Accepted date: 2011-06-04
Ranking (as of 2013-01-27): 47
Language: C++
Used the two variations of Graham's scan algorithm.
Although this solution was accepted, the below convex_hull function is a bit doubtful and still may include some other bugs.
/*
14.7.1 Herding Frosh
PC/UVa IDs: 111401/10135, Popularity: C, Success rate: average Level: 2
To build using Visucal Studio 2008:
cl -EHsc herding_frosh.cpp
*/
#include <iostream>
#include <vector>
#include <algorithm>
#include <cstdio>
#include <cfloat>
#include <cmath>
using namespace std;
#define EPSILON FLT_EPSILON /* DBL_EPSILON */
const double pi = 2.0 * acos(0.0); // 3.14159265358979323846
struct point {
double x, y;
point() : x(0.0), y(0.0) {}
point(double _x, double _y) : x(_x), y(_y) {}
point(const point &p) : x(p.x), y(p.y) {}
bool operator==(const point& p) const {return x == p.x && y == p.y;}
};
#ifdef DEBUG
ostream& operator<<(ostream& os, const point& p)
{
os << '(' << p.x << ", " << p.y << ')';
return os;
}
#endif
bool left_lower(const point& p1, const point& p2)
{
if (p1.x < p2.x)
return true;
else if (p1.x > p2.x)
return false;
else if (p1.y < p2.y)
return true;
else
return false;
}
void sort_and_remove_duplicates(vector<point>& points,
bool (*compare)(const point&, const point&) /* = left_lower */)
{
sort(points.begin(), points.end(), compare);
// sort the points in leftmost-lowest order
for (vector<point>::iterator i = points.begin(); i != points.end(); ) {
// remove the duplicate points
vector<point>::iterator j = i;
j++;
if (j != points.end() && *i == *j)
i = points.erase(i);
else
i++;
}
}
double signed_triangle_area(const point& a, const point& b, const point& c)
{
return (a.x * b.y - a.y * b.x + a.y * c.x -
a.x * c.y + b.x * c.y - c.x * b.y) / 2.0;
}
bool collinear(const point& a, const point& b, const point& c)
{
return fabs(signed_triangle_area(a, b, c)) <= EPSILON;
}
double euclidean_distance(const point& a, const point& b)
{
double dx = a.x - b.x, dy = a.y - b.y;
return sqrt(dx * dx + dy * dy);
}
bool ccw(const point& a, const point& b, const point& c)
{
// see if the point c is to the left of a -> b
// (or, a - b - c are counterclockwise)
return signed_triangle_area(a, b, c) > EPSILON;
}
bool cw(const point& a, const point& b, const point& c)
{
// see if the point c is to the right of a -> b
// (or, a - b - c are clockwise)
return signed_triangle_area(a, b, c) < -EPSILON;
}
struct smaller_angle {
const point& first;
smaller_angle(const point& _first) : first(_first) {}
bool operator() (const point& p1, const point& p2) const;
};
bool smaller_angle::operator() (const point& p1, const point& p2) const
{
if (collinear(first, p1, p2))
return euclidean_distance(first, p1) <= euclidean_distance(first, p2);
else
return ccw(first, p1, p2);
}
int convex_hull(vector<point>& points, vector<point>& hull)
{
sort_and_remove_duplicates(points, left_lower);
// sort the points in leftmost-lowest order
vector<point>::iterator i = points.begin();
i++;
sort(i, points.end(), smaller_angle(points[0]));
// sort the second and later points in increasing angular order
hull.resize(points.size());
hull[0] = points[0]; hull[1] = points[1];
int j = 1;
for (int i = 2; i < points.size(); ) {
if (cw(hull[j - 1], hull[j], points[i]))
j--; // remove hulll[j]
else {
if (!collinear(hull[j - 1], hull[j], points[i]))
j++;
hull[j] = points[i++];
}
}
if (cw(hull[j - 1], hull[j], points[0]))
;
else
j++;
hull.resize(j);
return hull.size();
}
#ifdef DEBUG
void print_polygon(const vector<point>& points)
