algorithm - 求 n 条线段与两条平行线上端点的交点数
问题描述
求两条平行线上的端点与 n 条线段的交点数。
设两组 n 个点:
A={p1,p2,…,pn} on y=0 B={q1,q2,…,qn} on y=1 每个点 pi 与其对应的点 qi 相连形成一条线段。
我需要使用分治算法编写代码,该算法返回所有 n 线段的交点数。
例如:
输入:
3
1 101
-234 234
567 765
输出:
1
我编码如下,但我有错误的答案。
任何人都可以帮助我使用此代码或为我提供另一个解决方案吗?
#include<iostream>
#include <vector>
#include<algorithm>
using namespace std;
void merge1(vector< pair <int, int> > vect, int l, int m, int r)
{
int n1 = m - l + 1;
int n2 = r - m;
vector< pair <int, int> > vect_c_l(n1);
vector< pair <int, int> > vect_c_r(n2);
for (int i = 0; i < n1; i++)
vect_c_l[i] = vect[l + i];
for (int j = 0; j < n2; j++)
vect_c_r[j] = vect[m + 1 + j];
int i = 0;
int j = 0;
int k = l;
while (i < n1 && j < n2) {
if (vect_c_l[i].first <= vect_c_r[j].first) {
vect[k] = vect_c_l[i];
i++;
}
else {
vect[k] = vect_c_r[j];
j++;
}
k++;
}
while (i < n1) {
vect[k] = vect_c_l[i];
i++;
k++;
}
while (j < n2) {
vect[k] = vect_c_r[j];
j++;
k++;
}
}
int merge2(vector< pair <int, int> > vect, int l, int m, int r)
{
int n1 = m - l + 1;
int n2 = r - m;
int inv_count = 0;
vector< pair <int, int> > vect_c_l(n1);
vector< pair <int, int> > vect_c_r(n2);
for (int i = 0; i < n1; i++)
vect_c_l[i] = vect[l + i];
for (int j = 0; j < n2; j++)
vect_c_r[j] = vect[m + 1 + j];
int i = 0;
int j = 0;
int k = l;
while (i < n1 && j < n2) {
if (vect_c_l[i].second < vect_c_r[j].second) {
vect[k] = vect_c_l[i];
i++;
}
else {
vect[k] = vect_c_r[j];
j++;
inv_count = inv_count + (m - i);
}
k++;
}
while (i < n1) {
vect[k] = vect_c_l[i];
i++;
k++;
}
while (j < n2) {
vect[k] = vect_c_r[j];
j++;
k++;
}
return inv_count;
}
void mergeSort1(vector< pair <int, int> > vect, int l, int r) {
if (l >= r) {
return;
}
int m = l + (r - l) / 2;
mergeSort1(vect, l, m);
mergeSort1(vect, m + 1, r);
merge1(vect, l, m, r);
}
int mergeSort2(vector< pair <int, int> > vect, int l, int r) {
int inv_count = 0;
if (r > l) {
int m = l + (r - l) / 2;
inv_count += mergeSort2(vect, l, m);
inv_count += mergeSort2(vect, m+ 1, r);
/*Merge the two parts*/
inv_count += merge2(vect, l, m + 1, r);
}
return inv_count;
}
int main() {
int n,c=0;
cin >> n;
int a, b;
vector< pair <int, int> > vect;
for (int i = 0;i < n;i++) {
cin >> a >> b;
vect.push_back(make_pair(a, b));
}
mergeSort1(vect,0,n-1);
cout << mergeSort2(vect,0, n - 1);
}
解决方案
我会利用这样的想法,即计算线段是否相交比计算它们相交的位置要简单得多。如果两个线段的 x 值在 y=1 和 y=0 上位于彼此的不同侧,则它们相交。(即,如果一个段上的两个 x 值都小于其他值,或者都大于其他值)。
对象使这很容易陈述。构建一个段对象,其主要工作是确定它是否与另一个实例相交。
class Segment {
constructor(x) {
this.x0 = x[0];
this.x1 = x[1];
}
// answer whether the reciever intersects the passed segment
intersects(segment) {
// this is ambiguous in the problem, but assume touching endpoints
// count as intersections
if (this.x0 === segment.x0 || this.x1 === segment.x1) return true;
let sort0 = this.x0 < segment.x0
let sort1 = this.x1 < segment.x1
return sort0 !== sort1
}
}
let input = [
[1, 101],
[-234, 234],
[567, 765]
];
let segments = input.map(x => new Segment(x))
// check segments with one another in pairs
let pairs = segments.map((v, i) => segments.slice(i + 1).map(w => [v, w])).flat();
let intersections = pairs.reduce((acc, p) => p[0].intersects(p[1]) ? acc + 1 : acc, 0)
console.log(intersections)