Problem:
MinimumCostPath

NxNの方眼がある。

(N=3の場合の図)

隣接するマス目に移動するにはコストが1かかる。ただし、斜めに移動することは出来ない。
また障害物があるマス目がM個あり、そのマスに入ることは許されない。
マス(1, 1)からマス(N, N)に最小のコストで移動する方法は何通りあるか。
答えは大きくなることがありうるので、1000000009(=109+9)でMODを取って出力せよ。

Input

入力は以下の形式で与えられる

N M
X1 Y1
X2 Y2
……
XM YM

Xi, Yiは(Xi, Yi)に障害物があることを表す

Constraints

2 ≤ N ≤ 106

0 ≤ M ≤ 50

1 ≤ Xi , Yi ≤ N

i ≠ jならば (Xi, Yi) ≠ (Xj, Yj)

(Xi, Yi) ≠ (1, 1)

(Xi, Yi) ≠ (N, N)

Output

1行に経路の総数を出力せよ。ただし、マス(1, 1)からマス(N, N)に移動する経路が存在しない場合は0と出力せよ。

Sample Input 1

3 0

Output for the Sample Input 1

6

以下の図に対応する。

Sample Input 2

3 1
2 2

Output for the Sample Input 2

2

以下の図に対応する。

Sample Input 3

5 8
4 3
2 1
4 2
4 4
3 4
2 2
2 4
1 4

Output for the Sample Input 3

1

以下の図に対応する。

Sample Input 4

1000 10
104 87
637 366
393 681
215 604
707 876
943 414
95 327
93 415
663 596
661 842

Output for the Sample Input 4

340340391

Sample Input 5

2 2
1 2
2 1

Output for the Sample Input 5

0

Solution in C# (judge verdict: Accepted)

using System;
using System.Linq;
using System.Collections.Generic;
using Debug = System.Diagnostics.Debug;
using StringBuilder = System.Text.StringBuilder;
using Number = System.Int64;
namespace Program
{
    public class Solver
    {
        public void Solve()
        {
            var n = sc.Integer();
            var m = sc.Integer();
            var a = Enumerate(m, x => sc.Integer(2));
            for (int i = 0; i < m; i++) { a[i][0]--; a[i][1]--; }

            const int MAX = 200;
            var dist = Enumerate(2, x => new Pair<int, ModInteger>[MAX, MAX]);
            for (int i = 0; i < 2; i++)
                for (int j = 0; j < MAX; j++)
                    for (int k = 0; k < MAX; k++)
                        dist[i][j, k] = new Pair<int, ModInteger>(1000000000, 0);
            for (int k = 0; k < 2; k++)
            {
                int[] dx = { 1, -1, 0, 0 };
                int[] dy = { 0, 0, 1, -1 };
                var d = dist[k];
                var s = new HashSet<long>();
                foreach (var p in a)
                    s.Add(p[0] * 2000000L + p[1]);
                var q = new Queue<int>();
                q.Enqueue(0);
                d[0, 0] = new Pair<int, ModInteger>(0, 1);
                while (q.Any())
                {
                    var p = q.Dequeue();
                    var x = p / MAX;
                    var y = p % MAX;
                    for (int t = 0; t < 4; t++)
                    {
                        var nx = x + dx[t];
                        var ny = y + dy[t];
                        if (s.Contains(nx * 2000000L + ny)) continue;
                        if (nx >= n || ny >= n) continue;
                        if (nx < 0 || nx >= MAX || ny < 0 || ny >= MAX) continue;
                        if (d[nx, ny].x > d[x, y].x + 1)
                        {
                            d[nx, ny].x = d[x, y].x + 1;
                            d[nx, ny].y = d[x, y].y;
                            q.Enqueue(nx * MAX + ny);
                        }
                        else if (d[nx, ny].x == d[x, y].x + 1)
                            d[nx, ny].y += d[x, y].y;

                    }
                }
                foreach (var x in a)
                {
                    x[0] = n - 1 - x[0];
                    x[1] = n - 1 - x[1];
                }
            }
            if (n <= MAX)
            {
                IO.Printer.Out.WriteLine(dist[0][n - 1, n - 1].y);
                return;
            }
            var table = new ModTable(3 * n);
            //Debug.WriteLine(table.Combination(2 * n - 2, n - 1));
            ModInteger ret = 0;
            var min = 1000000000;
            const int NEX = 100;
            for (int i = 0; i <= NEX; i++)
                for (int j = 0; j <= NEX; j++)
                {
                    var d = dist[0][i, NEX - i].x + dist[1][j, NEX - j].x;
                    var dx = (n - 1 - j) - i;
                    var dy = (n - 1 - (NEX - j)) - (NEX - i);
                    min = Math.Min(min, d + dx + dy);
                }
            Debug.WriteLine(min);
            for (int ii = 0; ii <= NEX; ii++)
                for (int jj = 0; jj <= NEX; jj++)
                {
                    {
                        var d = dist[0][ii, NEX - ii].x + dist[1][jj, NEX - jj].x;
                        var dx = (n - 1 - jj) - ii;
                        var dy = (n - 1 - (NEX - jj)) - (NEX - ii);
                        if (d + dx + dy != min) continue;
                    }
                    var p = new List<int[]>();
                    p.Add(new int[] { n - 1 - jj - ii, n - 1 - (NEX - jj) - (NEX - ii) });
                    foreach (var x in a)
                    {
                        var dx = ii;
                        var dy = NEX - ii;

