AGC002F Leftmost Ball
問題概要(原文)
$1,2,3, \ldots ,N$ という $N$ 色のボールが $K$ 個ずつある．これを一列に並べる． その後，各色のボールのうち，一番左側にあるボールを色 $0$ に塗る． 色の並びは何通りあるか $\mathrm{mod} 10^{9} + 7$ で求めよ．
考察
$K = 1$ ならば $1$ 通り．$K \geq 1$ とする．
最終的な並びにおいて，色 $1,2,3,\ldots, N$ の順番で出現したことにする． このとき，色の並びは何通りあるか？という問題は組合せをうまくやると解ける．
しかし，今回は色 $0$ があってややこしい． 色 $0$ と，各色の順番の関係について考えよう． まず色 $1,2,3, \ldots, N$ の順番で出現する． 色 $i$ が出現するのは $i$ 番目の $0$ が出現した以降である．
つまり，こんなグラフになる．
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このグラフでの並びの通り数を求めればよく，状態としては
- $0$ 番目が何回出現したか
- 何番目の色まで並べ終わったか
の $2$ つで $O(N^{2})$
遷移は以下の $2$ 通り
- 色 $0$ のボールを今置ける左端に置く
- 色 $i$ のボールを今置ける左端に $1$ つ， $K-2$ 個を残った場所に適当に置く
後者は置ける位置を $X$ 個として $\binom{X}{K-2}$ となる． これは前もって階乗を求めておけば $O(1)$ でできる．
全体として $O((N+K)N)$
ソースコード
using System;
using System.Linq;
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;
namespace Program
{
public class Solver
{
public void Solve()
{
var n = sc.Integer();
var m = sc.Integer();
if (m == 1)
{
IO.Printer.Out.WriteLine(1);
return;
}
var table = new ModTable(2001 * 2001);
var dp = Enumerate(n + 2, x => new ModInteger[n + 2]);
dp[0][0] = 1;
for (int i = 0; i <= n; i++)
for (int j = 0; j <= n; j++)
{
var rem = n - i + (m - 1) * (n - j);
dp[i + 1][j] += dp[i][j];
if (j + 1 <= i) dp[i][j + 1] += dp[i][j] * table.Combination(rem - 1, (m - 2));
}
IO.Printer.Out.WriteLine(dp[n][n] * table.perm[n]);
}
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 ModNumber
public partial struct ModInteger
{
public const long Mod = (long)1e9 + 7;
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 ModInteger operator ^(ModInteger l, long r) { return ModInteger.Pow(l, r); }
public static implicit operator ModInteger(long n) { return new ModInteger() { num = 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
コメント
- グラフを書いてみたけど，大変…
- 解法が分かるとコードは異常に短い
