Problem:
Score : 600 points

Problem Statement

On the xy-plane, Snuke is going to travel from the point (x_s, y_s) to the point (x_t, y_t).
He can move in arbitrary directions with speed 1.
Here, we will consider him as a point without size.

There are N circular barriers deployed on the plane.
The center and the radius of the i-th barrier are (x_i, y_i) and r_i, respectively.
The barriers may overlap or contain each other.

A point on the plane is exposed to cosmic rays if the point is not within any of the barriers.

Snuke wants to avoid exposure to cosmic rays as much as possible during the travel.
Find the minimum possible duration of time he is exposed to cosmic rays during the travel.

Constraints

All input values are integers.

-10^9 ≤ x_s, y_s, x_t, y_t ≤ 10^9

(x_s, y_s) ≠ (x_t, y_t)

1≤N≤1,000

-10^9 ≤ x_i, y_i ≤ 10^9

1 ≤ r_i ≤ 10^9

Input

The input is given from Standard Input in the following format:

x_s y_s x_t y_t
N
x_1 y_1 r_1
x_2 y_2 r_2
:
x_N y_N r_N

Output

Print the minimum possible duration of time Snuke is exposed to cosmic rays during the travel.
The output is considered correct if the absolute or relative error is at most 10^{-9}.

Sample Input 1

-2 -2 2 2
1
0 0 1

Sample Output 1

3.6568542495

An optimal route is as follows:

Sample Input 2

-2 0 2 0
2
-1 0 2
1 0 2

Sample Output 2

0.0000000000

An optimal route is as follows:

Sample Input 3

4 -2 -2 4
3
0 0 2
4 0 1
0 4 1

Sample Output 3

4.0000000000

An optimal route is as follows:

Solution in C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections;
using System.Collections.Generic;
using System.IO;
using System.Linq;

class Program
{
    static void Main(string[] args)
    {
        var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
        Console.SetOut(sw);
        Solve();
        Console.Out.Flush();
    }
    static void Solve()
    {
        var sx = sc.ReadLong();
        var sy = sc.ReadLong();
        var gx = sc.ReadLong();
        var gy = sc.ReadLong();
        var N = sc.ReadInt();
        var X = new long[N];
        var Y = new long[N];
        var R = new long[N];
        for (long i = 0; i < N; i++)
        {
            X[i] = sc.ReadLong();
            Y[i] = sc.ReadLong();
            R[i] = sc.ReadLong();
        }
        var edges = new List<Tuple<double, int, int>>();
        for (int i = 0; i < N; i++)
        {
            for (int j = i + 1; j < N; j++)
            {
                if (i == j) continue;
                var d = DistanceP(X[i], Y[i], X[j], Y[j]);
                d -= R[i] + R[j];
                d = Math.Max(d, 0.0);
                edges.Add(Tuple.Create(d, i, j));
            }
        }
        for (int i = 0; i < N; i++)
        {
            {
                var d = DistanceP(X[i], Y[i], sx, sy);
                d -= R[i];
                d = Math.Max(d, 0.0);
                edges.Add(Tuple.Create(d, i, N));
            }
            {
                var d = DistanceP(X[i], Y[i], gx, gy);
                d -= R[i];
                d = Math.Max(d, 0.0);
                edges.Add(Tuple.Create(d, i, N + 1));
            }
        }
        var sge = Tuple.Create(DistanceP(sx, sy, gx, gy), N, N + 1);
        edges.Add(sge);

        var dij = new Dijkstra(N + 2);
        foreach (var e in edges)
        {
            dij.Add(e.Item2, e.Item3, e.Item1);
            dij.Add(e.Item3, e.Item2, e.Item1);
        }
        var ans = dij.GetMinCost(N, N + 1);
        Console.WriteLine(ans);
    }

    class Dijkstra
    {
        public const double INF = double.MaxValue / 2;
        public int V { get; set; }
        public int E { get; set; }
        public List<Edge>[] G { get; set; }
        public Dijkstra(int n)
        {
            this.V = n;
            this.G = new List<Edge>[V];
            for (int i = 0; i < V; i++)
            {
                this.G[i] = new List<Edge>();
            }
        }
        public void Add(int a, int b, double cost = 1)
        {
            this.G[a].Add(new Edge(b, cost));
            this.E++;
        }
        public double GetMinCost(int start, int goal)
        {
            var cost = new double[V];
            for (int i = 0; i < V; i++) cost[i] = INF;
            cost[start] = 0;

