Problem p00771 (C++) — buggy submission, verdict: Time Limit Exceeded

#include <bits/stdc++.h>
using namespace std;
#define FOR(i, k, n) for (int i = (int)(k); i < (int)(n); i++)
#define REP(i, n) FOR(i, 0, n)
#define ALL(a) a.begin(), a.end()
#define MS(m, v) memset(m, v, sizeof(m))
#define D10 fixed << setprecision(10)
typedef long long ll;
typedef long double ld;
typedef vector<int> vi;
typedef vector<string> vs;
typedef pair<int, int> pii;
const int MOD = 1000000007;
const int INF = MOD + 1;
const ld EPS = 1e-12;
template <class T> T &chmin(T &a, const T &b) { return a = min(a, b); }
template <class T> T &chmax(T &a, const T &b) { return a = max(a, b); }

/*--------------------template--------------------*/

typedef long double ld;
const ld PI = acos(-1.0);
bool eq(ld a, ld b) { return abs(a - b) < EPS; }
typedef complex<ld> Point;
typedef vector<Point> Polygon;

namespace std {
bool operator<(const Point &a, const Point &b) {
  return real(a) != real(b) ? real(a) < real(b) : imag(a) < imag(b);
}
} // namespace std

struct Line {
  Point a, b;
  Line(Point p, Point q) : a(p), b(q){};
  Line(ld x1, ld y1, ld x2, ld y2) : a(Point(x1, y1)), b(Point(x2, y2)){};
};

struct Circle {
  Point p;
  ld r;
  Circle(Point a, ld b) : p(a), r(b){};
};

ld dot(Point a, Point b) { return real(conj(a) * b); }

ld cross(Point a, Point b) { return imag(conj(a) * b); }

int ccw(Point a, Point b, Point c) {
  b -= a;
  c -= a;
  if (cross(b, c) > EPS)
    return 1; // counter cloclwise
  if (cross(b, c) < -EPS)
    return -1; // cloclwise
  if (dot(b, c) < 0)
    return 2; // c--a--b on line
  if (norm(b) < norm(c))
    return -2; // a--b--c on line
  return 0;    // a--c--b on line
}

bool isis_ll(Line l, Line m) { return abs(cross(l.b - l.a, m.b - m.a)) > EPS; }

bool isis_ls(Line l, Line s) {
  return cross(l.b - l.a, s.a - l.a) * cross(l.b - l.a, s.b - l.a) < EPS;
}

bool isis_ss(Line s, Line t) {
  return (ccw(s.a, s.b, t.a) * ccw(s.a, s.b, t.b) <= 0 &&
          ccw(t.a, t.b, s.a) * ccw(t.a, t.b, s.b) <= 0);
}

bool isis_lp(Line l, Point p) { return (abs(cross(l.b - p, l.a - p)) < EPS); }

bool isis_sp(Line s, Point p) {
  return (abs(s.a - p) + abs(s.b - p) - abs(s.b - s.a)) < EPS;
}

Point projection(Line l, Point p) {
  Point base = l.b - l.a;
  ld r = dot(p - l.a, base) / norm(base);
  return l.a + base * r;
}

Point mirror(Line l, Point p) { return Point(2, 0) * projection(l, p) - p; }

ld dist_lp(Line l, Point p) { return abs(p - projection(l, p)); }

ld dist_ll(Line l, Line m) { return isis_ll(l, m) ? 0 : dist_lp(l, m.a); }

ld dist_ls(Line l, Line s) {
  if (isis_ls(l, s))
    return 0;
  return min(dist_lp(l, s.a), dist_lp(l, s.b));
}

ld dist_sp(Line s, Point p) {
  Point r = projection(s, p);
  if (isis_sp(s, r))
    return abs(r - p);
  return min(abs(s.a - p), abs(s.b - p));
}

ld dist_ss(Line s, Line t) {
  if (isis_ss(s, t))
    return 0;
  return min(min(dist_sp(s, t.a), dist_sp(s, t.b)),
             min(dist_sp(t, s.a), dist_sp(t, s.b)));
}

