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.Linq;
using System.Linq.Expressions;
using System.Collections.Generic;
using Debug = System.Diagnostics.Debug;
using StringBuilder = System.Text.StringBuilder;
using System.Numerics;
using Point = System.Numerics.Complex;
using Number = System.Int64;
using C = System.Int32;
using V = System.Int32;
namespace Program
{
    public class Solver
    {
        public void Solve()
        {
            var pp = new Point(sc.Integer(), sc.Integer());
            var qq = new Point(sc.Integer(), sc.Integer());
            var n = sc.Integer();
            var P = new Point[n + 2];
            P[n] = pp; P[n + 1] = qq;
            var R = new int[n + 2];
            for (int i = 0; i < n; i++)
            {
                P[i] = new Point(sc.Integer(), sc.Integer());
                R[i] = sc.Integer();
            }
            var dist = new double[n + 2];
            for (int i = 0; i < n + 2; i++)
                dist[i] = 1e18;
            dist[n] = 0;
            var pq = new BinaryHeapPriorityQueue<KeyValuePair<int, double>>((l, r) => l.Value.CompareTo(r.Value));
            pq.Enqueue(new KeyValuePair<C, double>(n, 0));
            var used = new bool[n + 2];
            foreach (var x in P)
                Debug.WriteLine(x);
            foreach (var x in R)
                Debug.WriteLine(x);
            while (pq.Count > 0)
            {
                var p = pq.Dequeue();
                if (used[p.Key]) continue;
                Debug.WriteLine(p);
                used[p.Key] = true;
                for (int i = 0; i < n + 2; i++)
                {
                    //if (used[i]) continue;
                    var cost = get(P[i], P[p.Key], R[i] + R[p.Key]);
                    if (dist[i] > p.Value + cost)
                    {
                        dist[i] = p.Value + cost;
                        pq.Enqueue(new KeyValuePair<C, double>(i, dist[i]));
                    }
                }

            }

            IO.Printer.Out.WriteLine(dist[n + 1]);
        }
        double get(Point p, Point q, double dd)
        {
            return Math.Max(0, (p - q).Magnitude - dd);
        }

        public IO.StreamScanner sc = new IO.StreamScanner(Console.OpenStandardInput());
        static T[] Enumerate<T>(int n, Func<int, T> f) { var a = new T[n]; for (int i = 0; i < n; ++i) a[i] = f(i); return a; }
        static public void Swap<T>(ref T a, ref T b) { var tmp = a; a = b; b = tmp; }
    }
}
#region main
static class Ex
{
    static public string AsString(this IEnumerable<char> ie) { return new string(System.Linq.Enumerable.ToArray(ie)); }
    static public string AsJoinedString<T>(this IEnumerable<T> ie, string st = " ") { return string.Join(st, ie); }
    static public void Main()
    {
        var solver = new Program.Solver();
        solver.Solve();
        Program.IO.Printer.Out.Flush();
    }
}
#endregion
#region Ex
namespace Program.IO
{
    using System.IO;
    using System.Text;
    using System.Globalization;
    public class Printer: StreamWriter
    {
        static Printer() { Out = new Printer(Console.OpenStandardOutput()) { AutoFlush = false }; }
        public static Printer Out { get; set; }
        public override IFormatProvider FormatProvider { get { return CultureInfo.InvariantCulture; } }
        public Printer(System.IO.Stream stream) : base(stream, new UTF8Encoding(false, true)) { }
        public Printer(System.IO.Stream stream, Encoding encoding) : base(stream, encoding) { }
        public void Write<T>(string format, T[] source) { base.Write(format, source.OfType<object>().ToArray()); }
        public void WriteLine<T>(string format, T[] source) { base.WriteLine(format, source.OfType<object>().ToArray()); }
    }
    public class StreamScanner
    {
        public StreamScanner(Stream stream) { str = stream; }
        public readonly Stream str;
        private readonly byte[] buf = new byte[1024];
        private int len, ptr;
        public bool isEof = false;
        public bool IsEndOfStream { get { return isEof; } }
        private byte read()
        {
            if (isEof) return 0;
            if (ptr >= len) { ptr = 0; if ((len = str.Read(buf, 0, 1024)) <= 0) { isEof = true; return 0; } }
            return buf[ptr++];
        }
        public char Char() { byte b = 0; do b = read(); while ((b < 33 || 126 < b) && !isEof); return (char)b; }

        public string Scan()
        {
            var sb = new StringBuilder();
            for (var b = Char(); b >= 33 && b <= 126; b = (char)read())
                sb.Append(b);
            return sb.ToString();
        }
        public string ScanLine()
        {
            var sb = new StringBuilder();
            for (var b = Char(); b != '\n'; b = (char)read())
                if (b == 0) break;
                else if (b != '\r') sb.Append(b);
            return sb.ToString();
        }
        public long Long()
        {
            if (isEof) return long.MinValue;
            long ret = 0; byte b = 0; var ng = false;
            do b = read();
            while (b != 0 && b != '-' && (b < '0' || '9' < b));
            if (b == 0) return long.MinValue;
            if (b == '-') { ng = true; b = read(); }
            for (; true; b = read())
            {
                if (b < '0' || '9' < b)
                    return ng ? -ret : ret;
                else ret = ret * 10 + b - '0';
            }
        }
        public int Integer() { return (isEof) ? int.MinValue : (int)Long(); }
        public double Double() { var s = Scan(); return s != "" ? double.Parse(s, CultureInfo.InvariantCulture) : double.NaN; }
        private T[] enumerate<T>(int n, Func<T> f)
        {
            var a = new T[n];
            for (int i = 0; i < n; ++i) a[i] = f();
            return a;
        }

        public char[] Char(int n) { return enumerate(n, Char); }
        public string[] Scan(int n) { return enumerate(n, Scan); }
        public double[] Double(int n) { return enumerate(n, Double); }
        public int[] Integer(int n) { return enumerate(n, Integer); }
        public long[] Long(int n) { return enumerate(n, Long); }
    }
}
#endregion
#region PriorityQueue by BinaryHeap
public class BinaryHeapPriorityQueue<T>
{
    private readonly List<T> heap;
    private readonly Comparison<T> compare;
    private int size;
    public BinaryHeapPriorityQueue() : this(Comparer<T>.Default) { }
    public BinaryHeapPriorityQueue(IComparer<T> comparer) : this(16, comparer.Compare) { }
    public BinaryHeapPriorityQueue(Comparison<T> comparison) : this(16, comparison) { }
    public BinaryHeapPriorityQueue(int capacity, Comparison<T> comparison)
    {
        heap = new List<T>(capacity);
        compare = comparison;
    }
    public void Enqueue(T item)
    {

        heap.Add(item);
        var i = size++;
        while (i > 0)
        {
            var p = (i - 1) >> 1;
            if (compare(heap[p], item) <= 0)
                break;
            heap[i] = heap[p];
            i = p;
        }
        heap[i] = item;

    }
    public T Dequeue()
    {
        var ret = heap[0];
        var x = heap[--size];
        var i = 0;
        while ((i << 1) + 1 < size)
        {
            var a = (i << 1) + 1;
            var b = (i << 1) + 2;
            if (b < size && compare(heap[b], heap[a]) < 0) a = b;
            if (compare(heap[a], x) >= 0)
                break;
            heap[i] = heap[a];
            i = a;
        }
        heap[i] = x;
        heap.RemoveAt(size);
        return ret;

    }
    public T Peek() { return heap[0]; }
    public int Count { get { return size; } }
    public bool Any() { return size > 0; }
}
#endregion