Problem:
Score : 900 points

Problem Statement

In the city of Nevermore, there are 10^8 streets and 10^8 avenues, both numbered from 0 to 10^8-1.
All streets run straight from west to east, and all avenues run straight from south to north.
The distance between neighboring streets and between neighboring avenues is exactly 100 meters.

Every street intersects every avenue. Every intersection can be described by pair (x, y), where x is avenue ID and y is street ID.

There are N fountains in the city, situated at intersections (X_i, Y_i).
Unlike normal intersections, there's a circle with radius 10 meters centered at the intersection, and there are no road parts inside this circle.

The picture below shows an example of how a part of the city with roads and fountains may look like.

City governors don't like encountering more than one fountain while moving along the same road.
Therefore, every street contains at most one fountain on it, as well as every avenue.

Citizens can move along streets, avenues and fountain perimeters.
What is the shortest distance one needs to cover in order to get from intersection (x_1, y_1) to intersection (x_2, y_2)?

Constraints

0 \leq x_1, y_1, x_2, y_2 < 10^8

1 \leq N \leq 200,000

0 \leq X_i, Y_i < 10^8

X_i \neq X_j for i \neq j

Y_i \neq Y_j for i \neq j

Intersections (x_1, y_1) and (x_2, y_2) are different and don't contain fountains.

All input values are integers.

Input

Input is given from Standard Input in the following format:

x_1 y_1 x_2 y_2
N
X_1 Y_1
X_2 Y_2
:
X_N Y_N

Output

Print the shortest possible distance one needs to cover in order to get from intersection (x_1, y_1) to intersection (x_2, y_2), in meters.
Your answer will be considered correct if its absolute or relative error doesn't exceed 10^{-11}.

Sample Input 1

1 1 6 5
3
3 2
5 3
2 4

Sample Output 1

891.415926535897938

One possible shortest path is shown on the picture below. The path starts at the blue point, finishes at the purple point and follows along the red line.

Sample Input 2

3 5 6 4
3
3 2
5 3
2 4

Sample Output 2

400.000000000000000

Sample Input 3

4 2 2 2
3
3 2
5 3
2 4

Sample Output 3

211.415926535897938

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

using System;
using System.Collections.Generic;
using System.Globalization;
using System.IO;
using System.Linq;
using System.Text;
using System.Threading;

// (づ°ω°)づﾐe★゜・。。・゜゜・。。・゜☆゜・。。・゜゜・。。・゜
public class Solver
{
    void Swap<T>(ref T a, ref T b)
    {
        T t = a;
        a = b;
        b = t;
    }

    class Bit
    {
        private readonly int size;
        private readonly int[] data;

        public Bit(int n)
        {
            size = n;
            data = new int[n];
        }

        public int PrefixSum(int r)
        {
            int ret = 0;
            for (; r >= 0; r = (r & r + 1) - 1)
                ret = Math.Max(ret, data[r]);
            return ret;
        }

        public void Update(int idx, int diff)
        {
            for (; idx < size; idx = (idx | idx + 1))
                data[idx] = Math.Max(data[idx], diff);
        }
    }

    public void Solve()
    {
        int r1 = ReadInt();
        int c1 = ReadInt();
        int r2 = ReadInt();
        int c2 = ReadInt();

        if (r1 > r2)
        {
            Swap(ref r1, ref r2);
            Swap(ref c1, ref c2);
        }

        var a = new List<int[]>();
        for (int n = ReadInt(); n > 0; n--)
        {
            int r = ReadInt();
            int c = ReadInt();
            if (r < r1 || r > r2 || c < Math.Min(c1, c2) || c > Math.Max(c1, c2))
                continue;
            a.Add(new [] { r - r1, Math.Abs(c - c1) });
        }

        if (r1 == r2)
        {
            Write(100.0 * Math.Abs(c1 - c2) + (a.Count > 0 ? 10 * (Math.PI - 2) : 0));
            return;
        }
        if (c1 == c2)
        {
            Write(100.0 * (r2 - r1) + (a.Count > 0 ? 10 * (Math.PI - 2) : 0));
            return;
        }

        double ans = 100.0 * (Math.Abs(r1 - r2) + Math.Abs(c1 - c2));
        if (a.Count == 0)
        {
            Write(ans);
            return;
        }

