結果

問題 No.886 Direct
ユーザー yuruhiya
提出日時 2021-07-10 19:21:54
言語 Crystal
(1.14.0)
結果
AC  
実行時間 543 ms / 4,000 ms
コード長 16,492 bytes
コンパイル時間 13,243 ms
コンパイル使用メモリ 295,040 KB
実行使用メモリ 183,232 KB
最終ジャッジ日時 2024-07-02 02:45:40
合計ジャッジ時間 18,166 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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ファイルパターン 結果
sample AC * 4
other AC * 32
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 #
プレゼンテーションモードにする

# require "/math/Mint"
# require "../atcoder/src/Math"
# ac-library.cr by hakatashi https://github.com/google/ac-library.cr
#
# Copyright 2021 Google LLC
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#      https://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
module AtCoder
  # Implements [ACL's Math library](https://atcoder.github.io/ac-library/master/document_en/math.html)
  module Math
    def self.extended_gcd(a, b)
      last_remainder, remainder = a.abs, b.abs
      x, last_x, y, last_y = 0_i64, 1_i64, 1_i64, 0_i64
      while remainder != 0
        new_last_remainder = remainder
        quotient, remainder = last_remainder.divmod(remainder)
        last_remainder = new_last_remainder
        x, last_x = last_x - quotient * x, x
        y, last_y = last_y - quotient * y, y
      end
      return last_remainder, last_x * (a < 0 ? -1 : 1)
    end
    # Implements atcoder::inv_mod(value, modulo).
    def self.inv_mod(value, modulo)
      gcd, inv = extended_gcd(value, modulo)
      if gcd != 1
        raise ArgumentError.new("#{value} and #{modulo} are not coprime")
      end
      inv % modulo
    end
    # Simplified AtCoder::Math.pow_mod with support of Int64
    def self.pow_mod(base, exponent, modulo)
      if exponent == 0
        return base.class.zero + 1
      end
      if base == 0
        return base
      end
      b = exponent > 0 ? base : inv_mod(base, modulo)
      e = exponent.abs
      ret = 1_i64
      while e > 0
        if e % 2 == 1
          ret = mul_mod(ret, b, modulo)
        end
        b = mul_mod(b, b, modulo)
        e //= 2
      end
      ret
    end
    # Caluculates a * b % mod without overflow detection
    @[AlwaysInline]
    def self.mul_mod(a : Int64, b : Int64, mod : Int64)
      if mod < Int32::MAX
        return a * b % mod
      end
      # 31-bit width
      a_high = (a >> 32).to_u64
      # 32-bit width
      a_low = (a & 0xFFFFFFFF).to_u64
      # 31-bit width
      b_high = (b >> 32).to_u64
      # 32-bit width
      b_low = (b & 0xFFFFFFFF).to_u64
      # 31-bit + 32-bit + 1-bit = 64-bit
      c = a_high * b_low + b_high * a_low
      c_high = c >> 32
      c_low = c & 0xFFFFFFFF
      # 31-bit + 31-bit
      res_high = a_high * b_high + c_high
      # 32-bit + 32-bit
      res_low = a_low * b_low
      res_low_high = res_low >> 32
      res_low_low = res_low & 0xFFFFFFFF
      # Overflow
      if res_low_high + c_low >= 0x100000000
        res_high += 1
      end
      res_low = (((res_low_high + c_low) & 0xFFFFFFFF) << 32) | res_low_low
      (((res_high.to_i128 << 64) | res_low) % mod).to_i64
    end
    @[AlwaysInline]
    def self.mul_mod(a, b, mod)
      typeof(mod).new(a.to_i64 * b % mod)
    end
    # Implements atcoder::crt(remainders, modulos).
    def self.crt(remainders, modulos)
      raise ArgumentError.new unless remainders.size == modulos.size
      total_modulo = 1_i64
      answer = 0_i64
      remainders.zip(modulos).each do |(remainder, modulo)|
        gcd, p = extended_gcd(total_modulo, modulo)
        if (remainder - answer) % gcd != 0
          return 0_i64, 0_i64
        end
        tmp = (remainder - answer) // gcd * p % (modulo // gcd)
