Problem p03866 (C++)

Buggy submission (status: Time Limit Exceeded):
#include <algorithm>
#include <cassert>
#include <cfloat>
#include <cmath>
#include <iomanip>
#include <iostream>
#include <queue>
#include <vector>
using namespace std;

#define EPS (1e-10)
#define equals(a, b) (fabs((a) - (b)) < EPS)

struct Point;
typedef Point Vector;
typedef vector<Point> Polygon;
struct Circle;
struct Segment;
typedef Segment Line;
double norm(Point a);
double abs(Point a);
double dot(Vector a, Vector b);
double cross(Vector a, Vector b);
double getDistance(Point a, Point b);
double getDistanceLP(Line l, Point p);
double getDistanceSP(Segment s, Point p);
double getDistance(Segment s1, Segment s2);
bool isOrthogonal(Vector a, Vector b);
bool isOrthogonal(Point a1, Point a2, Point b1, Point b2);
bool isOrthogonal(Segment s1, Segment s2);
bool isParallel(Vector a, Vector b);
bool isParallel(Point a1, Point a2, Point b1, Point b2);
bool isParallel(Segment s1, Segment s2);
int ccw(Point p0, Point p1, Point p2);
bool intersect(Point p1, Point p2, Point p3, Point p4);
bool intersect(Segment s1, Segment s2);
bool intersect(Circle c, Line l);     // 誤差の検証をしていない
bool intersect(Circle c1, Circle c2); // 誤差の検証をしていない

Point project(Segment s, Point p);
Point reflect(Segment s, Point p);
Point getCrossPoint(Segment s1, Segment s2);
pair<Point, Point> getCrossPoints(Circle c, Line l);
pair<Point, Point> getCrossPoints(Circle c1,
                                  Circle c2); // 誤差の検証をしていない
pair<Point, Point> getContactPoints(Circle c, Point p); // 接点 点は円の外部

double area(Polygon g); // convexでなくてもよい. absを消せば符号付き面積
bool isConvex(Polygon g); // O(n^2) 線形時間アルゴリズムが存在するらしい
int contains(Polygon g, Point p);

double arg(Vector p);             // 偏角
Vector polar(double a, double r); // 極座標系->ベクトル

Polygon andrewScan(
    Polygon
        g); // 凸包の辺上の点も含めたければ!=CLOCKWISEを==COUNTER_CLOCKWISEに
double convexDiameter(Polygon g); // gはconvex

struct Point {
  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); }

  double abs() { return sqrt(norm()); }
  double norm() { return x * x + y * y; }

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

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

typedef Point Vector;
typedef vector<Point> Polygon;

struct Circle {
  Point c;
  double r;
  Circle(Point c = Point(), double r = 0.0) : c(c), r(r) {}
};

struct Segment {
  Point p1, p2;
  Segment(Point p1, Point p2) : p1(p1), p2(p2) {}
};

typedef Segment Line;

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

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

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

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

double getDistance(Point a, Point b) { return abs(a - b); }

double getDistanceLP(Line l, Point p) {
  return abs(cross(l.p2 - l.p1, p - l.p1) / abs(l.p2 - l.p1));
}

double getDistanceSP(Segment s, Point p) {
  if (dot(s.p2 - s.p1, p - s.p1) < 0.0)
    return abs(p - s.p1);
  if (dot(s.p1 - s.p2, p - s.p2) < 0.0)
    return abs(p - s.p2);
  return getDistanceLP(s, p);
}

double getDistance(Segment s1, Segment s2) {
  if (intersect(s1, s2))
    return 0.0;
  return min({getDistanceSP(s1, s2.p1), getDistanceSP(s1, s2.p2),
              getDistanceSP(s2, s1.p1), getDistanceSP(s2, s1.p2)});
}

