解題區/解題區 - UVa

UVa 11768 - Lattice Point or Not

contents

  1. 1. Problem
  2. 2. Sample Input
  3. 3. Sample Output
  4. 4. Solution

Problem

給一個浮點數一位的線段，請問中間經過多少個整數點。

Sample Input

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3
10.1 10.1 11.2 11.2
10.2 100.3 300.3 11.1
1.0 1.0 2.0 2.0

Sample Output

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3
1
0
2

Solution

找到原本線段 ax + by = c 的線，為了使其落在整數座標上調整為 10a x + 10b y = c，在這個情況下找到符合的 (x, y) 整數解，但是其結果要被限制上線段上。

努力地做細部微調 …

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#include <stdio.h> 
#include <math.h>
#include <algorithm>
using namespace std;
long long gcd(long long x, long long y) {
    long long t;
    while (x%y)
        t = x, x = y, y = t%y; 
    return y;
}
long long exgcd(long long x, long long y, long long &a, long long &b) {
    // ax + by = gcd(x,y)
    int flag = 0;
    long long t, la = 1, lb = 0, ra = 0, rb = 1;
    while(x%y) {
        if(flag == 0)
            la -= x/y*ra, lb -= x/y*rb;
        else
            ra -= x/y*la, rb -= x/y*lb;
        t = x, x = y, y = t%y;
        flag = 1 - flag;
    }
    if(flag == 0)
        a = ra, b = rb;
    else
        a = la, b = lb;
    return y;
}
long long countSolution(long long n1, long long n2, long long n, 
					long long lx, long long rx, long long ly, long long ry) {
//	printf("%lld %lld %lld\n", n1, n2, n);
    if (lx > rx || ly > ry)	return 0;
    if (n1 == 0)	return rx - lx + 1;
    if (n2 == 0)	return ry - ly + 1;
    long long a, b, g;
    g = exgcd(n1, n2, a, b); // a*n1 + b*n2 = gcd(n1,2)
    if(n%g) return 0;
    long long k = n/g, k1, k2;
    a *= k, b *= k;// a*n1 + b*n2 = n
    // (a+F)*n1 + (b+G)*n2 = n =>  Fn1 + Gn2 = 0, F = lcm(n1, n2)/n1 * i, G = lcm(n1, n2)/n2 * i
    k1 = n1*n2/g/n1, k2 = n1*n2/g/n2;
    long long x1, x2, x3, x4, y1, y2, y3, y4;
    k = (a - lx)/k1;
    a -= k*k1, b += k*k2;
    while (a < lx) {
    	if (k1 < 0)	a -= k1, b += k2;
    	else		a += k1, b -= k2;
    }
    x1 = a, y1 = b;
    
    k = (b - ly)/k2;
    a += k*k1, b -= k*k2;  
    while (b < ly) {
    	if (k2 < 0)	a += k1, b -= k2;
    	else		a -= k1, b += k2;
    }
    x3 = a, y3 = b;  
    
    k = (a - rx)/k1;
    a -= k*k1, b += k*k2;
    while (a > rx) {
    	if (k1 < 0)	a += k1, b -= k2;
    	else		a -= k1, b += k2;
    }
    x2 = a, y2 = b;
    
    k = (b - ry)/k2;
    a += k*k1, b -= k*k2;
    while (b > ry) {
    	if (k2 < 0)	a -= k1, b += k2;
    	else		a += k1, b -= k2;
    }
    x4 = a, y4 = b;
    long long l1 = 0, r1 = (x2 - x1)/ k1;
    long long l2 = (x3 - x1)/k1, r2 = (x4 - x1)/k1;
    if (l2 > r2)	swap(l2, r2);
    if (x1 > x2 || y3 > y4)	return 0;
//    printf("%lld %lld %lld %lld\n", x1, y1, x2, y2);
//    printf("%lld %lld %lld %lld\n", x3, y3, x4, y4);
//    printf("%lld %lld %lld %lld\n", l1, r1, l2, r2);
    l1 = max(l1, l2), r1 = min(r1, r2);
    if (l1 <= r1)	return r1 - l1 + 1;
    return 0;
}
long long read1Float() {
    long long x, y;
    scanf("%lld.%lld", &x, &y);
    return x * 10 + y;
}
int main() {    
//	freopen("in.txt","r+t",stdin);
//  freopen("out2.txt","w+t",stdout);
    int testcase, cases = 0;
    long long sx, sy, ex, ey;
    scanf("%d", &testcase);
    while (testcase--) {
        sx = read1Float();
        sy = read1Float();
        ex = read1Float();
        ey = read1Float();
        long long minx, miny, maxx, maxy;
        minx = ceil(min(sx, ex)/10.0);
        maxx = floor(max(sx, ex)/10.0);
        miny = ceil(min(sy, ey)/10.0);
        maxy = floor(max(sy, ey)/10.0);
        long long a, b, c;
        a = (ey - sy), b = (sx - ex), c = (a * sx + b * sy);
        a = a * 10, b = b * 10;
//		printf("[%lld %lld] [%lld %lld]\n", minx, maxx, miny, maxy);
        printf("%lld\n", countSolution(a, b, c, minx, maxx, miny, maxy));
    }
    return 0;
}
/*
5
10.1 10.1 11.2 11.2
10.2 100.3 300.3 11.1
1.0 1.0 2.0 2.0
1.0 1.0 1.0 1.0
1.0 3.0 1.0 7.0
100
74779.7 170072.4 114261.9 224.6
12284.7 149345.2 89292.2 24355.6
119238.0 59748.9 153776.5 88822.9
146984.2 79235.1 116519.4 144150.1
75703.5 8776.5 147012.5 132491.5
93698.8 27585.7 137374.4 134649.2
100
142566.2 47373.1 4487.9 81130.6
122932.1 117873.7 7944.5 24862.7
164852.3 50346.6 58670.9 47341.7
140828.3 21325.9 141292.5 16763.8
144206.3 130562.5 96260.1 53203.7
100
175593.7 137910.8 151744.6 21975.7
66918.0 141841.9 170688.8 108941.3
110334.9 103217.3 145166.0 126201.0
40233.7 62521.1 116634.3 146758.8
187820.7 131930.2 69771.4 66737.3
*/