Problem p00260 (C++) — buggy submission, verdict: Time Limit Exceeded

#include <bits/stdc++.h>

#define REP(i, s, n) for (int i = s; i < n; i++)
#define rep(i, n) REP(i, 0, n)
#define EPS (1e-8)
#define COUNTER_CLOCKWISE 1
#define CLOCKWISE -1
#define ONLINE_BACK 2
#define ONLINE_FRONT -2
#define ON_SEGMENT 0
#define equals(a, b) (fabs((a) - (b)) < EPS)

using namespace std;

const int IINF = INT_MAX;

// template - begin
class Point {
public:
  double x, y;

  Point(double x = 0, double y = 0) : x(x), y(y) {}

  Point operator+(Point p) { return Point(x + p.x, y + p.y); }
  Point operator-(Point p) { return Point(x - p.x, y - p.y); }
  Point operator*(double a) { return Point(a * x, a * y); }
  Point operator/(double a) { return Point(x / a, y / a); }
  Point operator*(Point a) {
    return Point(x * a.x - y * a.y, x * a.y + y * a.x);
  }

  bool operator<(const Point &p) const {
    return !equals(x, p.x) ? x < p.x : (!equals(y, p.y) && y < p.y);
  }

  bool operator==(const Point &p) const {
    return fabs(x - p.x) < EPS && fabs(y - p.y) < EPS;
  }
};

struct Segment {
  Point p1, p2;
  Segment(Point p1 = Point(), Point p2 = Point()) : p1(p1), p2(p2) {}
  bool operator==(const Segment &p) const {
    return (p.p1 == p1 && p.p2 == p2) || (p.p1 == p2 && p.p2 == p1);
  }
  bool operator<(const Segment &s) const {
    return !(p1 == s.p1) ? p1 < s.p1 : p2 < s.p2;
  }
};

typedef Point Vector;
typedef Segment Line;
typedef vector<Point> Polygon;

ostream &operator<<(ostream &os, const Point &a) {
  os << "(" << a.x << "," << a.y << ")";
}

ostream &operator<<(ostream &os, const Segment &a) {
  os << "( " << a.p1 << " , " << a.p2 << " )";
}

double dot(Point a, Point b) { return a.x * b.x + a.y * b.y; }

double cross(Point a, Point b) { return a.x * b.y - a.y * b.x; }

double norm(Point a) { return a.x * a.x + a.y * a.y; }

double abs(Point a) { return sqrt(norm(a)); }

// rad は角度をラジアンで持たせること
Point rotate(Point a, double rad) {
  return Point(cos(rad) * a.x - sin(rad) * a.y,
               sin(rad) * a.x + cos(rad) * a.y);
}

// 度をラジアンに変換
double toRad(double agl) { return agl * M_PI / 180.0; }

// a => prev, b => cur, c=> next
// prev から cur へ行って next へ行く際の角度を求める
double getArg(Point a, Point b, Point c) {
  double arg1 = atan2(b.y - a.y, b.x - a.x);
  double arg2 = atan2(c.y - b.y, c.x - b.x);
  double arg = fabs(arg1 - arg2);
  while (arg > M_PI)
    arg -= 2.0 * M_PI;
  return fabs(arg);
}
// template - end

// ccw - begin
int ccw(Point p0, Point p1, Point p2) {
  Point a = p1 - p0;
  Point b = p2 - p0;
  if (cross(a, b) > EPS)
    return COUNTER_CLOCKWISE;
  if (cross(a, b) < -EPS)
    return CLOCKWISE;
  if (dot(a, b) < -EPS)
    return ONLINE_BACK;
  if (norm(a) < norm(b))
    return ONLINE_FRONT;
  return ON_SEGMENT;
}
// ccw - end

