2013-04-30 11:35:26Morris
[POJ][掃描線] 2489 - Line Segments
Description
Background
Line segments are a very common element in computational geometry. A line segment is the set of points forming the shortest path between two points (including those points). Although they are a very basic concept it can be hard to work with them if they appear in huge numbers unless you have an efficient algorithm.
Problem
Given a set of line segments, count how many distinct pairs of line segments are overlapping. Two line segments are said to be overlapping if they intersect in an infinite number of points.
Line segments are a very common element in computational geometry. A line segment is the set of points forming the shortest path between two points (including those points). Although they are a very basic concept it can be hard to work with them if they appear in huge numbers unless you have an efficient algorithm.
Problem
Given a set of line segments, count how many distinct pairs of line segments are overlapping. Two line segments are said to be overlapping if they intersect in an infinite number of points.
Input
The first line contains the number of scenarios.
Each scenario starts with the number n of line segments (1 <= n <= 100000). Then follow n lines consisting of four integers x1, y1, x2, y2 in the range [0, 1000000] each, representing a line segment that connects the points (x1, y1) and (x2, y2). It is guaranteed that a line segment does not degenerate to a single point.
Each scenario starts with the number n of line segments (1 <= n <= 100000). Then follow n lines consisting of four integers x1, y1, x2, y2 in the range [0, 1000000] each, representing a line segment that connects the points (x1, y1) and (x2, y2). It is guaranteed that a line segment does not degenerate to a single point.
Output
The
output for every scenario begins with a line containing "Scenario #i:",
where i is the number of the scenario starting at 1. Then print a
single line containing the number of distinct pairs of overlapping line
segments followed by an empty line.
Hint: The number of overlapping pairs may not fit into a 32-bit integer.
Hint: The number of overlapping pairs may not fit into a 32-bit integer.
Sample Input
2 8 1 1 2 2 2 2 3 3 1 3 3 1 10 0 20 0 20 0 30 0 15 0 25 0 50 0 100 0 70 0 80 0 1 0 0 1 1
Sample Output
Scenario #1: 3 Scenario #2: 0
Hint
Huge input,scanf is recommended.
Source
TUD Programming Contest 2005, Darmstadt, Germany
記住：交一點不算
#include <stdio.h>
#include <algorithm>
#include <queue>
#include <vector>
using namespace std;
struct Pt {
int x, y;
bool operator<(const Pt &a) const {
if(a.x != x)
return x < a.x;
return y < a.y;
}
bool operator>(const Pt &a) const {
if(a.x != x)
return x > a.x;
return y > a.y;
}
};
struct cmp {
bool operator() (const Pt &a, const Pt &b) const {
return a > b;
}
};
struct Seg {
int a, b;
int c; // ax + by = c maybe use long long
Pt s, e;
bool operator<(const Seg &A) const {
if(A.a != a) return a < A.a;
if(A.b != b) return b < A.b;
if(A.c != c) return c < A.c;
if(s < A.s) return true;
else return false;
}
};
int gcd(int x, int y) {
if(x == 0) return y;
if(y == 0) return x;
