UVa 10173 - Smallest Bounding Rectangle

Accepted date: 2015-02-01
Ranking (as of 2015-02-01): 180 out of 431
Language: C++

/*
  UVa 10173 - Smallest Bounding Rectangle

  To build using Visual Studio 2012:
    cl -EHsc -O2 UVa_10173_Smallest_Bounding_Rectangle.cpp
*/

#include <iostream>
#include <iomanip>
#include <vector>
#include <algorithm>
#include <limits>
#include <cmath>
using namespace std;

const double epsilon = numeric_limits<float>::epsilon();

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;}
};

ostream& operator<<(ostream& os, const point& p)
{
  os << '(' << p.x << ", " << p.y << ')';
  return os;
}

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);
#ifdef DEBUG
  for (int i = 0; i < hull.size(); i++) {
    if (i)
      cout << ' ';
    cout << hull[i];
  }
  cout << endl;
#endif
  return hull.size();
}

bool point_in_hull(point& p, const vector<point>& hull)
{
  int n = hull.size();
  for (int i = 0; i < n; i++)
    if (cw(hull[i], hull[(i + 1) % n], p))
      return false;
  return true;
}

double polygon_area(const vector<point>& polygon)
{
  double area = 0.0;
  for (int i = 0; i < polygon.size(); i++) {
    int j = (i + 1) % polygon.size();
    area += polygon[i].x * polygon[j].y - polygon[j].x * polygon[i].y;
  }
  return area / 2.0;
}

point rotate_point(const point& o, const point& p, double angle)
  // rotate p by angle around o
{
  if (fabs(angle) < epsilon)
    angle = 0.0;
  double x = p.x - o.x, y = p.y - o.y;
  return  point(o.x + x * cos(angle) - y * sin(angle),
    o.y + x * sin(angle) + y * cos(angle));
}

double min_bounding_rectangle_area(const vector<point>& hull)
{
/*
  for each edge of the convex hull:
    compute the edge orientation.
    rotate the convex hull using this orientation.
    calculate the bounding rectangle area with 
      min/max of x/y of the rotated convex hull.
*/
  int n = hull.size();
  double min_area = numeric_limits<double>::max();
  for (int i = 0; i < n; i++) {
    int j = (i + 1) % n;
    double angle = atan2(hull[j].y - hull[i].y, hull[j].x - hull[i].x);
    double min_x = hull[i].x, min_y = hull[i].y, max_x = hull[i].x, max_y = hull[i].y;
    for ( ; j != i; j = (j + 1) % n) {
      point p = rotate_point(hull[i], hull[j], -angle);
      min_x = min(min_x, p.x); min_y = min(min_y, p.y);
      max_x = max(max_x, p.x); max_y = max(max_y, p.y);
    }
#ifdef DEBUG
    cout << angle << ' ' << point(min_x, min_y) <<
      ' ' << point(max_x, max_y) << endl;
#endif
    min_area = min(min_area, (max_x - min_x) * (max_y - min_y));
  }
  return min_area;
}

int main()
{
  const int n_max = 1000;
  vector<point> points(n_max);
  while (true) {
    int n;
    cin >> n;
    if (!n)
      break;
    points.resize(n);
    for (int i = 0; i < n; i++)
      cin >> points[i].x >> points[i].y;
    double area = 0.0;
    if (n > 2) {
      vector<point> hull(n);
      convex_hull(points, hull);
      area = min_bounding_rectangle_area(hull);
    }
    cout << fixed << setprecision(4) << area << endl;
  }
  return 0;
}