// intersect - begin
bool intersectLL(Line l, Line m) {
  return abs(cross(l.p2 - l.p1, m.p2 - m.p1)) > EPS || // non-parallel
         abs(cross(l.p2 - l.p1, m.p1 - l.p1)) < EPS;   // same line
}
bool intersectLS(Line l, Line s) {
  return cross(l.p2 - l.p1, s.p1 - l.p1) * // s[0] is left of l
             cross(l.p2 - l.p1, s.p2 - l.p1) <
         EPS; // s[1] is right of l
}
bool intersectLP(Line l, Point p) {
  return abs(cross(l.p2 - p, l.p1 - p)) < EPS;
}
bool intersectSS(Line s, Line t) {
  return ccw(s.p1, s.p2, t.p1) * ccw(s.p1, s.p2, t.p2) <= 0 &&
         ccw(t.p1, t.p2, s.p1) * ccw(t.p1, t.p2, s.p2) <= 0;
}
bool intersectSP(Line s, Point p) {
  return abs(s.p1 - p) + abs(s.p2 - p) - abs(s.p2 - s.p1) <
         EPS; // triangle inequality
}

Point projection(Line l, Point p) {
  double t = dot(p - l.p1, l.p1 - l.p2) / norm(l.p1 - l.p2);
  return l.p1 + (l.p1 - l.p2) * t;
}
Point reflection(Line l, Point p) { return p + (projection(l, p) - p) * 2; }
double distanceLP(Line l, Point p) { return abs(p - projection(l, p)); }
double distanceLL(Line l, Line m) {
  return intersectLL(l, m) ? 0 : distanceLP(l, m.p1);
}

double distanceLS(Line l, Line s) {
  if (intersectLS(l, s))
    return 0;
  return min(distanceLP(l, s.p1), distanceLP(l, s.p2));
}
double distanceSP(Line s, Point p) {
  Point r = projection(s, p);
  if (intersectSP(s, r))
    return abs(r - p);
  return min(abs(s.p1 - p), abs(s.p2 - p));
}

double distanceSS(Line s, Line t) {
  if (intersectSS(s, t))
    return 0;
  return min(min(distanceSP(s, t.p1), distanceSP(s, t.p2)),
             min(distanceSP(t, s.p1), distanceSP(t, s.p2)));
}

Point crosspoint(Line l, Line m) {
  double A = cross(l.p2 - l.p1, m.p2 - m.p1);
  double B = cross(l.p2 - l.p1, l.p2 - m.p1);
  if (abs(A) < EPS && abs(B) < EPS) {
    vector<Point> vec;
    vec.push_back(l.p1), vec.push_back(l.p2), vec.push_back(m.p1),
        vec.push_back(m.p2);
    sort(vec.begin(), vec.end());
    assert(vec[1] == vec[2]); // 同じセグメントかもよ
    return vec[1];
    // return m.p1;
  }
  if (abs(A) < EPS)
    assert(false);
  return m.p1 + (m.p2 - m.p1) * (B / A);
}

// intersect - end

// inPolygonP - begin
// cross product of pq and pr
double cross3p(Point p, Point q, Point r) {
  return (r.x - q.x) * (p.y - q.y) - (r.y - q.y) * (p.x - q.x);
}

// returns true if point r is on the same line as the line pq
bool collinear(Point p, Point q, Point r) {
  return fabs(cross3p(p, q, r)) < EPS;
}

// returns true if point t is on the left side of line pq
bool ccwtest(Point p, Point q, Point r) {
  return cross3p(p, q, r) > 0; // can be modified to accept collinear points
}

bool onSegment(Point p, Point q, Point r) {
  return collinear(p, q, r) &&
         equals(sqrt(pow(p.x - r.x, 2) + pow(p.y - r.y, 2)) +
                    sqrt(pow(r.x - q.x, 2) + pow(r.y - q.y, 2)),
                sqrt(pow(p.x - q.x, 2) + pow(p.y - q.y, 2)));
}

bool isConvex(vector<Point> p) {
  int sz = (int)p.size();

  if (sz < 3)
    return false; // boundary case, we treat a point or a line as not convex

  bool isLeft = ccwtest(p[0], p[1], p[2]);

  for (int i = 1; i < (int)p.size(); i++)
    if (ccwtest(p[i], p[(i + 1) % sz], p[(i + 2) % sz]) != isLeft)
      return false;

  return true;
}

double angle(Point a, Point b, Point c) {
  double ux = b.x - a.x, uy = b.y - a.y;
  double vx = c.x - a.x, vy = c.y - a.y;
  return acos((ux * vx + uy * vy) /
              sqrt((ux * ux + uy * uy) * (vx * vx + vy * vy)));
}

