JAGSummer2014Day2B Dictionary
結構好き
問題概要(原文)
$N$ 個の英小文字のみからなる文字列がある．辞書順で $s_1 < s_2 < s_3 < \ldots < s_n$ であることが分かっている． 文字列のうち，いくつかの箇所は $\text{?}$ になってしまった．
もとの文字列たちとしてありうるものは何通りか？ $\mathrm{mod} \; 10^9 + 7$ で求めよ．
考察
辞書順の条件から $1$ 文字ずつ見ていく． すると，$i$ 文字目に着目しているとき，$N$ 個のうち，ある連続した区間 $[l,m)$が同じ文字 $x$ で，次の区間 $[m,r)$ は $y$ だったとすると，$x < y$でなければならず，さらに $[l,m)$ は最終的に $m-r =1$ となる必要がある．
状態としては，現在着目している区間の長さ，現在着目している文字のインデックス，区間の先頭の文字種類はどれ以上か．
遷移は次の文字種を試すか，先頭の区間の長さで試す，の $2$ 通り．
計算量は $O(N^3|s_i|X)$ となる ($X$ は文字の種類数)
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;
namespace Program
{
public class Solver
{
public void Solve()
{
const int L = 20;
const char ZERO = (char)('a' - 1);
var n = sc.Integer();
var s = Enumerate(n, x => sc.Scan());
for (int i = 0; i < n; i++)
while (s[i].Length < L) s[i] += ZERO;
//(i,j),k,[a,b]
var dp = new ModInteger[n, n, L, 27];
for (int i = 0; i < n; i++)
for (int j = 0; j < n; j++)
for (int k = 0; k < L; k++)
for (int l = 0; l < 27; l++)
dp[i, j, k, l].num = -1;
Func<int, int, int, int, ModInteger> dfs = null;
dfs = (l, r, i, a) =>
{
if (l > r) return 1;
if (i == L)
return r == l ? 1 : 0;
if (a >= 27) return 0;
if (dp[l, r, i, a].num != -1) return dp[l, r, i, a];
//Debug.WriteLine($"[{l} {r}] {i} {(char)(a + 'a')}");
ModInteger ret = 0;
ret += dfs(l, r, i, a + 1);
for (int m = l; m <= r; m++)
{
if (a == 0 && s[m][i] == '?') break;
if (s[m][i] != '?' && s[m][i] != a + ZERO) break;
ret += dfs(l, m, i + 1, 0) * dfs(m + 1, r, i, a + 1);
}
return dp[l, r, i, a] = ret;
};
var ans = dfs(0, n - 1, 0, 0);
IO.Printer.Out.WriteLine(ans);
}
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
コメント
- ちょっと面倒だけど着目の仕方が面白い
