     Memory Limit:4096KB     64bit IO Format:%I64d & %I64u

    Description

    There is an equation ax + by + c = 0. Given a,b,c,x1,x2,y1,y2 you must determine, how many integer roots of this equation are satisfy to the following conditions : x1<=x<=x2,   y1<=y<=y2. Integer root of this equation is a pair of integer numbers (x,y).

    Input

    Input contains integer numbers a,b,c,x1,x2,y1,y2 delimited by spaces and line breaks. All numbers are not greater than 108 by absolute value。

    Output

    Write answer to the output.

    Sample Input

    1 1 -3
    0 4
    0 4
    

    Sample Output

    4
    题意:求解ax+bx+c=0在[x1,x2][y1,y2]经过的整点个数。
    思路:扩展gcd解出一个特解,再根据[x1,x2][y1,y2]解出两个k的范围,取交集即可。一个细节,先取交集再取整WA,先取整再取交集就过了,不知为何。。一个技巧,对取整数闭区间可以对两个double边界,下界向上取整ceil,上界向下取整floor,即可取到整数边界的闭区间。
    #include<iostream>
    #include<cstdio>
    #include<cstring>
    #include<cstdlib>
    #include<algorithm>
    #include<vector>
    #include<stack>
    #include<queue>
    #include<set>
    #include<map>
    #include<string>
    #include<math.h>
    #include<cctype>
    #define ll long long
    #define REP(i,a,b) for(int (i)=(a);(i)<=(b);(i)++)
    #define REPP(i,a,b,t) for(int (i)=(a);(i)<=(b);(i)+=(t))
    #define PII pair<int,int>
    #define fst first
    #define snd second
    #define MP make_pair
    #define PB push_back
    #define RI(x) scanf("%d",&(x))
    #define RII(x,y) scanf("%d%d",&(x),&(y))
    #define RIII(x,y,z) scanf("%d%d%d",&(x),&(y),&(z))
    #define DRI(x) int (x);scanf("%d",&(x))
    #define DRII(x,y) int (x),(y);scanf("%d%d",&(x),&(y))
    #define DRIII(x,y,z) int (x),(y),(z);scanf("%d%d",&(x),&(y),&(z))
    #define RS(x) scanf("%s",s)
    #define RSS(x,y) scanf("%s%s",x,y)
    #define DRS(x) char x[maxn];scanf("%s",x)
    #define DRSS(x,y) char x[maxn],y[maxn];scanf("%s%s",x,y)
    #define MS0(a) memset((a),0,sizeof((a)))
    #define MS1(a) memset((a),-1,sizeof((a)))
    #define MS(a,b) memset((a),(b),sizeof((a)))
    #define ALL(v) v.begin(),v.end()
    #define SZ(v) (v).size()
    
    using namespace std;
    
    const int maxn=1000100;
    const int INF=(1<<29);
    const double EPS=0.0000000001;
    const double Pi=acos(-1.0);
    
    ll exgcd(ll a,ll b,ll &x,ll &y)
    {
        if(b==0){
            x=1;y=0;
            return a;
        }
        ll r=exgcd(b,a%b,x,y);
        ll t=y;
        y=x-a/b*y;
        x=t;
        return r;
    }
    
    int main()
    {
        ll a,b,c,x1,x2,y1,y2,x,y;
        while(cin>>a>>b>>c>>x1>>x2>>y1>>y2){
            c=-c;
            if(a==0&&b==0)
                if(c==0) cout<<(x2-x1+1)*(y2-y1+1)<<endl;
                else cout<<0<<endl;
            else if(a==0)
                if(c%b==0&&y1<=c/b&&c/b<=y2) cout<<x2-x1+1<<endl;
                else cout<<0<<endl;
            else if(b==0)
                if(c%a==0&&x1<=c/a&&c/a<=x2) cout<<y2-y1+1<<endl;
                else cout<<0<<endl;
            else{
                ll d=exgcd(a,b,x,y);
                if(c%d) cout<<0<<endl;
                else{
                    x*=c/d;y*=c/d;
                    double L1=(x1-x)*1.0*d/b,R1=(x2-x)*1.0*d/b;
                    if(L1>R1) swap(L1,R1);
                    double L2=(y-y1)*1.0*d/a,R2=(y-y2)*1.0*d/a;
                    if(L2>R2) swap(L2,R2);
                    ll Ld1=ceil(L1),Rd1=floor(R1);
                    ll Ld2=ceil(L2),Rd2=floor(R2);
                    ll L=max(Ld1,Ld2),R=min(Rd1,Rd2);
                    if(L>R) cout<<0<<endl;
                    else cout<<R-L+1<<endl;
                }
            }
        }
        return 0;
    }
    View Code
    没有AC不了的题,只有不努力的ACMER!
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  • 原文地址:https://www.cnblogs.com/--560/p/4583479.html
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