// 多角形poly内（線分上も含む）に点pが存在するかどうは判定する
bool inPolygon(Polygon poly, Point p) {
  if ((int)poly.size() == 0)
    return false;
  rep(i, poly.size()) if (onSegment(poly[i], poly[(i + 1) % poly.size()],
                                    p)) return true;
  double sum = 0;
  for (int i = 0; i < (int)poly.size(); i++) {
    if (equals(cross(poly[i] - p, poly[(i + 1) % poly.size()] - p), 0.0))
      continue; // ３点が平行だとangleがnanを返しsumがnanになり死ぬ
    if (cross3p(p, poly[i], poly[(i + 1) % poly.size()]) < 0)
      sum -= angle(p, poly[i], poly[(i + 1) % poly.size()]);
    else
      sum += angle(p, poly[i], poly[(i + 1) % poly.size()]);
  }
  // あまり誤差を厳しくしすぎると良くないので以下のほうが良い
  const double eps = 1e-5;
  return (fabs(sum - 2 * M_PI) < eps || fabs(sum + 2 * M_PI) < eps);
}
// inPolygonP - end

// inPolygonS - begin
bool inPolygon(const Polygon &poly, Segment seg) {
  vector<Point> vp;
  vp.push_back(seg.p1);
  vp.push_back(seg.p2);
  rep(i, (int)poly.size()) {
    Segment edge = Segment(poly[i], poly[(i + 1) % (int)poly.size()]);
    if (equals(cross(seg.p1 - seg.p2, edge.p1 - edge.p2), 0.0)) {
      if (onSegment(seg.p1, seg.p2, edge.p1))
        vp.push_back(edge.p1);
      if (onSegment(seg.p1, seg.p2, edge.p2))
        vp.push_back(edge.p2);
    } else {
      if (intersectSS(seg, edge))
        vp.push_back(crosspoint(seg, edge));
    }
  }
  sort(vp.begin(), vp.end());
  vp.erase(unique(vp.begin(), vp.end()), vp.end());
  rep(i, (int)vp.size() - 1) {
    Point middle_point = (vp[i] + vp[i + 1]) / 2.0;
    if (!inPolygon(poly, middle_point))
      return false;
  }
  return true;
}
// inPolygonS - end

const int MAX_V = 16 * 20 + 2000;
int n, bc[1 << 16];
Point ps[16];
Polygon poly;
vector<Segment> segs;

bool check(int bitmask) {
  int V = segs.size();
  assert(V < MAX_V);
  vector<Point> sp;
  rep(i, n) if ((bitmask >> i) & 1) sp.push_back(ps[i]);
  rep(i, V) {
    Point mp = (segs[i].p1 + segs[i].p2) / 2.0;
    bool ok = false;
    rep(j, bc[bitmask]) {
      Segment seg = Segment(sp[j], mp);
      Point middle = (seg.p1 + seg.p2) / 2.0;
      if (!inPolygon(poly, middle))
        continue;
      // if( !inPolygon(poly,seg) ) continue;
      bool failed = false;
      rep(k, n) {
        if (equals(cross(seg.p1 - seg.p2, poly[k] - poly[(k + 1) % n]), 0.0))
          continue;
        if (intersectSS(seg, Segment(poly[k], poly[(k + 1) % n]))) {
          if (onSegment(poly[k], poly[(k + 1) % n], seg.p1) ||
              onSegment(poly[k], poly[(k + 1) % n], seg.p2))
            continue;
          if (onSegment(seg.p1, seg.p2, poly[k]) ||
              onSegment(seg.p1, seg.p2, poly[(k + 1) % n]))
            continue;
          failed = true;
          break;
        }
      }
      if (!failed) {
        ok = true;
        break;
      }
    }
    if (!ok)
      return false;
  }

