#408058 (D) No.950 行列累乗

提出ソース

結果

問題	No.950 行列累乗
ユーザー	👑 hos.lyric
提出日時	2019-12-13 02:24:27
言語	D (dmd 2.106.1)
結果	AC
実行時間	196 ms / 2,000 ms
コード長	8,146 bytes
コンパイル時間	1,418 ms
コンパイル使用メモリ	141,288 KB
実行使用メモリ	19,956 KB
最終ジャッジ日時	2024-06-22 03:34:28
合計ジャッジ時間	5,580 ms
ジャッジサーバーID （参考情報）	judge2 / judge4

このコードへのチャレンジ
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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	11 ms 6,812 KB
testcase_01	AC	15 ms 6,944 KB
testcase_02	AC	1 ms 6,940 KB
testcase_03	AC	1 ms 6,944 KB
testcase_04	AC	1 ms 6,940 KB
testcase_05	AC	52 ms 11,564 KB
testcase_06	AC	1 ms 6,940 KB
testcase_07	AC	1 ms 6,944 KB
testcase_08	AC	2 ms 6,944 KB
testcase_09	AC	1 ms 6,940 KB
testcase_10	AC	1 ms 6,940 KB
testcase_11	AC	1 ms 6,944 KB
testcase_12	AC	1 ms 6,944 KB
testcase_13	AC	1 ms 6,944 KB
testcase_14	AC	1 ms 6,944 KB
testcase_15	AC	1 ms 6,944 KB
testcase_16	AC	1 ms 6,940 KB
testcase_17	AC	1 ms 6,944 KB
testcase_18	AC	1 ms 6,940 KB
testcase_19	AC	1 ms 6,940 KB
testcase_20	AC	1 ms 6,944 KB
testcase_21	AC	37 ms 6,940 KB
testcase_22	AC	110 ms 13,324 KB
testcase_23	AC	61 ms 12,440 KB
testcase_24	AC	119 ms 12,992 KB
testcase_25	AC	54 ms 8,916 KB
testcase_26	AC	79 ms 11,992 KB
testcase_27	AC	24 ms 6,940 KB
testcase_28	AC	97 ms 13,048 KB
testcase_29	AC	85 ms 13,152 KB
testcase_30	AC	134 ms 13,316 KB
testcase_31	AC	103 ms 13,276 KB
testcase_32	AC	138 ms 13,936 KB
testcase_33	AC	136 ms 13,872 KB
testcase_34	AC	141 ms 14,464 KB
testcase_35	AC	120 ms 13,396 KB
testcase_36	AC	1 ms 6,944 KB
testcase_37	AC	10 ms 6,944 KB
testcase_38	AC	11 ms 6,944 KB
testcase_39	AC	1 ms 6,944 KB
testcase_40	AC	91 ms 12,664 KB
testcase_41	AC	71 ms 13,656 KB
testcase_42	AC	34 ms 6,940 KB
testcase_43	AC	154 ms 14,316 KB
testcase_44	AC	146 ms 14,980 KB
testcase_45	AC	191 ms 19,832 KB
testcase_46	AC	160 ms 16,264 KB
testcase_47	AC	70 ms 12,532 KB
testcase_48	AC	155 ms 15,240 KB
testcase_49	AC	162 ms 16,180 KB
testcase_50	AC	181 ms 19,956 KB
testcase_51	AC	196 ms 19,456 KB
testcase_52	AC	21 ms 6,944 KB
testcase_53	AC	21 ms 6,940 KB
testcase_54	AC	1 ms 6,940 KB
testcase_55	AC	1 ms 6,944 KB
testcase_56	AC	13 ms 6,940 KB
testcase_57	AC	73 ms 12,564 KB
testcase_58	AC	1 ms 6,940 KB
testcase_59	AC	1 ms 6,940 KB
testcase_60	AC	1 ms 6,944 KB

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ソースコード

raw source code

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, core.stdc.stdlib;

