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 Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)


import java.util.Arrays;
import java.util.Scanner;

public class Main {
    MyScanner sc    = new MyScanner();
    Scanner   sc2   = new Scanner(System.in);
    long      start = System.currentTimeMillis();
    long      fin   = System.currentTimeMillis();
    final int MOD   = 1000000007;
    int[]     dx    = { 1, 0, 0, -1 };
    int[]     dy    = { 0, 1, -1, 0 };

    void run() {
        double xs = sc.nextDouble();
        double ys = sc.nextDouble();
        double xt = sc.nextDouble();
        double yt = sc.nextDouble();

        int N = sc.nextInt();
        P[] p = new P[N];
        for (int i = 0; i < N; i++) {
            p[i] = new P(sc.nextDouble(), sc.nextDouble(), sc.nextDouble());
        }
        double[][] g = new double[N + 2][N + 2];

        for (int i = 0; i < N + 2; i++)
            Arrays.fill(g[i], Long.MAX_VALUE);

        // start <-> goalの距離
        g[0][N + 1] = g[N + 1][0] = Math.hypot(xs - xt, ys - yt);
        // start <-> バリアの距離
        for (int i = 0; i < N; i++) {
            double dist = Math.hypot(xs - p[i].x, ys - p[i].y);
            g[0][i + 1] = g[i + 1][0] = Math.max(0, dist - p[i].r);
        }
        // バリア <-> goalの距離
        for (int i = 0; i < N; i++) {
            double dist = Math.hypot(xt - p[i].x, yt - p[i].y);
            g[N + 1][i + 1] = g[i + 1][N + 1] = Math.max(0, dist - p[i].r);
        }
        // バリア間の距離
        for (int i = 0; i < N; i++) {
            for (int j = i + 1; j < N; j++) {
                double dist = Math.hypot(p[j].x - p[i].x, p[j].y - p[i].y);
                if (dist <= p[j].r + p[i].r) {
                    g[i + 1][j + 1] = g[j + 1][i + 1] = 0;
                } else {
                    g[i + 1][j + 1] = g[j + 1][i + 1] = dist - (p[j].r + p[i].r);
                }
            }
        }

        // Dikstra
        double len[] = new double[N + 2]; // minimum graphance
        boolean v[] = new boolean[N + 2]; // decision flag
        for (int i = 0; i < N + 2; i++) {
            len[i] = Long.MAX_VALUE;
        }
        len[0] = 0;
        for (int i = 0; i < N + 2; i++) {
            // search minimum neighbor node
            int pp = 0;
            double min = Long.MAX_VALUE;
            for (int j = 0; j < N + 2; j++) {
                if (!v[j] && len[j] < min) {
                    pp = j;
                    min = len[j];
                }
            }
            v[pp] = true;
            for (int j = 0; j < N + 2; j++) {
                if ((len[pp] + g[pp][j]) < len[j]) {
                    len[j] = len[pp] + g[pp][j];
                }
            }
        }
        System.out.println(len[N + 1]);
    }

    class P {
        double x;
        double y;
        double r;

        public P(double x, double y, double r) {
            super();
            this.x = x;
            this.y = y;
            this.r = r;
        }
    }

    public static void main(String[] args) {
        new Main().run();
    }

    void debug(Object... o) {
        System.out.println(Arrays.deepToString(o));
    }

    void debug2(int[][] array) {
        for (int i = 0; i < array.length; i++) {
            for (int j = 0; j < array[i].length; j++) {
                System.out.print(array[i][j]);
            }
            System.out.println();
        }
    }

    boolean inner(int h, int w, int limH, int limW) {
        return 0 <= h && h < limH && 0 <= w && w < limW;
    }

    void swap(int[] x, int a, int b) {
        int tmp = x[a];
        x[a] = x[b];
        x[b] = tmp;
    }

    // find minimum i (k <= a[i])
    int lower_bound(int a[], int k) {
        int l = -1;
        int r = a.length;
        while (r - l > 1) {
            int mid = (l + r) / 2;
            if (k <= a[mid])
                r = mid;
            else
                l = mid;
        }
        // r = l + 1
        return r;
    }

    // find minimum i (k < a[i])
    int upper_bound(int a[], int k) {
        int l = -1;
        int r = a.length;
        while (r - l > 1) {
            int mid = (l + r) / 2;
            if (k < a[mid])
                r = mid;
            else
                l = mid;
        }
        // r = l + 1
        return r;
    }

    int gcd(int a, int b) {
        return a % b == 0 ? b : gcd(b, a % b);
    }

    int lcm(int a, int b) {
        return a * b / gcd(a, b);
    }

    boolean palindrome(String s) {
        for (int i = 0; i < s.length() / 2; i++) {
            if (s.charAt(i) != s.charAt(s.length() - 1 - i)) {
                return false;
            }
        }
        return true;
    }

    class MyScanner {
        int nextInt() {
            try {
                int c = System.in.read();
                while (c != '-' && (c < '0' || '9' < c))
                    c = System.in.read();
                if (c == '-')
                    return -nextInt();
                int res = 0;
                do {
                    res *= 10;
                    res += c - '0';
                    c = System.in.read();
                } while ('0' <= c && c <= '9');
                return res;
            } catch (Exception e) {
                return -1;
            }
        }

        double nextDouble() {
            return Double.parseDouble(next());
        }

        long nextLong() {
            return Long.parseLong(next());
        }

        String next() {
            try {
                StringBuilder res = new StringBuilder("");
                int c = System.in.read();
                while (Character.isWhitespace(c))
                    c = System.in.read();
                do {
                    res.append((char) c);
                } while (!Character.isWhitespace(c = System.in.read()));
                return res.toString();
            } catch (Exception e) {
                return null;
            }
        }

        int[] nextIntArray(int n) {
            int[] in = new int[n];
            for (int i = 0; i < n; i++) {
                in[i] = nextInt();
            }
            return in;
        }

        int[][] nextInt2dArray(int n, int m) {
            int[][] in = new int[n][m];
            for (int i = 0; i < n; i++) {
                in[i] = nextIntArray(m);
            }
            return in;
        }

        double[] nextDoubleArray(int n) {
            double[] in = new double[n];
            for (int i = 0; i < n; i++) {
                in[i] = nextDouble();
            }
            return in;
        }

        long[] nextLongArray(int n) {
            long[] in = new long[n];
            for (int i = 0; i < n; i++) {
                in[i] = nextLong();
            }
            return in;
        }

        char[][] nextCharField(int n, int m) {
            char[][] in = new char[n][m];
            for (int i = 0; i < n; i++) {
                String s = sc.next();
                for (int j = 0; j < m; j++) {
                    in[i][j] = s.charAt(j);
                }
            }
            return in;
        }
    }
}