bool isOrthogonal(Vector a, Vector b) { return equals(dot(a, b), 0.0); }
bool isOrthogonal(Point a1, Point a2, Point b1, Point b2) {
  return isOrthogonal(a1 - a2, b1 - b2);
}
bool isOrthogonal(Segment s1, Segment s2) {
  return equals(dot(s1.p2 - s1.p1, s2.p2 - s2.p1), 0.0);
}
bool isParallel(Vector a, Vector b) { return equals(cross(a, b), 0.0); }
bool isParallel(Point a1, Point a2, Point b1, Point b2) {
  return isParallel(a1 - a2, b1 - b2);
}
bool isParallel(Segment s1, Segment s2) {
  return equals(cross(s1.p2 - s1.p1, s2.p2 - s2.p1), 0.0);
}

static const int COUNTER_CLOCKWISE = 1;
static const int CLOCKWISE = -1;
static const int ONLINE_BACK = 2;   // p2->p0->p1
static const int ONLINE_FRONT = -2; // p0->p1->p2
static const int ON_SEGMENT = 0;    // p0->p2->p1
int ccw(Point p0, Point p1, Point p2) {
  Vector a = p1 - p0;
  Vector 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;
}

bool intersect(Point p1, Point p2, Point p3, Point p4) {
  return (ccw(p1, p2, p3) * ccw(p1, p2, p4) <= 0 &&
          ccw(p3, p4, p1) * ccw(p3, p4, p2) <= 0);
}
bool intersect(Segment s1, Segment s2) {
  return intersect(s1.p1, s1.p2, s2.p1, s2.p2);
}
bool intersect(Circle c, Line l) { return getDistanceLP(l, c.c) < c.r + EPS; }
bool intersect(Circle c1, Circle c2) {
  return abs(c1.r - c2.r) <= getDistance(c1.c, c2.c) &&
         getDistance(c1.c, c2.c) < c1.r + c2.r + EPS;
}

Point project(Segment s, Point p) {
  Vector base = s.p2 - s.p1;
  double r = dot(p - s.p1, base) / norm(base);
  return s.p1 + base * r;
}

Point reflect(Segment s, Point p) { return p + (project(s, p) - p) * 2.0; }

Point getCrossPoint(Segment s1, Segment s2) {
  Vector base = s2.p2 - s2.p1;
  double d1 = abs(cross(base, s1.p1 - s2.p1));
  double d2 = abs(cross(base, s1.p2 - s2.p1));
  double t = d1 / (d1 + d2);
  return s1.p1 + (s1.p2 - s1.p1) * t;
}

pair<Point, Point> getCrossPoints(Circle c, Line l) {
  assert(intersect(c, l));
  Vector pr = project(l, c.c);
  Vector e = (l.p2 - l.p1) / abs(l.p2 - l.p1);
  double base = sqrt(c.r * c.r - norm(pr - c.c));
  return make_pair(pr + e * base, pr - e * base);
}

pair<Point, Point> getCrossPoints(Circle c1, Circle c2) {
  assert(intersect(c1, c2));
  double d = abs(c1.c - c2.c);
  double a = acos((c1.r * c1.r + d * d - c2.r * c2.r) / (2 * c1.r * d));
  double t = arg(c2.c - c1.c);
  return make_pair(c1.c + polar(c1.r, t + a), c1.c + polar(c1.r, t - a));
}

pair<Point, Point> getContactPoints(Circle c, Point p) {
  assert(c.r < getDistance(c.c, p));
  double d = getDistance(c.c, p);
  return getCrossPoints(c, Circle(p, sqrt(d * d - c.r * c.r)));
}

double area(Polygon g) {
  if (g.size() < 3)
    return 0;
  int n = g.size();
  Point o(0.0, 0.0);
  double s = 0.0;
  for (int i = 0; i < n; i++)
    s += cross(g[i] - o, g[(i + 1) % n] - o);
  return abs(s) / 2.0;
}

bool isConvex(Polygon g) {
  bool ret = true;
  int n = g.size();
  for (int i = 0; i < n; i++) {
    for (int j = i + 1; j < n; j++) {
      if (cross(g[i] - g[(i + n - 1) % n], g[j] - g[(i + n - 1) % n]) < -EPS ||
          cross(g[(i + 1) % n] - g[i], g[j] - g[i]) < -EPS) {
        ret = false;
      }
    }
  }
  return ret;
}

