結果

問題 No.2503 Typical Path Counting Problem on a Grid
ユーザー 👑 hos.lyrichos.lyric
提出日時 2023-10-13 22:32:08
言語 D
(dmd 2.106.1)
結果
AC  
実行時間 1,306 ms / 2,000 ms
コード長 5,346 bytes
コンパイル時間 3,935 ms
コンパイル使用メモリ 153,216 KB
実行使用メモリ 45,824 KB
最終ジャッジ日時 2024-09-15 18:17:13
合計ジャッジ時間 13,297 ms
ジャッジサーバーID
（参考情報） 		judge2 / judge4
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テストケース

テストケース表示
入力 結果 実行時間
実行使用メモリ
testcase_00 AC 299 ms
41,620 KB
testcase_01 AC 401 ms
45,824 KB
testcase_02 AC 330 ms
45,824 KB
testcase_03 AC 830 ms
45,824 KB
testcase_04 AC 1,089 ms
45,824 KB
testcase_05 AC 601 ms
45,824 KB
testcase_06 AC 1,304 ms
45,824 KB
testcase_07 AC 1,306 ms
45,824 KB
testcase_08 AC 1,149 ms
45,696 KB
testcase_09 AC 1,102 ms
45,696 KB
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ソースコード

diff # 

import std.conv, std.functional, std.range, std.stdio, std.string;
import std.algorithm, std.array, std.bigint, std.bitmanip, std.complex, std.container, std.math, std.mathspecial, std.numeric, std.regex, std.typecons;
import core.bitop;

class EOFException : Throwable { this() { super("EOF"); } }
string[] tokens;
string readToken() { for (; tokens.empty; ) { if (stdin.eof) { throw new EOFException; } tokens = readln.split; } auto token = tokens.front; tokens.popFront; return token; }
int readInt() { return readToken.to!int; }
long readLong() { return readToken.to!long; }

string COLOR(string s = "") { return "\x1b[" ~ s ~ "m"; }

bool chmin(T)(ref T t, in T f) { if (t > f) { t = f; return true; } else { return false; } }
bool chmax(T)(ref T t, in T f) { if (t < f) { t = f; return true; } else { return false; } }

int binarySearch(alias pred, T)(in T[] as) { int lo = -1, hi = cast(int)(as.length); for (; lo + 1 < hi; ) { const mid = (lo + hi) >> 1; (unaryFun!pred(as[mid]) ? hi : lo) = mid; } return hi; }
int lowerBound(T)(in T[] as, T val) { return as.binarySearch!(a => (a >= val)); }
int upperBound(T)(in T[] as, T val) { return as.binarySearch!(a => (a > val)); }

struct ModInt(uint M_) {
  import std.conv : to;
  alias M = M_;
  uint x;
  this(ModInt a) { x = a.x; }
  this(uint x_) { x = x_ % M; }
  this(ulong x_) { x = cast(uint)(x_ % M); }
  this(int x_) { x = ((x_ %= cast(int)(M)) < 0) ? (x_ + cast(int)(M)) : x_; }
  this(long x_) { x = cast(uint)(((x_ %= cast(long)(M)) < 0) ? (x_ + cast(long)(M)) : x_); }
  ref ModInt opAssign(T)(inout(T) a) if (is(T == uint) || is(T == ulong) || is(T == int) || is(T == long)) { return this = ModInt(a); }
  ref ModInt opOpAssign(string op, T)(T a) {
    static if (is(T == ModInt)) {
      static if (op == "+") { x = ((x += a.x) >= M) ? (x - M) : x; }
      else static if (op == "-") { x = ((x -= a.x) >= M) ? (x + M) : x; }
      else static if (op == "*") { x = cast(uint)((cast(ulong)(x) * a.x) % M); }
      else static if (op == "/") { this *= a.inv(); }
      else static assert(false);
      return this;
    } else static if (op == "^^") {
      if (a < 0) return this = inv()^^(-a);
      ModInt b = this, c = 1U;
      for (long e = a; e; e >>= 1) { if (e & 1) c *= b; b *= b; }
      return this = c;
    } else {
      return mixin("this " ~ op ~ "= ModInt(a)");
    }
  }
  ModInt inv() const {
    uint a = M, b = x; int y = 0, z = 1;
    for (; b; ) { const q = a / b; const c = a - q * b; a = b; b = c; const w = y - cast(int)(q) * z; y = z; z = w; }
    assert(a == 1); return ModInt(y);
  }
  ModInt opUnary(string op)() const {
    static if (op == "+") { return this; }
    else static if (op == "-") { ModInt a; a.x = x ? (M - x) : 0U; return a; }
    else static assert(false);
  }
  ModInt opBinary(string op, T)(T a) const { return mixin("ModInt(this) " ~ op ~ "= a"); }
  ModInt opBinaryRight(string op, T)(T a) const { return mixin("ModInt(a) " ~ op ~ "= this"); }
  bool opCast(T: bool)() const { return (x != 0U); }
  string toString() const { return x.to!string; }
}