            var capacity = Math.Max(1, E);
            var q = new PriorityQueue<P>(false);
            q.Enqueue(new P(start, 0));

            while (q.Count > 0)
            {
                var p = q.Dequeue();
                if (p.Cost != cost[p.A]) continue;
                foreach (var e in this.G[p.A])
                {
                    if (cost[e.To] > p.Cost + e.Cost)
                    {
                        cost[e.To] = p.Cost + e.Cost;
                        q.Enqueue(new P(e.To, cost[e.To]));
                    }
                }
            }
            return cost[goal] == long.MaxValue ? -1 : cost[goal];
        }
        public double[] Calc(int start)
        {
            var visited = new bool[V];
            var cost = new double[V];
            for (int i = 0; i < V; i++) cost[i] = INF;
            cost[start] = 0;

            var capacity = Math.Max(1, E);
            var q = new PriorityQueue<P>(false);
            q.Enqueue(new P(start, 0));

            while (q.Count > 0)
            {
                var p = q.Dequeue();
                visited[p.A] = true;
                if (p.Cost != cost[p.A]) continue;
                foreach (var e in this.G[p.A])
                {
                    if (cost[e.To] > p.Cost + e.Cost && !visited[e.To])
                    {
                        cost[e.To] = p.Cost + e.Cost;
                        q.Enqueue(new P(e.To, cost[e.To]));
                    }
                }
            }
            return cost;
        }
        public struct Edge
        {
            public int To;
            public double Cost;
            public Edge(int to, double cost)
            {
                this.To = to;
                this.Cost = cost;
            }
        }
        public struct P : IComparable<P>
        {
            public int A;
            public double Cost;
            public P(int a, double cost)
            {
                this.A = a;
                this.Cost = cost;
            }
            public int CompareTo(P other)
            {
                return this.Cost.CompareTo(other.Cost);
            }
        }
        class PriorityQueue<T>
        {
            public int Count;
            private int Capacity;
            private bool IsMaxHeap;
            private T[] Buffer;
            private IComparer<T> Cmp;
            public PriorityQueue(bool isMaxHeap = true, IComparer<T> cmp = null)
            {
                Count = 0;
                Capacity = 1 << 10;
                IsMaxHeap = isMaxHeap;
                Buffer = new T[Capacity];
                if (cmp != null) Cmp = cmp;
                else Cmp = Comparer<T>.Default;
            }
            public PriorityQueue(int capacity, bool isMaxHeap = true, IComparer<T> cmp = null)
            {
                Count = 0;
                Capacity = Math.Max(16, capacity);
                IsMaxHeap = isMaxHeap;
                Buffer = new T[Capacity];
                if (cmp != null) Cmp = cmp;
                else Cmp = Comparer<T>.Default;
            }
            private void Resize()
            {
                Capacity <<= 1;
                Array.Resize(ref Buffer, Capacity);
            }
            public void Enqueue(T value)
            {
                if (Count == Capacity) Resize();
                Buffer[Count] = value;
                Count++;
                UpHeap();
            }
            public T Dequeue()
            {
                var res = Buffer[0];
                Swap(0, Count - 1);
                Count--;
                DownHeap();
                return res;
            }
            public T Peek() { return Buffer[0]; }
            private void Swap(int i, int j)
            {
                var tmp = Buffer[i];
                Buffer[i] = Buffer[j];
                Buffer[j] = tmp;
            }
            private void UpHeap()
            {
                var n = this.Count - 1;
                while (n != 0)
                {
                    var parent = (n - 1) / 2;
                    if (Compare(Buffer[n], Buffer[parent]) > 0)
                    {
                        Swap(n, parent);
                        n = parent;
                    }
                    else break;
                }
            }
            private void DownHeap()
            {
                var parent = 0;
                while (true)
                {
                    var child = 2 * parent + 1;
                    if (child > this.Count - 1) break;
                    if (child < Count - 1 && Compare(Buffer[child], Buffer[child + 1]) < 0)
                    {
                        child += 1;
                    }
                    if (Compare(Buffer[parent], Buffer[child]) < 0)
                    {
                        Swap(parent, child);
                        parent = child;
                    }
                    else break;
                }
            }
            private int Compare(T x, T y)
            {
                var res = Cmp.Compare(x, y);
                if (!IsMaxHeap) res *= -1;
                return res;
            }
        }
    }
    static double DistanceP(long x1, long y1, long x2, long y2)
    {
        return Math.Sqrt((x1 - x2) * (x1 - x2) + (y1 - y2) * (y1 - y2));
    }