                        if (x[0] + x[1] <= NEX) continue;
                        if (x[0] < dx || x[1] < dy) continue;
                        if ((n - 1 - x[0]) + (n - 1 - x[1]) <= NEX) continue;
                        p.Add(new int[] { x[0] - dx, x[1] - dy });
                    }
                    p.Sort((l, r) => (l[0] + l[1]).CompareTo(r[0] + r[1]));
                    var dp = new ModInteger[p.Count];
                    var k = p.Count;
                    for (int i = 0; i < k; i++)
                    {
                        if (p[i][0] < 0 || p[i][1] < 0) continue;
                        if (p[i][0] > p[k - 1][0] || p[i][1] > p[k - 1][1]) continue;
                        dp[i] = table.Combination(p[i][0] + p[i][1], p[i][0]);
                        for (int j = 0; j < i; j++)
                        {
                            var dx = p[i][0] - p[j][0];
                            var dy = p[i][1] - p[j][1];
                            if (dx < 0 || dy < 0) continue;
                            dp[i] -= table.Combination(dx + dy, dx) * dp[j];
                        }
                    }
                    //Debug.WriteLine(p[k - 1].AsJoinedString());
                    ret += dist[0][ii, NEX - ii].y * dist[1][jj, NEX - jj].y * dp.Last();

                }
            IO.Printer.Out.WriteLine(ret);

        }

        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 Compair
public struct Pair<FT, ST>
{
    public FT x;
    public ST y;
    public Pair(FT f, ST s) : this() { x = f; y = s; }

    public override string ToString() { return string.Format("{0} {1}", x, y); }
}
#endregion
#region ModNumber
public partial struct ModInteger
{
    public const long Mod = (long)1e9 + 9;
    public long num;
    public ModInteger(long n) : this() { num = n % Mod; if (num < 0) num += Mod; }
    public override string ToString() { return num.ToString(); }
    public static ModInteger operator +(ModInteger l, ModInteger r) { var n = l.num + r.num; if (n >= Mod) n -= Mod; return new ModInteger() { num = n }; }
    public static ModInteger operator -(ModInteger l, ModInteger r) { var n = l.num + Mod - r.num; if (n >= Mod) n -= Mod; return new ModInteger() { num = n }; }
    public static ModInteger operator *(ModInteger l, ModInteger r) { return new ModInteger(l.num * r.num); }
    public static implicit operator ModInteger(long n) { return new ModInteger(n); }
    public static ModInteger Pow(ModInteger v, long k)
    {
        ModInteger ret = 1;
        var n = k;
        for (; n > 0; n >>= 1, v *= v)
        {
            if ((n & 1) == 1)
                ret = ret * v;
        }
        return ret;
    }
}
#endregion
#region Inverse
public partial struct ModInteger
{
    static public ModInteger Inverse(ModInteger v)
    {
        long p, q;
        ExGCD(v.num, Mod, out p, out q);
        return new ModInteger(p % Mod + Mod);
    }
    static public long ExGCD(long a, long b, out long x, out long y)
    {
        var u = new long[] { a, 1, 0 };
        var v = new long[] { b, 0, 1 };
        while (v[0] != 0)
        {
            var t = u[0] / v[0];
            for (int i = 0; i < 3; i++)
            {
                var tmp = u[i] - t * v[i];
                u[i] = v[i];
                v[i] = tmp;
            }
        }
        x = u[1];
        y = u[2];
        if (u[0] > 0)
            return u[0];
        for (int i = 0; i < 3; i++)
            u[i] = -u[i];
        return u[0];
    }
}
#endregion
#region ModTable
public class ModTable
{
    public ModInteger[] perm, invp;
    public ModTable(int n)
    {
        perm = new ModInteger[n + 1];
        invp = new ModInteger[n + 1];
        perm[0] = 1;
        for (int i = 1; i <= n; i++)
            perm[i] = perm[i - 1] * i;
        invp[n] = ModInteger.Inverse(perm[n]);
        for (int i = n - 1; i >= 0; i--)
            invp[i] = invp[i + 1] * (i + 1);
        invp[0] = invp[1];
    }
    public ModInteger Inverse(int k) { return invp[k]; }
    public ModInteger Permutation(int n, int k)
    {
        if (n < 0 || n >= perm.Length)
            return 0;
        if (k < 0 || k >= n)
            return 0;
        return perm[n] * invp[n - k];
    }
    public ModInteger Combination(int n, int r)
    {
        if (n < 0 || n >= perm.Length || r < 0 || r > n) return 0;
        return perm[n] * invp[n - r] * invp[r];
    }
    public ModInteger RepeatedCombination(int n, int k)
    {
        if (k == 0) return 1;
        return Combination(n + k - 1, k);
    }
}
#endregion