Point is_ll(Line s, Line t) {
  ld a = cross(s.b - s.a, t.b - t.a);
  ld b = cross(s.b - s.a, s.b - t.a);
  return t.a + b / a * (t.b - t.a);
}

vector<Point> is_cc(Circle c1, Circle c2) {
  vector<Point> res;
  ld d = abs(c1.p - c2.p);
  ld rc = (d * d + pow(c1.r, 2) - pow(c2.r, 2)) / (2 * d);
  ld dfr = pow(c1.r, 2) - rc * rc;
  if (abs(dfr) < EPS)
    dfr = 0;
  if (dfr < 0.0)
    return res;
  ld rs = sqrt(dfr);
  Point diff = (c2.p - c1.p) / d;
  res.push_back(c1.p + diff * Point(rc, rs));
  if (dfr != 0.0)
    res.push_back(c1.p + diff * Point(rc, -rs));
  return res;
}

int isis_cc(Circle c1, Circle c2) {
  ld d = abs(c1.p - c2.p);
  if (d - c1.r - c2.r < -EPS)
    return -2; // separate
  if (abs(d - c1.r - c2.r) < EPS)
    return -1; // circumscribe
  if (c1.r < c2.r)
    swap(c1, c2);
  if (abs(d - c1.r + c2.r) < EPS)
    return 1; // inscribe
  if (c1.r - d - c2.r > EPS)
    return 2; // involve
  else
    return 0; // intersect
}

bool isis_cl(Circle c, Line l) {
  ld d = dist_lp(l, c.p);
  return d - c.r < -EPS;
}

vector<Point> is_cl(Circle c, Line l) {
  vector<Point> res;
  ld d = dist_lp(l, c.p);
  if (d > c.r + EPS)
    return res;
  ld len = (d > c.r) ? 0 : sqrt(c.r * c.r - d * d);
  Point nor = (l.a - l.b) / abs(l.a - l.b);
  res.push_back(projection(l, c.p) + len * nor);
  if (abs(len) > EPS)
    res.push_back(projection(l, c.p - len * nor));
  return res;
}

vector<Point> is_cs(Circle c, Line s) {
  vector<Point> v = is_cl(c, s), res;
  REP(i, v.size()) {
    if (ccw(s.a, v[i], s.b) == -2)
      res.push_back(v[i]);
  }
  return res;
}

vector<Line> tangent_cp(Circle c, Point p) {
  vector<Line> res;
  Point v = c.p - p;
  ld d = abs(v);
  ld l = sqrt(norm(v) - c.r * c.r);
  if (isnan(l)) {
    return res;
  }
  Point v1 = v * Point(l / d, c.r / d);
  Point v2 = v * Point(l / d, -c.r / d);
  res.push_back(Line(p, p + v1));
  if (l < EPS)
    return res;
  res.push_back(Line(p, p + v2));
  return res;
}

vector<Line> tangent_cc(Circle c1, Circle c2) {
  vector<Line> res;
  if (abs(c1.p - c2.p) - (c1.r + c2.r) > -EPS) {
    Point center = (c1.p * c2.r + c2.p * c1.r) / (c1.r + c2.r);
    res = tangent_cp(c1, center);
  }
  if (abs(c1.r - c2.r) > EPS) {
    Point out = (-c1.p * c2.r + c2.p * c1.r) / (c1.r - c2.r);
    vector<Line> nres = tangent_cp(c1, out);
    res.insert(res.end(), nres.begin(), nres.end());
  } else {
    Point v = c2.p - c1.p;
    v /= abs(v);
    Point q1 = c1.p + v * Point(0, 1) * c1.r;
    Point q2 = c1.p + v * Point(0, -1) * c1.r;
    res.push_back(Line(q1, q1 + v));
    res.push_back(Line(q2, q2 + v));
  }
  return res;
}

bool is_in(Circle c, Point p) {
  ld dist = abs(c.p - p);
  return c.r - dist > -EPS;
}

typedef pair<Point, ld> Data;

int main() {
  cin.sync_with_stdio(false);
  int n;
  while (cin >> n, n) {
    vector<Data> v;
    REP(i, n) {
      ld x, y, l;
      cin >> x >> y >> l;
      v.emplace_back(Point(x, y), l);
    }
    ld lb = 0, ub = INF;
    REP(loop, 200) {
      ld height = (lb + ub) / 2.0;
      vector<Circle> vc;
      REP(i, n) {
        ld r = sqrt((v[i].second * v[i].second) - height * height);
        vc.emplace_back(v[i].first, r);
      }
      vector<Point> ps;
      REP(i, n) ps.emplace_back(vc[i].p);
      REP(i, n) REP(j, i) {
        if (isis_cc(vc[i], vc[j])) {
          vector<Point> tmp = is_cc(vc[i], vc[j]);
          for (auto k : tmp)
            ps.push_back(k);
        }
      }
      bool f = false;
      REP(i, ps.size()) {
        bool ff = true;
        REP(j, vc.size()) {
          if (!is_in(vc[j], ps[i]))
            ff = false;
        }
        if (ff)
          f = true;
      }
      if (f)
        lb = height;
      else
        ub = height;
    }
    cout << D10 << lb << endl;
  }
  return 0;
}