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)); }


// floor(a^(1/k))
ulong floorKthRoot(ulong a, ulong k) {
  import core.bitop : bsr;
  import std.algorithm : min;
  if (a == 0) {
    return 0;
  } else if (k <= 1) {
    return a;
  } else if (k == 2) {
    ulong b = a, x = 0, y = 0;
    for (int e = bsr(a) & ~1; e >= 0; e -= 2) {
      x <<= 1;
      y <<= 1;
      if (b >= (y | 1) << e) {
        b -= (y | 1) << e;
        x |= 1;
        y += 2;
      }
    }
    return x;
  } else if (k <= 40) {
    // min x s.t. x^k >= 2^64
    enum ulong[] HIS =
        [0, 0, 4294967296UL, 2642246, 65536, 7132, 1626, 566, 256, 139, 85, 57,
         41, 31, 24, 20, 16, 14, 12, 11, 10, 9, 8, 7, 7, 6, 6, 6, 5, 5, 5, 5,
         4, 4, 4, 4, 4, 4, 4, 4, 4];
    ulong lo = 1UL << (bsr(a) / k);
    ulong hi = min(1UL << (bsr(a) / k + 1), HIS[cast(size_t)(k)]);
    for (; lo + 1 < hi; ) {
      const ulong mid = (lo + hi) / 2;
      ulong b = mid * mid;
      foreach (i; 2 .. k) b *= mid;
      ((b <= a) ? lo : hi) = mid;
    }
    return lo;
  } else if (k <= 63) {
    return ((1UL << k) <= a) ? 2 : 1;
  } else {
    return 1;
  }
}


// a^-1 (mod m)
long modInv(long a, long m)
in {
  assert(m > 0, "modInv: m > 0 must hold");
}
do {
  long b = m, x = 1, y = 0, t;
  for (; ; ) {
    t = a / b; a -= t * b;
    if (a == 0) {
      assert(b == 1 || b == -1, "modInv: gcd(a, m) != 1");
      if (b == -1) y = -y;
      return (y < 0) ? (y + m) : y;
    }
    x -= t * y;
    t = b / a; b -= t * a;
    if (b == 0) {
      assert(a == 1 || a == -1, "modInv: gcd(a, m) != 1");
      if (a == -1) x = -x;
      return (x < 0) ? (x + m) : x;
    }
    y -= t * x;
  }
}


Int mod(Int)(Int a, Int m) {
  if ((a %= m) < 0) a += m; return a;
}
Int gcd(Int)(Int a, Int b) { return (b != 0) ? gcd(b, a % b) : a; }
Int lcm(Int)(Int a, Int b) { return a / gcd(a, b) * b; }
Int gojo(Int)(Int a, Int b, out Int x, out Int y) {
  if (b != 0) { Int g = gojo(b, a % b, y, x); y -= (a / b) * x; return g; }
  x = 1; y = 0; return a;
}
Int modInv(Int)(Int a, Int m) { Int x, y; gojo(a, m, x, y); return mod(x, m); }
long modLogP(long a, long b, long m) {
	a = mod(a, m); b = mod(b, m);
	if (m == 1) return 0;
	long k, al = 1;
	Tuple!(long,long)[] as;
	for (k = 0; k * k < m; ++k) {
		as ~= tuple(al, k);
		al = (al * a) % m;
	}
	as.sort;
	al = modInv(al, m);
	for (long i = 0; i < k; ++i) {
		int pos = as.lowerBound(tuple(b, 0L));
		if (pos < as.length && as[pos][0] == b) return i * k + as[pos][1];
		b = (b * al) % m;
	}
	return -1;
}


long P;
long[][] A, B;

long[][] div(long[][] a, long t) {
  const invT = modInv(t, P);
  auto b = new long[][](2, 2);
  foreach (i; 0 .. 2) foreach (j; 0 .. 2) {
    b[i][j] = mod(invT * a[i][j], P);
  }
  return b;
}
long[][] inv(long[][] a) {
  const det = mod(a[0][0] * a[1][1] - a[0][1] * a[1][0], P);
  return div([[a[1][1], -a[0][1]], [-a[1][0], a[0][0]]], det);
}
long[][] mul(long[][] a, long[][] b) {
  auto c = new long[][](2, 2);
  foreach (i; 0 .. 2) foreach (k; 0 .. 2) foreach (j; 0 .. 2) {
    c[i][j] = mod(c[i][j] + a[i][k] * b[k][j], P);
  }
  return c;
}
long[][] power(long[][] a, long e) {
  long[][] b = a, c = [[1, 0], [0, 1]];
  for (; e; e >>= 1) {
    if (e & 1) c = mul(c, b);
    b = mul(b, b);
  }
  return c;
}

