AGC001E BBQ Hard
考察の本質さえ分かってしまえばあっさり．
問題概要(原文)
$N(1 \leq N \leq 2 \times 10^5)$ 個の皿があり， $A_i(1 \leq A_i \leq 2000)$ 個の肉と $B_i (1 \leq B_i \leq 2000)$ 個の野菜がある．皿を $2$ 枚選んで， 串に好きなように具材を並べる．串は区別できるが肉どうし，野菜どうしは区別できない．
何通りのやり方があるか？ MOD $10^9+7$ で求めよ．
考察
求めれば良い(ものに近い)もの $X$ は以下の式で表せる．
\[ X = \sum_{i = 1}^{N} \sum_{ j = 1 }^{N} \binom{A[i]+B[i]+A[j]+B[j]}{A[i]+A[j]} \]
上の式をそのまま求めると $O(N^2)$ かかる． ここで，上記の二項係数の部分は二次元グリッドを移動する経路の数で表せることを思い出す．
さらに， $(0,0)$ から $(A[i]+A[j],B[i]+B[j])$ への経路ではなく， $(-A[i],-B[i])$ から $(A[j],B[j])$ への経路と言い換えても一般性を失わない．
以下のような二次元グリッドでの $S-T$ パスの総数を求めよう
- スーパーソース $S$ から $(-A[i],-B[i])$ へと辺が伸びている
- $(A[i],B[i])$ からシンク $T$ へと辺が伸びている
- それ以外は普通の二次元グリッド
さて，さっきの式では本当に求めたいものとは少し違うのであった．自分自身を選ぶような経路の数を差し引いてやる．
最後に$(i,j)$ が入換え可能なパターンを考慮して $2$ で割った数が答え．
計算量は $O(N+(A_i+B_i)^2)$
ソースコード
using System;
using System.Linq;
using System.Linq.Expressions;
using System.Collections.Generic;
using Debug = System.Diagnostics.Debug;
using StringBuilder = System.Text.StringBuilder;
using System.Numerics;
using Number = System.Int64;
namespace Program
{
public class Solver
{
public void Solve()
{
var n = sc.Integer();
var A = new int[n];
var B = new int[n];
const int ADD = 2100;
var dp = Enumerate(4200, x => new ModInteger[4200]);
for (int i = 0; i < n; i++)
{
A[i] = sc.Integer();
B[i] = sc.Integer();
dp[ADD - A[i]][ADD - B[i]] += 1;
}
for (int i = 1; i < 4200; i++)
for (int j = 1; j < 4200; j++)
dp[i][j] += dp[i - 1][j] + dp[i][j - 1];
ModInteger sum = 0;
for (int i = 0; i < n; i++)
sum += dp[ADD + A[i]][ADD + B[i]];
var fact = new ModInteger[8200];
var inv = new ModInteger[8200];
fact[0] = inv[0] = 1;
for (int i = 1; i < 8200; i++)
{
fact[i] = fact[i - 1] * i;
inv[i] = ModInteger.Inverse(fact[i]);
}
for (int i = 0; i < n; i++)
sum -= fact[2 * A[i] + 2 * B[i]] * inv[2 * A[i]] * inv[2 * B[i]];
sum *= ModInteger.Inverse(2);
IO.Printer.Out.WriteLine(sum);
}
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(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
コメント
- 考察さえできれば実装はあっさり
- 対称な点からの経路数と言い換えるのがポイント