        answer += total_modulo * tmp
        total_modulo *= modulo // gcd
      end
      return answer % total_modulo, total_modulo
    end
    # Implements atcoder::floor_sum(n, m, a, b).
    def self.floor_sum(n, m, a, b)
      n, m, a, b = n.to_i64, m.to_i64, a.to_i64, b.to_i64
      res = 0_i64
      if a < 0
        a2 = a % m
        res -= n * (n - 1) // 2 * ((a2 - a) // m)
        a = a2
      end
      if b < 0
        b2 = b % m
        res -= n * ((b2 - b) // m)
        b = b2
      end
      res + floor_sum_unsigned(n, m, a, b)
    end
    private def self.floor_sum_unsigned(n, m, a, b)
      res = 0_i64
      loop do
        if a >= m
          res += n * (n - 1) // 2 * (a // m)
          a = a % m
        end
        if b >= m
          res += n * (b // m)
          b = b % m
        end
        y_max = a * n + b
        break if y_max < m
        n = y_max // m
        b = y_max % m
        m, a = a, m
      end
      res
    end
  end
end
macro static_modint(name, mod)
  struct {{name}}
    MOD = {{mod}}i64
    def self.zero
      new
    end
    def self.raw(value : Int64)
      result = new
      result.value = value
      result
    end
    getter value : Int64
    def initialize
      @value = 0i64
    end
    def initialize(value)
      @value = value.to_i64 % MOD
    end
    def initialize(m : self)
      @value = m.value
    end
    protected def value=(value : Int64)
      @value = value
    end
    def ==(m : self)
      value == m.value
    end
    def ==(m)
      value == m
    end
    def + : self
      self
    end
    def - : self
      self.class.raw(value != 0 ? MOD &- value : 0i64)
    end
    def +(v)
      self + self.class.new(v)
    end
    def +(m : self)
      x = value &+ m.value
      x &-= MOD if x >= MOD
      self.class.raw(x)
    end
    def -(v)
      self - self.class.new(v)
    end
    def -(m : self)
      x = value &- m.value
      x &+= MOD if x < 0
      self.class.raw(x)
    end
    def *(v)
      self * self.class.new(v)
    end
    def *(m : self)
      self.class.new(value &* m.value)
    end
    def /(v)
      self / self.class.new(v)
    end
    def /(m : self)
      raise DivisionByZeroError.new if m.value == 0
      a, b, u, v = m.value, MOD, 1i64, 0i64
      while b != 0
        t = a // b
        a &-= t &* b
        a, b = b, a
        u &-= t &* v
        u, v = v, u
      end
      self.class.new(value &* u)
    end
    def //(v)
      self / v
    end
    def **(exponent : Int)
      t, res = self, self.class.raw(1i64)
      while exponent > 0
        res *= t if exponent & 1 == 1
        t *= t
        exponent >>= 1
      end
      res
    end
    {% for op in %w[< <= > >=] %}
      def {{op.id}}(other)
        raise NotImplementedError.new({{op}})
      end
    {% end %}
    def inv
      self.class.raw AtCoder::Math.inv_mod(value, MOD)
    end
    def succ
      self.class.raw(value != MOD &- 1 ? value &+ 1 : 0i64)
    end
    def pred
      self.class.raw(value != 0 ? value &- 1 : MOD &- 1)
    end
    def abs
      self
    end
    def abs2
      self * self
    end
    def to_i64 : Int64
      value
    end
    delegate to_s, to: @value
    delegate inspect, to: @value
  end
  {% to = ("to_" + name.stringify.downcase.gsub(/mint|modint/, "m")).id %}
  struct Int
    {% for op in %w[+ - * / //] %}
      def {{op.id}}(value : {{name}})
        {{to}} {{op.id}} value
      end
    {% end %}
    {% for op in %w[< <= > >=] %}
      def {{op.id}}(m : {{name}})
        raise NotImplementedError.new({{op}})
      end
    {% end %}
    def {{to}} : {{name}}
      {{name}}.new(self)
    end
  end
  class String
    def {{to}} : {{name}}
      {{name}}.new(self)
    end
  end
end
static_modint(Mint, 1000000007)
static_modint(Mint2, 998244353)
# require "/math/GCD"
# require "/atcoder/src/Prime"
# ac-library.cr by hakatashi https://github.com/google/ac-library.cr
#
# Copyright 2021 Google LLC
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#      https://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