long solveBSGS(long[][] a, long[][] b) {
  const m = cast(int)(floorKthRoot(2 * P, 2) + 2);
  auto als = new Tuple!(long[][], int)[m];
  long[][] al = [[1, 0], [0, 1]];
  foreach (l; 0 .. m) {
    als[l] = tuple(al, l);
    al = mul(al, a);
  }
  als.sort;
  al = inv(al);
  long[][] amk = b;
  foreach (k; 0 .. m) {
    const pos = als.lowerBound(tuple(amk, 0));
    if (pos < m && als[pos][0] == amk) {
      return k * cast(long)(m) + als[pos][1];
    }
    amk = mul(amk, al);
  }
  return -1;
}


void main() {
  try {
    for (; ; ) {
      P = readLong();
      A = new long[][](2, 2);
      foreach (i; 0 .. 2) foreach (j; 0 .. 2) {
        A[i][j] = readLong();
      }
      B = new long[][](2, 2);
      foreach (i; 0 .. 2) foreach (j; 0 .. 2) {
        B[i][j] = readLong();
      }
      
      long ans;
      const detA = mod(A[0][0] * A[1][1] - A[0][1] * A[1][0], P);
      const detB = mod(B[0][0] * B[1][1] - B[0][1] * B[1][0], P);
      if (detA == 0) {
        if (detB == 0) {
          // eigenvalues: 0, r
          const r = mod(A[0][0] + A[1][1], P);
          if (r == 0) {
            if (A == B) {
              ans = 1;
            } else if (mul(A, A) == B) {
              ans = 2;
            } else {
              ans = -1;
            }
          } else {
            /*
              C^-1 A C = [[r, 0], [0, 0]]
              
              a00 c00 + a01 c10 = r c00
              a10 c00 + a11 c10 = r c10
              a00 c01 + a01 c11 = 0
              a10 c01 + a11 c11 = 0
            */
            auto c = new long[][](2, 2);
            c[0][0] = mod(-A[0][1], P);
            c[1][0] = mod(A[0][0] - r, P);
            if (c[0][0] == 0 && c[1][0] == 0) {
              c[0][0] = mod(-(A[1][1] - r), P);
              c[1][0] = mod(A[1][0], P);
            }
            if (c[0][0] == 0 && c[1][0] == 0) {
              assert(false);
            }
            c[0][1] = mod(-A[0][1], P);
            c[1][1] = mod(A[0][0], P);
            if (c[0][1] == 0 && c[1][1] == 0) {
              c[0][1] = mod(-A[1][1], P);
              c[1][1] = mod(A[1][0], P);
            }
            if (c[0][1] == 0 && c[1][1] == 0) {
              assert(false);
            }
            debug {
              writeln("r = ", r);
              writeln("c = ", c);
            }
            auto a = mul(mul(inv(c), A), c);
            auto b = mul(mul(inv(c), B), c);
            debug {
              writeln("a = ", a);
              writeln("b = ", b);
            }
            assert(a == [[r, 0L], [0L, 0L]]);
            if (b[0][1] == 0 && b[1][0] == 0 && b[1][1] == 0) {
              const res = modLogP(r, mod(b[0][0] * modInv(r, P), P), P);
              if (res == -1) {
                ans = -1;
              } else {
                ans = res + 1;
              }
            } else {
              ans = -1;
            }
          }
        } else {
          ans = -1;
        }
      } else {
        const period = modLogP(detA, modInv(detA, P), P) + 1;
        long e0 = modLogP(detA, detB, P);
        debug {
          writeln("detA = ", detA, ", detB = ", detB);
          writeln("period = ", period);
          writeln("e0 = ", e0);
        }
        if (e0 == -1) {
          ans = -1;
        } else {
          if (e0 == 0) {
            e0 += period;
          }
          auto a = power(A, period);
          auto b = mul(B, inv(power(A, e0)));
          debug {
            writeln("a = ", a);
            writeln("b = ", b);
          }
          const res = solveBSGS(a, b);
          if (res == -1) {
            ans = -1;
          } else {
            ans = e0 + period * res;
          }
        }
      }
      writeln(ans);
    }
  } catch (EOFException e) {
  }
}

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結果

テストケース

ソースコード