結果

問題 No.1873 Bracket Swapping
コンテスト
ユーザー hos.lyric
提出日時 2022-03-17 02:22:33
言語 D
(dmd 2.109.1)
結果
AC  
実行時間 1,179 ms / 2,000 ms
コード長 7,463 bytes
コンパイル時間 1,586 ms
コンパイル使用メモリ 160,324 KB
実行使用メモリ 19,132 KB
最終ジャッジ日時 2024-06-22 14:37:46
合計ジャッジ時間 12,496 ms
ジャッジサーバーID
（参考情報） 		judge4 / judge2
このコードへのチャレンジ
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ファイルパターン 結果
sample AC * 3
other AC * 27
権限があれば一括ダウンロードができます

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; }
real readReal() { return readToken.to!real; }

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 = 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;

enum LIM = 410;
Mint[] inv, fac, invFac;
void prepare() {
  inv = new Mint[LIM];
  fac = new Mint[LIM];
  invFac = new Mint[LIM];
  inv[1] = 1;
  foreach (i; 2 .. LIM) {
    inv[i] = -((Mint.M / i) * inv[cast(size_t)(Mint.M % i)]);
  }
  fac[0] = invFac[0] = 1;
  foreach (i; 1 .. LIM) {
    fac[i] = fac[i - 1] * i;
    invFac[i] = invFac[i - 1] * inv[i];
  }
}
Mint binom(long n, long k) {
  if (n < 0) {
    if (k >= 0) {
      return (-1)^^(k & 1) * binom(-n + k - 1, k);
    } else if (n - k >= 0) {
      return (-1)^^((n - k) & 1) * binom(-k - 1, n - k);
    } else {
      return Mint(0);
    }
  } else {
    if (0 <= k && k <= n) {
      assert(n < LIM);
      return fac[cast(size_t)(n)] * invFac[cast(size_t)(k)] * invFac[cast(size_t)(n - k)];
    } else {
      return Mint(0);
    }
  }
}


// det(x I + a)
// O(n^3)
T[] charPoly(T)(const(T[][]) a) {
  import std.algorithm.mutation : swap;
  const n = cast(int)(a.length);
  auto b = new T[][](n, n);
  foreach (i; 0 .. n) b[i][] = a[i][];
  // upper Hessenberg
  foreach (j; 0 .. n - 2) {
    foreach (i; j + 1 .. n) {
      if (b[i][j]) {
        swap(b[j + 1], b[i]);
        foreach (ii; 0 .. n) swap(b[ii][j + 1], b[ii][i]);
        break;
      }
    }
    if (b[j + 1][j]) {
      const r = 1 / b[j + 1][j];
      foreach (i; j + 2 .. n) {
        const s = r * b[i][j];
        foreach (jj; j .. n) b[i][jj] -= s * b[j + 1][jj];
        foreach (ii; 0 .. n) b[ii][j + 1] += s * b[ii][i];
      }
    }
  }
  // fss[i] := det(x I_i + b[0..i][0..i])
  auto fss = new T[][n + 1];
  fss[0] = [T(1)];
  foreach (i; 0 .. n) {
    fss[i + 1] = new T[i + 2];
    foreach (k; 0 .. i + 1) fss[i + 1][k + 1] = fss[i][k];
    foreach (k; 0 .. i + 1) fss[i + 1][k] += b[i][i] * fss[i][k];
    T prod = 1;
    foreach_reverse (j; 0 .. i) {
      prod *= -b[j + 1][j];
      const s = prod * b[j][i];
      foreach (k; 0 .. j + 1) fss[i + 1][k] += s * fss[j][k];
    }
  }
  return fss[n];
}


Mint solve(Mint[] as, Mint[] cs, long N) {
  const d = cast(int)(cs.length) - 1;
  Mint[] mul(Mint[] fs, 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 (long e = N; e; e >>= 1) {
    if (e & 1) ys = mul(ys, xs);
    xs = mul(xs, xs);
  }
  Mint ans;
  foreach (i; 0 .. d) {
    ans += as[i] * ys[i];
  }
  return ans;
}


void main() {
  prepare;
  
  try {
    for (; ; ) {
      const S = readToken;
      const K = readLong;
      const N = cast(int)(S.length) / 2;
      
      // pos, depth, state
      auto dp = new Mint[][][](2 * N + 1, N + 2, N + 2);
      dp[0][0][0] = 1;
      foreach (i; 0 .. 2 * N) {
        foreach (j; 0 .. N + 1) foreach (u; 0 .. N + 1) {
          dp[i + 1][j + 1][u] += dp[i][j][u];
          if (j > 0) {
            dp[i + 1][j - 1][u + ((S[i] == '(') ? 1 : 0)] += dp[i][j][u];
          }
        }
      }
      debug {
        writeln("dp[2 N][0] = ", dp[2 * N][0]);
      }
      
      
      // initial state, final state
      auto mat = new Mint[][](N + 1, N + 1);
      foreach (u; 0 .. N + 1) {
        mat[u][u] += ((2 * N) * (2 * N - 1) / 2 - (N - u)^^2 - u^^2);
        if (u < N) mat[u][u + 1] += (N - u)^^2;
        if (u > 0) mat[u][u - 1] += u^^2;
      }
      auto cs = charPoly(mat);
      cs.reverse;
      for (int i = 1; i <= N + 1; i += 2) {
        cs[i] *= -1;
      }
      debug {
        writeln("mat = ", mat);
        writeln("cs = ", cs);
      }
      
      // # of ops, state
      auto DP = new Mint[][](N + 1, N + 1);
      DP[0][0] = 1;
      foreach (k; 0 .. N) {
        foreach (u; 0 .. N + 1) foreach (v; 0 .. N + 1) {
          DP[k + 1][v] += DP[k][u] * mat[u][v];
        }
      }
      auto gs = new Mint[N + 1];
      foreach (u; 0 .. N + 1) {
        auto as = iota(N + 1).map!(k => DP[k][u]).array;
        gs[u] = solve(as, cs, K);
        gs[u] /= binom(N, u)^^2;
      }
      debug {
        writeln("gs = ", gs);
      }
      
      
      Mint ans;
      foreach (u; 0 .. N + 1) {
        ans += dp[2 * N][0][u] * gs[u];
      }
      
      writeln(ans);
    }
  } catch (EOFException e) {
  }
}
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