# require "./Math.cr"
module AtCoder
  # Implements [Ruby's Prime library](https://ruby-doc.com/stdlib/libdoc/prime/rdoc/Prime.html).
  #
  # ```
  # AtCoder::Prime.first(7) # => [2, 3, 5, 7, 11, 13, 17]
  # ```
  module Prime
    extend self
    include Enumerable(Int64)
    @@primes = [
      2_i64, 3_i64, 5_i64, 7_i64, 11_i64, 13_i64, 17_i64, 19_i64,
      23_i64, 29_i64, 31_i64, 37_i64, 41_i64, 43_i64, 47_i64,
      53_i64, 59_i64, 61_i64, 67_i64, 71_i64, 73_i64, 79_i64,
      83_i64, 89_i64, 97_i64, 101_i64,
    ]
    def each
      index = 0
      loop do
        yield get_nth_prime(index)
        index += 1
      end
    end
    def prime_division(value : Int)
      raise DivisionByZeroError.new if value == 0
      int = typeof(value)
      factors = [] of Tuple(typeof(value), typeof(value))
      if value < 0
        value = value.abs
        factors << {int.new(-1), int.new(1)}
      end
      until prime?(value) || value == 1
        factor = value
        until prime?(factor)
          factor = find_factor(factor)
        end
        count = 0
        while value % factor == 0
          value //= factor
          count += 1
        end
        factors << {int.new(factor), int.new(count)}
      end
      if value > 1
        factors << {value, int.new(1)}
      end
      factors.sort_by! { |(factor, _)| factor }
    end
    private def find_factor(n : Int)
      # Factor of 4 cannot be discovered by Pollard's Rho with f(x) = x^x+1
      if n == 4
        typeof(n).new(2)
      else
        pollard_rho(n).not_nil!
      end
    end
    # Get single factor by Pollard's Rho Algorithm
    private def pollard_rho(n : Int)
      typeof(n).new(1).upto(n) do |i|
        x = i
        y = pollard_random_f(x, n)
        loop do
          x = pollard_random_f(x, n)
          y = pollard_random_f(pollard_random_f(y, n), n)
          gcd = (x - y).gcd(n)
          if gcd == n
            break
          end
          if gcd != 1
            return gcd
          end
        end
      end
    end
    private def pollard_random_f(n : Int, mod : Int)
      (AtCoder::Math.mul_mod(n, n, mod) + 1) % mod
    end
    private def extract_prime_division_base(prime_divisions_class : Array({T, T}).class) forall T
      T
    end
    def int_from_prime_division(prime_divisions : Array({Int, Int}))
      int_class = extract_prime_division_base(prime_divisions.class)
      prime_divisions.reduce(int_class.new(1)) { |i, (factor, exponent)| i * factor ** exponent }
    end
    def prime?(value : Int)
      # Obvious patterns
      return false if value < 2
      return true if value <= 3
      return false if value.even?
      return true if value < 9
      if value < 0xffff
        return false unless typeof(value).new(30).gcd(value % 30) == 1
        7.step(by: 30, to: value) do |base|
          break if base * base > value
          if {0, 4, 6, 10, 12, 16, 22, 24}.any? { |i| value % (base + i) == 0 }
            return false
          end
        end
        return true
      end
      miller_rabin(value.to_i64)
    end
    private def miller_rabin(value)
      d = value - 1
      s = 0_i64
      until d.odd?
        d >>= 1
        s += 1
      end
      miller_rabin_bases(value).each do |base|
        next if base == value
        x = AtCoder::Math.pow_mod(base.to_i64, d, value)
        next if x == 1 || x == value - 1
        is_composite = s.times.all? do
          x = AtCoder::Math.mul_mod(x, x, value)
          x != value - 1
        end
        return false if is_composite
      end
      true
    end
    # We can reduce time complexity of Miller-Rabin tests by testing against
    # predefined bases which is enough to test against primarity in the given range.
    # https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test
    # ameba:disable Metrics/CyclomaticComplexity
    private def miller_rabin_bases(value)
      case
      when value < 1_373_653_i64
        [2, 3]
      when value < 9_080_191_i64
        [31, 73]
      when value < 25_326_001_i64
        [2, 3, 5]
      when value < 3_215_031_751_i64
        [2, 3, 5, 7]
      when value < 4_759_123_141_i64
        [2, 7, 61]
      when value < 1_122_004_669_633_i64
        [2, 13, 23, 1662803]
      when value < 2_152_302_898_747_i64
        [2, 3, 5, 7, 11]
      when value < 3_474_749_660_383_i64
        [2, 3, 5, 7, 11, 13]
      when value < 341_550_071_728_321_i64
        [2, 3, 5, 7, 11, 13, 17]
      when value < 3_825_123_056_546_413_051_i64
        [2, 3, 5, 7, 11, 13, 17, 19, 23]
      else
        [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37]
      end
    end
    private def get_nth_prime(n)
      while @@primes.size <= n
        generate_primes
      end
      @@primes[n]