{
for (vector<point>::const_iterator i = points.begin();
i != points.end(); i++)
cout << *i << endl;
}
#endif
struct smaller_polar_angle {
point p;
double angle;
smaller_polar_angle(const point& _p) : p(_p), angle(atan2(p.y, p.x)) {}
bool operator() (const point& p1, const point& p2) const;
};
bool smaller_polar_angle::operator() (const point& p1, const point& p2) const
{
double a1 = atan2(p1.y, p1.x) - angle, a2 = atan2(p2.y, p2.x) - angle;
if (fabs(a1 - a2) <= EPSILON)
return euclidean_distance(p, p1) < euclidean_distance(p, p2);
if (a1 < 0)
a1 += pi * 2.0;
if (a2 < 0)
a2 += pi * 2.0;
return a1 < a2;
}
int convex_hull_ex(vector<point>& points, vector<point>& hull)
{
point original_point;
hull.resize(points.size());
hull[0] = points[0]; hull[1] = points[1];
int j = 1;
for (int i = 2; i < points.size(); ) {
if (!j)
hull[++j] = points[i++];
else if (cw(hull[j - 1], hull[j], points[i])) {
if (hull[j] == original_point)
hull[++j] = points[i++];
else
j--; // remove hulll[j]
}
else {
/*
if (hull[j] == original_point ||
!collinear(hull[j - 1], hull[j], points[i]))
*/
j++;
hull[j] = points[i++];
}
}
for ( ; j; j--) {
if (hull[j] == original_point)
break;
else if (cw(hull[j - 1], hull[j], points[0]))
;
else
break;
}
j++;
hull.resize(j);
return hull.size();
}
double polygon_perimeter(const vector<point>& polygon)
{
double perimeter = 0.0;
for (int i = 0; i < polygon.size(); i++) {
int j = (i + 1) % polygon.size();
perimeter += euclidean_distance(polygon[i], polygon[j]);
}
return perimeter;
}
int main(int /* argc */, char** /* argv */)
{
int nr_cases;
cin >> nr_cases;
for (int c = 0; c < nr_cases; c++) {
int nr_frosh;
cin >> nr_frosh;
point original_point;
vector<point> frosh(nr_frosh + 1);
frosh[0] = original_point;
for (int i = 1; i <= nr_frosh; i++)
cin >> frosh[i].y >> frosh[i].x;
if (nr_frosh < 2) { // special cases
double perimeter = (nr_frosh) ?
2.0 /* one extra meter at each end */ +
euclidean_distance(frosh[0], frosh[1]) * 2.0 : 2.0;
printf("%.2f\n", perimeter);
if (c < nr_cases - 1)
cout << endl; // print a blank line between the two consecutive cases
continue;
}
vector<point> hull;
convex_hull(frosh, hull);
// if a frosh is at the original point,
// the convex_hull function removes the duplicate points.
#ifdef DEBUG
cout << endl; print_polygon(hull);
#endif
double min_perimeter = DBL_MAX;
if (find(hull.begin(), hull.end(), original_point) != hull.end())
// the original point is one of the vertices of the convex hull
min_perimeter = polygon_perimeter(hull);
else {
frosh.erase(find(frosh.begin(), frosh.end(), original_point));
nr_frosh = frosh.size();
vector<point>::iterator i = frosh.begin();
i++;
// sort the 2nd and and later points by their polar angles,
// being the angle of the 1st point as the reference one
sort(i, frosh.end(), smaller_polar_angle(frosh[0]));
#ifdef DEBUG
cout << endl; print_polygon(frosh);
#endif
for (int j = 0; j < nr_frosh; j++) {
i = frosh.insert(i, original_point);
// calculate the convex hull
// except for the original point and its neigbors
convex_hull_ex(frosh, hull);
double perimeter = polygon_perimeter(hull);
#ifdef DEBUG
cout << endl; print_polygon(hull);
cout << perimeter << endl;
#endif
min_perimeter = min(min_perimeter, perimeter);
i = frosh.erase(i);
if (i != frosh.end())
i++;
}
}
printf("%.2f\n", 2.0 /* one extra meter at each end */ + min_perimeter);
if (c < nr_cases - 1)
cout << endl; // print a blank line between the two consecutive cases
}
return 0;
}