static const int IN = 2;
static const int ON = 1;
static const int OUT = 0;
int contains(Polygon g, Point p) {
  int n = g.size();
  bool x = false;
  for (int i = 0; i < n; i++) {
    Point a = g[i] - p, b = g[(i + 1) % n] - p;
    if (abs(cross(a, b)) < EPS && dot(a, b) < EPS)
      return ON;
    if (a.y > b.y)
      swap(a, b);
    if (a.y < EPS && EPS < b.y && cross(a, b) > EPS)
      x = !x;
  }
  return x ? IN : OUT;
}

double arg(Vector p) { return atan2(p.y, p.x); }

Vector polar(double a, double r) { return Point(a * cos(r), a * sin(r)); }

Polygon andrewScan(Polygon g) {
  Polygon u, l;
  if (g.size() < 3)
    return g;
  sort(g.begin(), g.end());
  u.push_back(g[0]);
  u.push_back(g[1]);
  l.push_back(g[g.size() - 1]);
  l.push_back(g[g.size() - 2]);

  // upper
  for (int i = 2; i < g.size(); i++) {
    for (int n = u.size(); n >= 2 && ccw(u[n - 2], u[n - 1], g[i]) != CLOCKWISE;
         n--) {
      u.pop_back();
    }
    u.push_back(g[i]);
  }

  // lower
  for (int i = g.size() - 3; i >= 0; i--) {
    for (int n = l.size(); n >= 2 && ccw(l[n - 2], l[n - 1], g[i]) != CLOCKWISE;
         n--) {
      l.pop_back();
    }
    l.push_back(g[i]);
  }

  reverse(l.begin(), l.end());
  for (int i = u.size() - 2; i >= 1; i--)
    l.push_back(u[i]);

  return l;
}

double convexDiameter(Polygon g) {
  double d = 0.0;
  int n = g.size();
  int is = 0, js = 0;
  for (int i = 1; i < n; i++) {
    if (g[i].y > g[is].y)
      is = i;
    if (g[i].y < g[js].y)
      js = i;
  }
  d = getDistance(g[is], g[js]);

  int i = is, j = js, maxi = is, maxj = js;
  do {
    if (cross(g[(i + 1) % n] - g[i], g[(j + 1) % n] - g[j]) >= 0.0)
      j = (j + 1) % n;
    else
      i = (i + 1) % n;
    if (getDistance(g[i], g[j]) > d) {
      d = getDistance(g[i], g[j]);
      maxi = i, maxj = j;
    }
  } while (i != is || j != js);

  return d; // farthest pair is (maxi, maxj).
}

int main() {
  double sx, sy, tx, ty;
  cin >> sx >> sy >> tx >> ty;
  Point s(sx, sy), t(tx, ty);
  int n;
  cin >> n;
  vector<Circle> v;
  for (int i = 0; i < n; i++) {
    double x, y, r;
    cin >> x >> y >> r;
    v.push_back(Circle(Point(x, y), r));
  }
  vector<double> dp(n, DBL_MAX);
  double ans = getDistance(s, t);
  priority_queue<pair<double, int>> pq;
  for (int i = 0; i < n; i++) {
    if (getDistance(s, v[i].c) <= v[i].r)
      dp[i] = 0;
    else
      dp[i] = getDistance(s, v[i].c) - v[i].r;
    pq.push({-dp[i], i});
  }
  while (!pq.empty()) {
    auto p = pq.top();
    pq.pop();
    int i = p.second;
    if (dp[i] != -p.first)
      continue;
    for (int j = 0; j < n; j++) {
      if (i == j)
        continue;
      double cost = 0;
      if (!intersect(v[i], v[j])) {
        cost = getDistance(v[i].c, v[j].c) - v[i].r - v[j].r;
      }
      if (dp[i] + cost < dp[j]) {
        dp[j] = dp[i] + cost;
        pq.push({-dp[j], j});
      }
    }
  }
  for (int i = 0; i < n; i++) {
    ans = min(ans, dp[i] + max(0.0, getDistance(v[i].c, t) - v[i].r));
  }
  cout << fixed << setprecision(12) << ans << endl;
  return 0;
}