Error: TLE

Fixed submission:

#include <bits/stdc++.h>
using namespace std;
#define FOR(i, k, n) for (int i = (int)(k); i < (int)(n); i++)
#define REP(i, n) FOR(i, 0, n)
#define ALL(a) a.begin(), a.end()
#define MS(m, v) memset(m, v, sizeof(m))
#define D10 fixed << setprecision(10)
typedef long long ll;
typedef long double ld;
typedef vector<int> vi;
typedef vector<string> vs;
typedef pair<int, int> pii;
const int MOD = 1000000007;
const int INF = MOD + 1;
const ld EPS = 1e-12;
template <class T> T &chmin(T &a, const T &b) { return a = min(a, b); }
template <class T> T &chmax(T &a, const T &b) { return a = max(a, b); }

/*--------------------template--------------------*/

typedef long double ld;
const ld PI = acos(-1.0);
bool eq(ld a, ld b) { return abs(a - b) < EPS; }
typedef complex<ld> Point;
typedef vector<Point> Polygon;

namespace std {
bool operator<(const Point &a, const Point &b) {
  return real(a) != real(b) ? real(a) < real(b) : imag(a) < imag(b);
}
} // namespace std

struct Line {
  Point a, b;
  Line(Point p, Point q) : a(p), b(q){};
  Line(ld x1, ld y1, ld x2, ld y2) : a(Point(x1, y1)), b(Point(x2, y2)){};
};

struct Circle {
  Point p;
  ld r;
  Circle(Point a, ld b) : p(a), r(b){};
};

ld dot(Point a, Point b) { return real(conj(a) * b); }

ld cross(Point a, Point b) { return imag(conj(a) * b); }

int ccw(Point a, Point b, Point c) {
  b -= a;
  c -= a;
  if (cross(b, c) > EPS)
    return 1; // counter cloclwise
  if (cross(b, c) < -EPS)
    return -1; // cloclwise
  if (dot(b, c) < 0)
    return 2; // c--a--b on line
  if (norm(b) < norm(c))
    return -2; // a--b--c on line
  return 0;    // a--c--b on line
}

bool isis_ll(Line l, Line m) { return abs(cross(l.b - l.a, m.b - m.a)) > EPS; }

bool isis_ls(Line l, Line s) {
  return cross(l.b - l.a, s.a - l.a) * cross(l.b - l.a, s.b - l.a) < EPS;
}

bool isis_ss(Line s, Line t) {
  return (ccw(s.a, s.b, t.a) * ccw(s.a, s.b, t.b) <= 0 &&
          ccw(t.a, t.b, s.a) * ccw(t.a, t.b, s.b) <= 0);
}

bool isis_lp(Line l, Point p) { return (abs(cross(l.b - p, l.a - p)) < EPS); }

bool isis_sp(Line s, Point p) {
  return (abs(s.a - p) + abs(s.b - p) - abs(s.b - s.a)) < EPS;
}

Point projection(Line l, Point p) {
  Point base = l.b - l.a;
  ld r = dot(p - l.a, base) / norm(base);
  return l.a + base * r;
}

Point mirror(Line l, Point p) { return Point(2, 0) * projection(l, p) - p; }

ld dist_lp(Line l, Point p) { return abs(p - projection(l, p)); }

ld dist_ll(Line l, Line m) { return isis_ll(l, m) ? 0 : dist_lp(l, m.a); }

ld dist_ls(Line l, Line s) {
  if (isis_ls(l, s))
    return 0;
  return min(dist_lp(l, s.a), dist_lp(l, s.b));
}

ld dist_sp(Line s, Point p) {
  Point r = projection(s, p);
  if (isis_sp(s, r))
    return abs(r - p);
  return min(abs(s.a - p), abs(s.b - p));
}

ld dist_ss(Line s, Line t) {
  if (isis_ss(s, t))
    return 0;
  return min(min(dist_sp(s, t.a), dist_sp(s, t.b)),
             min(dist_sp(t, s.a), dist_sp(t, s.b)));
}