    end
    # Doubles the size of the cached prime array and performs the
    # Sieve of Eratosthenes on it.
    private def generate_primes
      new_primes_size = @@primes.size < 1_000_000 ? @@primes.size : 1_000_000
      new_primes = Array(Int64).new(new_primes_size) { |i| @@primes.last + (i + 1) * 2 }
      new_primes_max = new_primes.last
      @@primes.each do |prime|
        next if prime == 2
        break if prime * prime > new_primes_max
        # Here I use the technique of the Sieve of Sundaram. We can
        # only test against the odd multiple of the given prime.
        # min_composite is the minimum number that is greater than
        # the last confirmed prime, and is an odd multiple of
        # the given prime.
        min_multiple = ((@@primes.last // prime + 1) // 2 * 2 + 1) * prime
        min_multiple.step(by: prime * 2, to: new_primes_max) do |multiple|
          index = new_primes_size - (new_primes_max - multiple) // 2 - 1
          new_primes[index] = 0_i64
        end
      end
      @@primes.concat(new_primes.reject(0_i64))
    end
    private struct EachDivisor(T)
      include Enumerable(T)
      def initialize(@exponential_factors : Array(Array(T)))
      end
      def each
        Array.each_product(@exponential_factors) do |factors|
          yield factors.reduce { |a, b| a * b }
        end
      end
    end
    # Returns an enumerator that iterates through the all positive divisors of
    # the given number. **The order is not guaranteed.**
    # Not in the original Ruby's Prime library.
    #
    # ```
    # AtCoder::Prime.each_divisor(20) do |n|
    #   puts n
    # end # => Puts 1, 2, 4, 5, 10, and 20
    #
    # AtCoder::Prime.each_divisor(10).map { |n| 1.0 / n }.to_a # => [1.0, 0.5, 0.2, 0.1]
    # ```
    def each_divisor(value : Int)
      raise ArgumentError.new unless value > 0
      factors = prime_division(value)
      if value == 1
        exponential_factors = [[value]]
      else
        exponential_factors = factors.map do |(factor, count)|
          cnt = typeof(value).zero + 1
          Array(typeof(value)).new(count + 1) do |i|
            cnt_copy = cnt
            if i < count
              cnt *= factor
            end
            cnt_copy
          end
        end
      end
      EachDivisor(typeof(value)).new(exponential_factors)
    end
    # :ditto:
    def each_divisor(value : T, &block : T ->)
      each_divisor(value).each(&block)
    end
  end
end
struct Int
  def prime?
    AtCoder::Prime.prime?(self)
  end
end
class Array(T)
  # result[i] = Sum_{n | i} a[i]
  def gcd_zeta!
    AtCoder::Prime.each do |p|
      break if p >= size
      i, k = size.pred // p, size.pred // p * p
      while k > 0
        self[i] += self[i * p]
        i -= 1; k -= p
      end
    end
    self
  end
  # :ditto:
  def gcd_zeta
    dup.gcd_zeta!
  end
  # a[i] = Sum_{n | i} result[i]
  def gcd_mobius!
    AtCoder::Prime.each do |p|
      break if p >= size
      i, k = 1, p
      while k < size
        self[i] -= self[k]
        i += 1; k += p
      end
    end
    self
  end
  # :ditto:
  def gcd_mobius
    dup.gcd_mobius!
  end
end
module GCD
  extend self
  # result[n] = Sum_{gcd(i, j) = n} f[i] * g[j]
  def convolution(f : Array(T), g : Array(T)) forall T
    f.gcd_zeta.zip?(g.gcd_zeta).map { |x, y| (x || T.zero) * (y || T.zero) }.gcd_mobius!
  end
end
h, w = read_line.split.map(&.to_i64)
hh = (0...h).map { |i| (h - i).to_m }
ww = (0...w).map { |i| (w - i).to_m }
one = h * w.pred + h.pred * w
puts GCD.convolution(hh, ww).fetch(1, Mint.zero) * 2 + one
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