Fixed submission:
#include <algorithm>
#include <cassert>
#include <cfloat>
#include <cmath>
#include <iomanip>
#include <iostream>
#include <queue>
#include <vector>
using namespace std;

#define EPS (1e-10)
#define equals(a, b) (fabs((a) - (b)) < EPS)

struct Point;
typedef Point Vector;
typedef vector<Point> Polygon;
struct Circle;
struct Segment;
typedef Segment Line;
double norm(Point a);
double abs(Point a);
double dot(Vector a, Vector b);
double cross(Vector a, Vector b);
double getDistance(Point a, Point b);
double getDistanceLP(Line l, Point p);
double getDistanceSP(Segment s, Point p);
double getDistance(Segment s1, Segment s2);
bool isOrthogonal(Vector a, Vector b);
bool isOrthogonal(Point a1, Point a2, Point b1, Point b2);
bool isOrthogonal(Segment s1, Segment s2);
bool isParallel(Vector a, Vector b);
bool isParallel(Point a1, Point a2, Point b1, Point b2);
bool isParallel(Segment s1, Segment s2);
int ccw(Point p0, Point p1, Point p2);
bool intersect(Point p1, Point p2, Point p3, Point p4);
bool intersect(Segment s1, Segment s2);
bool intersect(Circle c, Line l);     // 誤差の検証をしていない
bool intersect(Circle c1, Circle c2); // 誤差の検証をしていない

Point project(Segment s, Point p);
Point reflect(Segment s, Point p);
Point getCrossPoint(Segment s1, Segment s2);
pair<Point, Point> getCrossPoints(Circle c, Line l);
pair<Point, Point> getCrossPoints(Circle c1,
                                  Circle c2); // 誤差の検証をしていない
pair<Point, Point> getContactPoints(Circle c, Point p); // 接点 点は円の外部

double area(Polygon g); // convexでなくてもよい. absを消せば符号付き面積
bool isConvex(Polygon g); // O(n^2) 線形時間アルゴリズムが存在するらしい
int contains(Polygon g, Point p);

double arg(Vector p);             // 偏角
Vector polar(double a, double r); // 極座標系->ベクトル

Polygon andrewScan(
    Polygon
        g); // 凸包の辺上の点も含めたければ!=CLOCKWISEを==COUNTER_CLOCKWISEに
double convexDiameter(Polygon g); // gはconvex

struct Point {
  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); }

  double abs() { return sqrt(norm()); }
  double norm() { return x * x + y * y; }

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

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

typedef Point Vector;
typedef vector<Point> Polygon;

struct Circle {
  Point c;
  double r;
  Circle(Point c = Point(), double r = 0.0) : c(c), r(r) {}
};

struct Segment {
  Point p1, p2;
  Segment(Point p1, Point p2) : p1(p1), p2(p2) {}
};

typedef Segment Line;

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

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

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

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

double getDistance(Point a, Point b) { return abs(a - b); }

double getDistanceLP(Line l, Point p) {
  return abs(cross(l.p2 - l.p1, p - l.p1) / abs(l.p2 - l.p1));
}

double getDistanceSP(Segment s, Point p) {
  if (dot(s.p2 - s.p1, p - s.p1) < 0.0)
    return abs(p - s.p1);
  if (dot(s.p1 - s.p2, p - s.p2) < 0.0)
    return abs(p - s.p2);
  return getDistanceLP(s, p);
}

double getDistance(Segment s1, Segment s2) {
  if (intersect(s1, s2))
    return 0.0;
  return min({getDistanceSP(s1, s2.p1), getDistanceSP(s1, s2.p2),
              getDistanceSP(s2, s1.p1), getDistanceSP(s2, s1.p2)});
}

bool isOrthogonal(Vector a, Vector b) { return equals(dot(a, b), 0.0); }
bool isOrthogonal(Point a1, Point a2, Point b1, Point b2) {
  return isOrthogonal(a1 - a2, b1 - b2);
}
bool isOrthogonal(Segment s1, Segment s2) {
  return equals(dot(s1.p2 - s1.p1, s2.p2 - s2.p1), 0.0);
}
bool isParallel(Vector a, Vector b) { return equals(cross(a, b), 0.0); }
bool isParallel(Point a1, Point a2, Point b1, Point b2) {
  return isParallel(a1 - a2, b1 - b2);
}
bool isParallel(Segment s1, Segment s2) {
  return equals(cross(s1.p2 - s1.p1, s2.p2 - s2.p1), 0.0);
}