enum MO = 998244353;
alias Mint = ModInt!MO;


// x^e mod rev(cs)
Mint[] powerRev(const(Mint[]) cs, long e) {
  assert(!cs.empty);
  assert(cs[0].x == 1);
  const d = cast(int)(cs.length) - 1;
  if (d == 0) {
    return [];
  } else if (d == 1) {
    return [(-cs[1])^^e];
  }
  Mint[] mul(const(Mint[]) fs, const(Mint[]) gs) {
    auto hs = new Mint[d + d - 1];
    foreach (i; 0 .. d) foreach (j; 0 .. d) {
      hs[i + j] += fs[i] * gs[j];
    }
    foreach_reverse (i; d .. d + d - 1) {
      foreach (j; 1 .. d + 1) {
        hs[i - j] -= cs[j] * hs[i];
      }
    }
    hs.length = d;
    return hs;
  }
  auto xs = new Mint[d];
  auto ys = new Mint[d];
  xs[1] = 1;
  ys[0] = 1;
  for (; ; xs = mul(xs, xs)) {
    if (e & 1) ys = mul(ys, xs);
    if (!(e >>= 1)) break;
  }
  return ys;
}

Mint linearRecurrenceAt(const(Mint[]) as, const(Mint[]) cs, long e) {
  assert(!cs.empty);
  assert(cs[0].x == 1);
  const d = cast(int)(cs.length) - 1;
  assert(as.length >= d);
  const fs = powerRev(cs, e);
  Mint ans;
  foreach (i; 0 .. d) {
    ans += fs[i] * as[i];
  }
  return ans;
}


void main() {
  enum LIM = 10^^7 + 5;
  auto dp = new Mint[LIM];
  dp[0] = 1;
  foreach (n; 0 .. LIM) {
    if (n + 1 < LIM) dp[n + 1] += dp[n] * (n + 1);
    if (n + 1 < LIM) dp[n + 1] += dp[n] * (n + 1);
    if (n + 2 < LIM) dp[n + 2] += dp[n] * (n + 1);
  }
  debug {
    writeln(dp[0 .. 100]);
  }
  
  try {
    for (; ; ) {
      const numCases = readInt;
      foreach (caseId; 0 .. numCases) {
        long M = readLong;
        long N = readLong;
        if (M > N) {
          swap(M, N);
        }
        
        // 1 / (1 - p x - q x^2)
        const Mint p = 2 * M + 1;
        const Mint q = M;
        const as = [Mint(1), p, p * p + q];
        const cs = [Mint(1), -p, -q];
        
        const f = dp[M];
        const g = M ? (dp[M - 1] * M) : Mint(0);
        Mint ans;
        if (M >= 1 && N - M == 0) {
          ans += dp[M - 1]^^2 * M;
        }
        if (N - M >= 0) ans +=     f * f * linearRecurrenceAt(as, cs, N - M);
        if (N - M >= 1) ans += 2 * f * g * linearRecurrenceAt(as, cs, N - M - 1);
        if (N - M >= 2) ans +=     g * g * linearRecurrenceAt(as, cs, N - M - 2);
        writeln(ans);
      }
    }
  } catch (EOFException e) {
  }
}
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