  return true;
}

Point vir;
bool comp(const Point &p1, const Point &p2) {
  return !equals(abs(vir - p1), abs(vir - p2)) && abs(vir - p1) < abs(vir - p2);
}

vector<Segment> makeGraph() {
  vector<Segment> vs;
  rep(i, n) vs.push_back(Segment(ps[i], ps[(i + 1) % n]));
  vector<Segment> ret;
  rep(i, vs.size()) {
    vector<Point> vp;
    vp.push_back(vs[i].p1);
    vp.push_back(vs[i].p2);
    rep(j, n) REP(k, j + 1, n) {
      Segment seg = Segment(ps[j], ps[k]);
      if (equals(cross(vs[i].p1 - vs[i].p2, seg.p1 - seg.p2), 0.0)) {
        if (onSegment(vs[i].p1, vs[i].p2, seg.p1))
          vp.push_back(seg.p1);
        if (onSegment(vs[i].p1, vs[i].p2, seg.p2))
          vp.push_back(seg.p2);
      } else if (intersectLS(seg, vs[i]))
        vp.push_back(crosspoint(vs[i], seg));
    }
    vir = vs[i].p1;
    sort(vp.begin(), vp.end(), comp);
    vp.erase(unique(vp.begin(), vp.end()), vp.end());
    rep(j, (int)vp.size() - 1) ret.push_back(Segment(vp[j], vp[j + 1]));
  }
  return ret;
}

int main() {
  rep(i, 1 << 16) bc[i] = __builtin_popcount(i);
  while (cin >> n, n) {
    rep(i, n) cin >> ps[i].x >> ps[i].y;
    poly.resize(n);
    rep(i, n) poly[i] = ps[i];
    segs = makeGraph();
    // rep(i,segs.size()) cout << segs[i] << endl;
    // cout << check (552) << endl;
    // return 0;
    int ans = n;
    rep(i, (1 << n)) {
      if (bc[i] >= ans)
        continue;
      if (check(i))
        ans = bc[i];
    }
    cout << ans << endl;
  }
  return 0;
}

Error: TLE

Fixed submission:

#include <bits/stdc++.h>

#define REP(i, s, n) for (int i = s; i < n; i++)
#define rep(i, n) REP(i, 0, n)
#define EPS (1e-8)
#define COUNTER_CLOCKWISE 1
#define CLOCKWISE -1
#define ONLINE_BACK 2
#define ONLINE_FRONT -2
#define ON_SEGMENT 0
#define equals(a, b) (fabs((a) - (b)) < EPS)

using namespace std;

const int IINF = INT_MAX;

// template - begin
class Point {
public:
  double x, y;

  Point(double x = 0, double y = 0) : x(x), y(y) {}

  Point operator+(Point p) { return Point(x + p.x, y + p.y); }
  Point operator-(Point p) { return Point(x - p.x, y - p.y); }
  Point operator*(double a) { return Point(a * x, a * y); }
  Point operator/(double a) { return Point(x / a, y / a); }
  Point operator*(Point a) {
    return Point(x * a.x - y * a.y, x * a.y + y * a.x);
  }

  bool operator<(const Point &p) const {
    return !equals(x, p.x) ? x < p.x : (!equals(y, p.y) && y < p.y);
  }

  bool operator==(const Point &p) const {
    return fabs(x - p.x) < EPS && fabs(y - p.y) < EPS;
  }
};

struct Segment {
  Point p1, p2;
  Segment(Point p1 = Point(), Point p2 = Point()) : p1(p1), p2(p2) {}
  bool operator==(const Segment &p) const {
    return (p.p1 == p1 && p.p2 == p2) || (p.p1 == p2 && p.p2 == p1);
  }
  bool operator<(const Segment &s) const {
    return !(p1 == s.p1) ? p1 < s.p1 : p2 < s.p2;
  }
};

typedef Point Vector;
typedef Segment Line;
typedef vector<Point> Polygon;

ostream &operator<<(ostream &os, const Point &a) {
  os << "(" << a.x << "," << a.y << ")";
}