static const int COUNTER_CLOCKWISE = 1;
static const int CLOCKWISE = -1;
static const int ONLINE_BACK = 2;   // p2->p0->p1
static const int ONLINE_FRONT = -2; // p0->p1->p2
static const int ON_SEGMENT = 0;    // p0->p2->p1
int ccw(Point p0, Point p1, Point p2) {
  Vector a = p1 - p0;
  Vector 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;
}

bool intersect(Point p1, Point p2, Point p3, Point p4) {
  return (ccw(p1, p2, p3) * ccw(p1, p2, p4) <= 0 &&
          ccw(p3, p4, p1) * ccw(p3, p4, p2) <= 0);
}
bool intersect(Segment s1, Segment s2) {
  return intersect(s1.p1, s1.p2, s2.p1, s2.p2);
}
bool intersect(Circle c, Line l) { return getDistanceLP(l, c.c) < c.r + EPS; }
bool intersect(Circle c1, Circle c2) {
  return abs(c1.r - c2.r) <= getDistance(c1.c, c2.c) &&
         getDistance(c1.c, c2.c) < c1.r + c2.r + EPS;
}

Point project(Segment s, Point p) {
  Vector base = s.p2 - s.p1;
  double r = dot(p - s.p1, base) / norm(base);
  return s.p1 + base * r;
}

Point reflect(Segment s, Point p) { return p + (project(s, p) - p) * 2.0; }

Point getCrossPoint(Segment s1, Segment s2) {
  Vector base = s2.p2 - s2.p1;
  double d1 = abs(cross(base, s1.p1 - s2.p1));
  double d2 = abs(cross(base, s1.p2 - s2.p1));
  double t = d1 / (d1 + d2);
  return s1.p1 + (s1.p2 - s1.p1) * t;
}

pair<Point, Point> getCrossPoints(Circle c, Line l) {
  assert(intersect(c, l));
  Vector pr = project(l, c.c);
  Vector e = (l.p2 - l.p1) / abs(l.p2 - l.p1);
  double base = sqrt(c.r * c.r - norm(pr - c.c));
  return make_pair(pr + e * base, pr - e * base);
}

pair<Point, Point> getCrossPoints(Circle c1, Circle c2) {
  assert(intersect(c1, c2));
  double d = abs(c1.c - c2.c);
  double a = acos((c1.r * c1.r + d * d - c2.r * c2.r) / (2 * c1.r * d));
  double t = arg(c2.c - c1.c);
  return make_pair(c1.c + polar(c1.r, t + a), c1.c + polar(c1.r, t - a));
}

pair<Point, Point> getContactPoints(Circle c, Point p) {
  assert(c.r < getDistance(c.c, p));
  double d = getDistance(c.c, p);
  return getCrossPoints(c, Circle(p, sqrt(d * d - c.r * c.r)));
}

double area(Polygon g) {
  if (g.size() < 3)
    return 0;
  int n = g.size();
  Point o(0.0, 0.0);
  double s = 0.0;
  for (int i = 0; i < n; i++)
    s += cross(g[i] - o, g[(i + 1) % n] - o);
  return abs(s) / 2.0;
}

bool isConvex(Polygon g) {
  bool ret = true;
  int n = g.size();
  for (int i = 0; i < n; i++) {
    for (int j = i + 1; j < n; j++) {
      if (cross(g[i] - g[(i + n - 1) % n], g[j] - g[(i + n - 1) % n]) < -EPS ||
          cross(g[(i + 1) % n] - g[i], g[j] - g[i]) < -EPS) {
        ret = false;
      }
    }
  }
  return ret;
}