Point is_ll(Line s, Line t) {
  ld a = cross(s.b - s.a, t.b - t.a);
  ld b = cross(s.b - s.a, s.b - t.a);
  return t.a + b / a * (t.b - t.a);
}

vector<Point> is_cc(Circle c1, Circle c2) {
  vector<Point> res;
  ld d = abs(c1.p - c2.p);
  ld rc = (d * d + pow(c1.r, 2) - pow(c2.r, 2)) / (2 * d);
  ld dfr = pow(c1.r, 2) - rc * rc;
  if (abs(dfr) < EPS)
    dfr = 0;
  if (dfr < 0.0)
    return res;
  ld rs = sqrt(dfr);
  Point diff = (c2.p - c1.p) / d;
  res.push_back(c1.p + diff * Point(rc, rs));
  if (dfr != 0.0)
    res.push_back(c1.p + diff * Point(rc, -rs));
  return res;
}

int isis_cc(Circle c1, Circle c2) {
  ld d = abs(c1.p - c2.p);
  if (d - c1.r - c2.r < -EPS)
    return -2; // separate
  if (abs(d - c1.r - c2.r) < EPS)
    return -1; // circumscribe
  if (c1.r < c2.r)
    swap(c1, c2);
  if (abs(d - c1.r + c2.r) < EPS)
    return 1; // inscribe
  if (c1.r - d - c2.r > EPS)
    return 2; // involve
  else
    return 0; // intersect
}

bool isis_cl(Circle c, Line l) {
  ld d = dist_lp(l, c.p);
  return d - c.r < -EPS;
}

vector<Point> is_cl(Circle c, Line l) {
  vector<Point> res;
  ld d = dist_lp(l, c.p);
  if (d > c.r + EPS)
    return res;
  ld len = (d > c.r) ? 0 : sqrt(c.r * c.r - d * d);
  Point nor = (l.a - l.b) / abs(l.a - l.b);
  res.push_back(projection(l, c.p) + len * nor);
  if (abs(len) > EPS)
    res.push_back(projection(l, c.p - len * nor));
  return res;
}

vector<Point> is_cs(Circle c, Line s) {
  vector<Point> v = is_cl(c, s), res;
  REP(i, v.size()) {
    if (ccw(s.a, v[i], s.b) == -2)
      res.push_back(v[i]);
  }
  return res;
}

vector<Line> tangent_cp(Circle c, Point p) {
  vector<Line> res;
  Point v = c.p - p;
  ld d = abs(v);
  ld l = sqrt(norm(v) - c.r * c.r);
  if (isnan(l)) {
    return res;
  }
  Point v1 = v * Point(l / d, c.r / d);
  Point v2 = v * Point(l / d, -c.r / d);
  res.push_back(Line(p, p + v1));
  if (l < EPS)
    return res;
  res.push_back(Line(p, p + v2));
  return res;
}

vector<Line> tangent_cc(Circle c1, Circle c2) {
  vector<Line> res;
  if (abs(c1.p - c2.p) - (c1.r + c2.r) > -EPS) {
    Point center = (c1.p * c2.r + c2.p * c1.r) / (c1.r + c2.r);
    res = tangent_cp(c1, center);
  }
  if (abs(c1.r - c2.r) > EPS) {
    Point out = (-c1.p * c2.r + c2.p * c1.r) / (c1.r - c2.r);
    vector<Line> nres = tangent_cp(c1, out);
    res.insert(res.end(), nres.begin(), nres.end());
  } else {
    Point v = c2.p - c1.p;
    v /= abs(v);
    Point q1 = c1.p + v * Point(0, 1) * c1.r;
    Point q2 = c1.p + v * Point(0, -1) * c1.r;
    res.push_back(Line(q1, q1 + v));
    res.push_back(Line(q2, q2 + v));
  }
  return res;
}

bool is_in(Circle c, Point p) {
  ld dist = abs(c.p - p);
  return c.r - dist > -EPS;
}

typedef pair<Point, ld> Data;

int main() {
  cin.sync_with_stdio(false);
  int n;
  while (cin >> n, n) {
    vector<Data> v;
    REP(i, n) {
      ld x, y, l;
      cin >> x >> y >> l;
      v.emplace_back(Point(x, y), l);
    }
    ld lb = 0, ub = INF;
    REP(loop, 1000) {
      ld height = (lb + ub) / 2.0;
      vector<Circle> vc;
      REP(i, n) {
        ld r = sqrt((v[i].second * v[i].second) - height * height);
        vc.emplace_back(v[i].first, r);
      }
      vector<Point> ps;
      REP(i, n) ps.emplace_back(vc[i].p);
      REP(i, n) REP(j, i) {
        if (isis_cc(vc[i], vc[j])) {
          vector<Point> tmp = is_cc(vc[i], vc[j]);
          for (auto k : tmp)
            ps.push_back(k);
        }
      }
      bool f = false;
      REP(i, ps.size()) {
        bool ff = true;
        REP(j, vc.size()) {
          if (!is_in(vc[j], ps[i]))
            ff = false;
        }
        if (ff)
          f = true;
      }
      if (f)
        lb = height;
      else
        ub = height;
    }
    cout << D10 << lb << endl;
  }
  return 0;
}