ostream &operator<<(ostream &os, const Segment &a) {
  os << "( " << a.p1 << " , " << a.p2 << " )";
}

double dot(Point a, Point b) { return a.x * b.x + a.y * b.y; }

double cross(Point a, Point b) { return a.x * b.y - a.y * b.x; }

double norm(Point a) { return a.x * a.x + a.y * a.y; }

double abs(Point a) { return sqrt(norm(a)); }

// rad は角度をラジアンで持たせること
Point rotate(Point a, double rad) {
  return Point(cos(rad) * a.x - sin(rad) * a.y,
               sin(rad) * a.x + cos(rad) * a.y);
}

// 度をラジアンに変換
double toRad(double agl) { return agl * M_PI / 180.0; }

// a => prev, b => cur, c=> next
// prev から cur へ行って next へ行く際の角度を求める
double getArg(Point a, Point b, Point c) {
  double arg1 = atan2(b.y - a.y, b.x - a.x);
  double arg2 = atan2(c.y - b.y, c.x - b.x);
  double arg = fabs(arg1 - arg2);
  while (arg > M_PI)
    arg -= 2.0 * M_PI;
  return fabs(arg);
}
// template - end

// ccw - begin
int ccw(Point p0, Point p1, Point p2) {
  Point a = p1 - p0;
  Point b = p2 - p0;
  if (cross(a, b) > EPS)
    return COUNTER_CLOCKWISE;
  if (cross(a, b) < -EPS)
    return CLOCKWISE;
  if (dot(a, b) < -EPS)
    return ONLINE_BACK;
  if (norm(a) < norm(b))
    return ONLINE_FRONT;
  return ON_SEGMENT;
}
// ccw - end

// intersect - begin
bool intersectLL(Line l, Line m) {
  return abs(cross(l.p2 - l.p1, m.p2 - m.p1)) > EPS || // non-parallel
         abs(cross(l.p2 - l.p1, m.p1 - l.p1)) < EPS;   // same line
}
bool intersectLS(Line l, Line s) {
  return cross(l.p2 - l.p1, s.p1 - l.p1) * // s[0] is left of l
             cross(l.p2 - l.p1, s.p2 - l.p1) <
         EPS; // s[1] is right of l
}
bool intersectLP(Line l, Point p) {
  return abs(cross(l.p2 - p, l.p1 - p)) < EPS;
}
bool intersectSS(Line s, Line t) {
  return ccw(s.p1, s.p2, t.p1) * ccw(s.p1, s.p2, t.p2) <= 0 &&
         ccw(t.p1, t.p2, s.p1) * ccw(t.p1, t.p2, s.p2) <= 0;
}
bool intersectSP(Line s, Point p) {
  return abs(s.p1 - p) + abs(s.p2 - p) - abs(s.p2 - s.p1) <
         EPS; // triangle inequality
}

Point projection(Line l, Point p) {
  double t = dot(p - l.p1, l.p1 - l.p2) / norm(l.p1 - l.p2);
  return l.p1 + (l.p1 - l.p2) * t;
}
Point reflection(Line l, Point p) { return p + (projection(l, p) - p) * 2; }
double distanceLP(Line l, Point p) { return abs(p - projection(l, p)); }
double distanceLL(Line l, Line m) {
  return intersectLL(l, m) ? 0 : distanceLP(l, m.p1);
}

double distanceLS(Line l, Line s) {
  if (intersectLS(l, s))
    return 0;
  return min(distanceLP(l, s.p1), distanceLP(l, s.p2));
}
double distanceSP(Line s, Point p) {
  Point r = projection(s, p);
  if (intersectSP(s, r))
    return abs(r - p);
  return min(abs(s.p1 - p), abs(s.p2 - p));
}

double distanceSS(Line s, Line t) {
  if (intersectSS(s, t))
    return 0;
  return min(min(distanceSP(s, t.p1), distanceSP(s, t.p2)),
             min(distanceSP(t, s.p1), distanceSP(t, s.p2)));
}