static const int IN = 2;
static const int ON = 1;
static const int OUT = 0;
int contains(Polygon g, Point p) {
  int n = g.size();
  bool x = false;
  for (int i = 0; i < n; i++) {
    Point a = g[i] - p, b = g[(i + 1) % n] - p;
    if (abs(cross(a, b)) < EPS && dot(a, b) < EPS)
      return ON;
    if (a.y > b.y)
      swap(a, b);
    if (a.y < EPS && EPS < b.y && cross(a, b) > EPS)
      x = !x;
  }
  return x ? IN : OUT;
}

double arg(Vector p) { return atan2(p.y, p.x); }

Vector polar(double a, double r) { return Point(a * cos(r), a * sin(r)); }

Polygon andrewScan(Polygon g) {
  Polygon u, l;
  if (g.size() < 3)
    return g;
  sort(g.begin(), g.end());
  u.push_back(g[0]);
  u.push_back(g[1]);
  l.push_back(g[g.size() - 1]);
  l.push_back(g[g.size() - 2]);

  // upper
  for (int i = 2; i < g.size(); i++) {
    for (int n = u.size(); n >= 2 && ccw(u[n - 2], u[n - 1], g[i]) != CLOCKWISE;
         n--) {
      u.pop_back();
    }
    u.push_back(g[i]);
  }

  // lower
  for (int i = g.size() - 3; i >= 0; i--) {
    for (int n = l.size(); n >= 2 && ccw(l[n - 2], l[n - 1], g[i]) != CLOCKWISE;
         n--) {
      l.pop_back();
    }
    l.push_back(g[i]);
  }

  reverse(l.begin(), l.end());
  for (int i = u.size() - 2; i >= 1; i--)
    l.push_back(u[i]);

  return l;
}

double convexDiameter(Polygon g) {
  double d = 0.0;
  int n = g.size();
  int is = 0, js = 0;
  for (int i = 1; i < n; i++) {
    if (g[i].y > g[is].y)
      is = i;
    if (g[i].y < g[js].y)
      js = i;
  }
  d = getDistance(g[is], g[js]);

  int i = is, j = js, maxi = is, maxj = js;
  do {
    if (cross(g[(i + 1) % n] - g[i], g[(j + 1) % n] - g[j]) >= 0.0)
      j = (j + 1) % n;
    else
      i = (i + 1) % n;
    if (getDistance(g[i], g[j]) > d) {
      d = getDistance(g[i], g[j]);
      maxi = i, maxj = j;
    }
  } while (i != is || j != js);

  return d; // farthest pair is (maxi, maxj).
}

int main() {
  double sx, sy, tx, ty;
  cin >> sx >> sy >> tx >> ty;
  Point s(sx, sy), t(tx, ty);
  int n;
  cin >> n;
  vector<Circle> v;
  for (int i = 0; i < n; i++) {
    double x, y, r;
    cin >> x >> y >> r;
    v.push_back(Circle(Point(x, y), r));
  }
  vector<double> dp(n, DBL_MAX);
  double ans = getDistance(s, t);
  priority_queue<pair<double, int>> pq;
  for (int i = 0; i < n; i++) {
    if (getDistance(s, v[i].c) <= v[i].r)
      dp[i] = 0;
    else
      dp[i] = getDistance(s, v[i].c) - v[i].r;
    pq.push({-dp[i], i});
  }
  while (!pq.empty()) {
    auto p = pq.top();
    pq.pop();
    int i = p.second;
    if (dp[i] != -p.first)
      continue;
    for (int j = 0; j < n; j++) {
      if (i == j)
        continue;
      double cost = 0;
      if (getDistance(v[i].c, v[j].c) > abs(v[i].r - v[j].r) &&
          !intersect(v[i], v[j])) {
        cost = getDistance(v[i].c, v[j].c) - v[i].r - v[j].r;
      }
      if (dp[i] + cost < dp[j]) {
        dp[j] = dp[i] + cost;
        pq.push({-dp[j], j});
      }
    }
  }
  for (int i = 0; i < n; i++) {
    ans = min(ans, dp[i] + max(0.0, getDistance(v[i].c, t) - v[i].r));
  }
  cout << fixed << setprecision(12) << ans << endl;
  return 0;
}