    static Scanner sc = new Scanner();
}
class Scanner
{
    string[] S = new string[0];
    int Index = 0;
    char[] Separators = new char[] { ' ' };

    public string Next()
    {
        if (this.Index < this.S.Length) return this.S[this.Index++];
        var line = "";
        while (line == "") line = Console.ReadLine();
        this.S = line.Split(this.Separators, StringSplitOptions.RemoveEmptyEntries);
        if (this.S.Length == 0) return this.Next();
        this.Index = 0;
        return this.S[this.Index++];
    }
    public string ReadStr()
    {
        return this.Next();
    }
    public char ReadChar()
    {
        return this.Next()[0];
    }
    public int ReadInt()
    {
        return int.Parse(this.Next());
    }
    public uint ReadUInt()
    {
        return uint.Parse(this.Next());
    }
    public long ReadLong()
    {
        return long.Parse(this.Next());
    }
    public double ReadDouble()
    {
        return double.Parse(this.Next());
    }
    public Tuple<int, int> ReadTup(int add = 0)
    {
        return Tuple.Create(this.ReadInt() + add, this.ReadInt() + add);
    }
    public Tuple<long, long> ReadTupLong(int add = 0)
    {
        return Tuple.Create(this.ReadLong() + add, this.ReadLong() + add);
    }
    public Tuple<int, int, int> ReadTup3(int add = 0)
    {
        return Tuple.Create(this.ReadInt() + add, this.ReadInt() + add, this.ReadInt() + add);
    }
    public Tuple<int, int, int, int> ReadTup4(int add = 0)
    {
        return Tuple.Create(this.ReadInt() + add, this.ReadInt() + add, this.ReadInt() + add, this.ReadInt() + add);
    }
    public int[] ReadIntArray(int n)
    {
        var array = new int[n];
        for (int i = 0; i < array.Length; i++)
        {
            array[i] = this.ReadInt();
        }
        return array;
    }
    public long[] ReadLongArray(int n)
    {
        var array = new long[n];
        for (int i = 0; i < array.Length; i++)
        {
            array[i] = this.ReadLong();
        }
        return array;
    }
    public double[] ReadDoubleArray(int n)
    {
        var array = new double[n];
        for (int i = 0; i < array.Length; i++)
        {
            array[i] = this.ReadDouble();
        }
        return array;
    }
    public char[] ReadCharArray(int n)
    {
        var array = new char[n];
        for (int i = 0; i < array.Length; i++)
        {
            array[i] = this.ReadChar();
        }
        return array;
    }
    public string[] ReadStrArray(int n)
    {
        var array = new string[n];
        for (int i = 0; i < array.Length; i++)
        {
            array[i] = this.ReadStr();
        }
        return array;
    }
    public Tuple<long, long>[] ReadTupLongArray(int n, int add = 0)
    {
        var array = new Tuple<long, long>[n];
        for (int i = 0; i < n; i++)
        {
            array[i] = this.ReadTupLong(add);
        }
        return array;
    }
    public Tuple<int, int>[] ReadTupArray(int n, int add = 0)
    {
        var array = new Tuple<int, int>[n];
        for (int i = 0; i < n; i++)
        {
            array[i] = this.ReadTup(add);
        }
        return array;
    }
    public Tuple<int, int, int>[] ReadTup3Array(int n, int add = 0)
    {
        var array = new Tuple<int, int, int>[n];
        for (int i = 0; i < n; i++)
        {
            array[i] = this.ReadTup3(add);
        }
        return array;
    }
    public Tuple<int, int, int, int>[] ReadTup4Array(int n, int add = 0)
    {
        var array = new Tuple<int, int, int, int>[n];
        for (int i = 0; i < n; i++)
        {
            array[i] = this.ReadTup4(add);
        }
        return array;
    }
}