Point crosspoint(Line l, Line m) {
  double A = cross(l.p2 - l.p1, m.p2 - m.p1);
  double B = cross(l.p2 - l.p1, l.p2 - m.p1);
  if (abs(A) < EPS && abs(B) < EPS) {
    vector<Point> vec;
    vec.push_back(l.p1), vec.push_back(l.p2), vec.push_back(m.p1),
        vec.push_back(m.p2);
    sort(vec.begin(), vec.end());
    assert(vec[1] == vec[2]); // 同じセグメントかもよ
    return vec[1];
    // return m.p1;
  }
  if (abs(A) < EPS)
    assert(false);
  return m.p1 + (m.p2 - m.p1) * (B / A);
}

// intersect - end

// inPolygonP - begin
// cross product of pq and pr
double cross3p(Point p, Point q, Point r) {
  return (r.x - q.x) * (p.y - q.y) - (r.y - q.y) * (p.x - q.x);
}

// returns true if point r is on the same line as the line pq
bool collinear(Point p, Point q, Point r) {
  return fabs(cross3p(p, q, r)) < EPS;
}

// returns true if point t is on the left side of line pq
bool ccwtest(Point p, Point q, Point r) {
  return cross3p(p, q, r) > 0; // can be modified to accept collinear points
}

bool onSegment(Point p, Point q, Point r) {
  return collinear(p, q, r) &&
         equals(sqrt(pow(p.x - r.x, 2) + pow(p.y - r.y, 2)) +
                    sqrt(pow(r.x - q.x, 2) + pow(r.y - q.y, 2)),
                sqrt(pow(p.x - q.x, 2) + pow(p.y - q.y, 2)));
}

bool isConvex(vector<Point> p) {
  int sz = (int)p.size();

  if (sz < 3)
    return false; // boundary case, we treat a point or a line as not convex

  bool isLeft = ccwtest(p[0], p[1], p[2]);

  for (int i = 1; i < (int)p.size(); i++)
    if (ccwtest(p[i], p[(i + 1) % sz], p[(i + 2) % sz]) != isLeft)
      return false;

  return true;
}

double angle(Point a, Point b, Point c) {
  double ux = b.x - a.x, uy = b.y - a.y;
  double vx = c.x - a.x, vy = c.y - a.y;
  return acos((ux * vx + uy * vy) /
              sqrt((ux * ux + uy * uy) * (vx * vx + vy * vy)));
}

// 多角形poly内（線分上も含む）に点pが存在するかどうは判定する
bool inPolygon(Polygon poly, Point p) {
  if ((int)poly.size() == 0)
    return false;
  rep(i, poly.size()) if (onSegment(poly[i], poly[(i + 1) % poly.size()],
                                    p)) return true;
  double sum = 0;
  for (int i = 0; i < (int)poly.size(); i++) {
    if (equals(cross(poly[i] - p, poly[(i + 1) % poly.size()] - p), 0.0))
      continue; // ３点が平行だとangleがnanを返しsumがnanになり死ぬ
    if (cross3p(p, poly[i], poly[(i + 1) % poly.size()]) < 0)
      sum -= angle(p, poly[i], poly[(i + 1) % poly.size()]);
    else
      sum += angle(p, poly[i], poly[(i + 1) % poly.size()]);
  }
  // あまり誤差を厳しくしすぎると良くないので以下のほうが良い
  const double eps = 1e-5;
  return (fabs(sum - 2 * M_PI) < eps || fabs(sum + 2 * M_PI) < eps);
}
// inPolygonP - end

// inPolygonS - begin
bool inPolygon(const Polygon &poly, Segment seg) {
  vector<Point> vp;
  vp.push_back(seg.p1);
  vp.push_back(seg.p2);
  rep(i, (int)poly.size()) {
    Segment edge = Segment(poly[i], poly[(i + 1) % (int)poly.size()]);
    if (equals(cross(seg.p1 - seg.p2, edge.p1 - edge.p2), 0.0)) {
      if (onSegment(seg.p1, seg.p2, edge.p1))
        vp.push_back(edge.p1);
      if (onSegment(seg.p1, seg.p2, edge.p2))
        vp.push_back(edge.p2);
    } else {
      if (intersectSS(seg, edge))
        vp.push_back(crosspoint(seg, edge));
    }
  }
  sort(vp.begin(), vp.end());
  vp.erase(unique(vp.begin(), vp.end()), vp.end());
  rep(i, (int)vp.size() - 1) {
    Point middle_point = (vp[i] + vp[i + 1]) / 2.0;
    if (!inPolygon(poly, middle_point))
      return false;
  }
  return true;
}
// inPolygonS - end