        a.Sort((a1, a2) => a1[0].CompareTo(a2[0]));
        if (a.Count == r2 - r1 + 1)
        {
            bool f = true;
            for (int i = 1; i < a.Count; i++)
                if (a[i][1] < a[i - 1][1])
                    f = false;
            if (f)
            {
                Write(ans - (20 - 5 * Math.PI) * (r2 - r1) + 10 * (Math.PI - 2));
                return;
            }
        }
        if (a.Count == Math.Abs(c2 - c1) + 1)
        {
            bool f = true;
            for (int i = 1; i < a.Count; i++)
                if (a[i][1] < a[i - 1][1])
                    f = false;
            if (f)
            {
                Write(ans - (20 - 5 * Math.PI) * Math.Abs(c2 - c1) + 10 * (Math.PI - 2));
                return;
            }
        }

        int m = a.Count;
        var list = new List<int>();
        for (int i = 0; i < m; i++)
            list.Add(a[i][1]);
        list.Sort();
        
        var bit = new Bit(m);
        for (int i = 0; i < m; i++)
        {
            int y = list.BinarySearch(a[i][1]);
            int v = bit.PrefixSum(y);
            bit.Update(y, v + 1);
        }

        ans -= (20 - 5 * Math.PI) * bit.PrefixSum(m - 1);
        Write(ans);
    }

    #region Main

    protected static TextReader reader;
    protected static TextWriter writer;
    static void Main()
    {
#if DEBUG
        reader = new StreamReader("..\\..\\input.txt");
        //reader = new StreamReader(Console.OpenStandardInput());
        writer = Console.Out;
        //writer = new StreamWriter("..\\..\\output.txt");
#else
        reader = new StreamReader(Console.OpenStandardInput());
        writer = new StreamWriter(Console.OpenStandardOutput());
        //reader = new StreamReader("input.txt");
        //writer = new StreamWriter("output.txt");
#endif
        try
        {
            new Solver().Solve();
            //var thread = new Thread(new Solver().Solve, 1024 * 1024 * 128);
            //thread.Start();
            //thread.Join();
        }
        catch (Exception ex)
        {
#if DEBUG
            Console.WriteLine(ex);
#else
            throw;
#endif
        }
        reader.Close();
        writer.Close();
    }

    #endregion

    #region Read / Write
    private static Queue<string> currentLineTokens = new Queue<string>();
    private static string[] ReadAndSplitLine() { return reader.ReadLine().Split(new[] { ' ', '\t', }, StringSplitOptions.RemoveEmptyEntries); }
    public static string ReadToken() { while (currentLineTokens.Count == 0)currentLineTokens = new Queue<string>(ReadAndSplitLine()); return currentLineTokens.Dequeue(); }
    public static int ReadInt() { return int.Parse(ReadToken()); }
    public static long ReadLong() { return long.Parse(ReadToken()); }
    public static double ReadDouble() { return double.Parse(ReadToken(), CultureInfo.InvariantCulture); }
    public static int[] ReadIntArray() { return ReadAndSplitLine().Select(int.Parse).ToArray(); }
    public static long[] ReadLongArray() { return ReadAndSplitLine().Select(long.Parse).ToArray(); }
    public static double[] ReadDoubleArray() { return ReadAndSplitLine().Select(s => double.Parse(s, CultureInfo.InvariantCulture)).ToArray(); }
    public static int[][] ReadIntMatrix(int numberOfRows) { int[][] matrix = new int[numberOfRows][]; for (int i = 0; i < numberOfRows; i++)matrix[i] = ReadIntArray(); return matrix; }
    public static int[][] ReadAndTransposeIntMatrix(int numberOfRows)
    {
        int[][] matrix = ReadIntMatrix(numberOfRows); int[][] ret = new int[matrix[0].Length][];
        for (int i = 0; i < ret.Length; i++) { ret[i] = new int[numberOfRows]; for (int j = 0; j < numberOfRows; j++)ret[i][j] = matrix[j][i]; } return ret;
    }
    public static string[] ReadLines(int quantity) { string[] lines = new string[quantity]; for (int i = 0; i < quantity; i++)lines[i] = reader.ReadLine().Trim(); return lines; }
    public static void WriteArray<T>(IEnumerable<T> array) { writer.WriteLine(string.Join(" ", array)); }
    public static void Write(params object[] array) { WriteArray(array); }
    public static void WriteLines<T>(IEnumerable<T> array) { foreach (var a in array)writer.WriteLine(a); }
    private class SDictionary<TKey, TValue> : Dictionary<TKey, TValue>
    {
        public new TValue this[TKey key]
        {
            get { return ContainsKey(key) ? base[key] : default(TValue); }
            set { base[key] = value; }
        }
    }
    private static T[] Init<T>(int size) where T : new() { var ret = new T[size]; for (int i = 0; i < size; i++)ret[i] = new T(); return ret; }
    #endregion
}