if(x < 0) x = -x;
if(y < 0) y = -y;
long long t;
while(x%y)
t = x, x = y, y = t%y;
return y;
}
Seg D[100000];
int main() {
int testcase, cases = 0;
int n;
int i, j, k;
scanf("%d", &testcase);
while(testcase--) {
scanf("%d", &n);
int g;
for(i = 0; i < n; i++) {
scanf("%d %d", &D[i].s.x, &D[i].s.y);
scanf("%d %d", &D[i].e.x, &D[i].e.y);
if(D[i].e < D[i].s)
swap(D[i].e, D[i].s);
D[i].a = D[i].s.y-D[i].e.y;
D[i].b = -D[i].s.x+D[i].e.x;
g = gcd(D[i].a, D[i].b);
if(g == 0) {D[i].a = D[i].b = 0;}
else D[i].a /= g, D[i].b /= g;
if(D[i].a < 0) D[i].a *= -1, D[i].b *= -1;
D[i].c = D[i].a*D[i].s.x + D[i].b*D[i].s.y;
}
sort(D, D+n);
priority_queue<Pt, vector<Pt>, cmp> pQ;
long long ret = 0;
for(i = 0; i < n; ) {
j = i;
while(i < n && D[i].a == D[j].a && D[i].b == D[j].b && D[i].c == D[j].c)
i++;
while(!pQ.empty()) //<pQ.clear()>
pQ.pop();
for(k = j; k < i; k++) {
while(!pQ.empty()) {
if(D[k].s.x == D[k].e.x) {
if(pQ.top().y <= D[k].s.y)
pQ.pop();
else break;
} else {
if(pQ.top().x <= D[k].s.x)
pQ.pop();
else break;
}
}
ret += pQ.size();
pQ.push(D[k].e);
}
}
printf("Scenario #%d:\n%I64d\n", ++cases, ret);
if(testcase)
puts("");
}
return 0;
}
記住：交一點不算
算出每個線段的方程 ax+by = c 將同一方程分堆，在同一堆中使用掃描線算法， 即等價於計算一維空間的區間的 overlap pair. O(nlogn) // 860 ms
#include <stdio.h>
#include <algorithm>
#include <queue>
#include <vector>
using namespace std;
struct Pt {
int x, y;
bool operator<(const Pt &a) const {
if(a.x != x)
return x < a.x;
return y < a.y;
}
bool operator>(const Pt &a) const {
if(a.x != x)
return x > a.x;
return y > a.y;
}
};
struct cmp {
bool operator() (const Pt &a, const Pt &b) const {
return a > b;
}
};
struct Seg {
int a, b;
int c; // ax + by = c maybe use long long
Pt s, e;
bool operator<(const Seg &A) const {
if(A.a != a) return a < A.a;
if(A.b != b) return b < A.b;
if(A.c != c) return c < A.c;
if(s < A.s) return true;
else return false;
}
};
int gcd(int x, int y) {
if(x == 0) return y;
if(y == 0) return x;
if(x < 0) x = -x;
if(y < 0) y = -y;
long long t;
while(x%y)
t = x, x = y, y = t%y;
return y;
}
Seg D[100000];
int main() {
int testcase, cases = 0;
int n;
int i, j, k;
scanf("%d", &testcase);
while(testcase--) {
scanf("%d", &n);
int g;
for(i = 0; i < n; i++) {
scanf("%d %d", &D[i].s.x, &D[i].s.y);
scanf("%d %d", &D[i].e.x, &D[i].e.y);
if(D[i].e < D[i].s)
swap(D[i].e, D[i].s);
D[i].a = D[i].s.y-D[i].e.y;
D[i].b = -D[i].s.x+D[i].e.x;
g = gcd(D[i].a, D[i].b);
if(g == 0) {D[i].a = D[i].b = 0;}
else D[i].a /= g, D[i].b /= g;
if(D[i].a < 0) D[i].a *= -1, D[i].b *= -1;
D[i].c = D[i].a*D[i].s.x + D[i].b*D[i].s.y;
}
sort(D, D+n);
priority_queue<Pt, vector<Pt>, cmp> pQ;
long long ret = 0;
for(i = 0; i < n; ) {
j = i;
while(i < n && D[i].a == D[j].a && D[i].b == D[j].b && D[i].c == D[j].c)
i++;
while(!pQ.empty()) //<pQ.clear()>
pQ.pop();
for(k = j; k < i; k++) {
while(!pQ.empty()) {
if(D[k].s.x == D[k].e.x) {
if(pQ.top().y <= D[k].s.y)
pQ.pop();
else break;
} else {
if(pQ.top().x <= D[k].s.x)
pQ.pop();
else break;
}
}
ret += pQ.size();
pQ.push(D[k].e);
}
}
printf("Scenario #%d:\n%I64d\n", ++cases, ret);
if(testcase)
puts("");
}
return 0;
}
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