const int MAX_V = 16 * 20 + 2000;
int n, bc[1 << 16];
Point ps[16];
Polygon poly;
bool saw[MAX_V];
vector<Segment> segs;

bool check(int bitmask) {
  int V = segs.size();
  assert(V < MAX_V);
  rep(i, V) saw[i] = false;
  vector<Point> sp;
  rep(i, n) if ((bitmask >> i) & 1) sp.push_back(ps[i]);
  rep(i, V) {
    Point mp = (segs[i].p1 + segs[i].p2) / 2.0;
    // cout << segs[i] << " +> " << mp << endl;
    bool ok = false;
    rep(j, bc[bitmask]) {
      Segment seg = Segment(sp[j], mp);
      if (!inPolygon(poly, seg))
        continue;
      bool failed = false;
      rep(k, V) {
        if (equals(cross(seg.p1 - seg.p2, segs[k].p1 - segs[k].p2), 0.0))
          continue;
        if (intersectSS(seg, segs[k])) {
          if (onSegment(segs[k].p1, segs[k].p2, seg.p1) ||
              onSegment(segs[k].p1, segs[k].p2, seg.p2))
            continue;
          if (onSegment(seg.p1, seg.p2, segs[k].p1) ||
              onSegment(seg.p1, seg.p2, segs[k].p2))
            continue;
          failed = true;
          break;
        }
      }
      if (!failed) {
        ok = true;
        break;
      }
    }
    if (!ok)
      return false;
  }

  return true;
  rep(i, V) if (!saw[i]) return false;
  return true;
}

Point vir;
bool comp(const Point &p1, const Point &p2) {
  return !equals(abs(vir - p1), abs(vir - p2)) && abs(vir - p1) < abs(vir - p2);
}

vector<Segment> makeGraph() {
  vector<Segment> vs;
  rep(i, n) vs.push_back(Segment(ps[i], ps[(i + 1) % n]));
  vector<Segment> ret;
  rep(i, vs.size()) {
    vector<Point> vp;
    vp.push_back(vs[i].p1);
    vp.push_back(vs[i].p2);
    rep(j, n) REP(k, j + 1, n) {
      Segment seg = Segment(ps[j], ps[k]);
      if (equals(cross(vs[i].p1 - vs[i].p2, seg.p1 - seg.p2), 0.0)) {
        if (onSegment(vs[i].p1, vs[i].p2, seg.p1))
          vp.push_back(seg.p1);
        if (onSegment(vs[i].p1, vs[i].p2, seg.p2))
          vp.push_back(seg.p2);
      } else if (intersectLS(seg, vs[i]))
        vp.push_back(crosspoint(vs[i], seg));
    }
    vir = vs[i].p1;
    sort(vp.begin(), vp.end(), comp);
    vp.erase(unique(vp.begin(), vp.end()), vp.end());
    rep(j, (int)vp.size() - 1) ret.push_back(Segment(vp[j], vp[j + 1]));
  }
  return ret;
}

int main() {
  rep(i, 1 << 16) bc[i] = __builtin_popcount(i);
  while (cin >> n, n) {
    rep(i, n) cin >> ps[i].x >> ps[i].y;
    poly.resize(n);
    rep(i, n) poly[i] = ps[i];
    segs = makeGraph();
    // rep(i,segs.size()) cout << segs[i] << endl;
    // cout << check (552) << endl;
    // return 0;
    int ans = n;
    rep(i, (1 << n)) {
      if (bc[i] >= ans)
        continue;
      if (check(i))
        ans = bc[i];
    }
    cout << ans << endl;
  }
  return 0;
}