高精算法 - 整数

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记录一下高精算法模板

Addition

原理

没什么好说的，小学数学——竖式计算，位数不足补零，当然实现里为了方便起见会将原数倒序，以便进位

实现

Addition
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string add(string a, string b) {
  bool neg = false;
  if (a[0] == '+') a = a.substr(1);
  if (b[0] == '+') b = b.substr(1);
  if (a[0] == '-' && b[0] == '-') {
    neg = true, a = a.substr(1), b = b.substr(1);
  } else if (a[0] == '-' && b[0] != '-')
    return sub(b, a.substr(1));
  else if (a[0] != '-' && b[0] == '-')
    return sub(a, b.substr(1));
  ull maxlen = max(a.size(), b.size()), l = 0, r = a.size() - 1;
  string ans(maxlen + 1, '0');
  while (l < r) (swap(a[l], a[r]), l++, r--);

  l = 0, r = b.size() - 1;
  while (l < r) (swap(b[l], b[r]), l++, r--);
  a += string(maxlen - a.size(), '0'), b += string(maxlen - b.size(), '0');
  for (ull i = 0; i < maxlen; ++i) {
    ans[i + 1] = (ans[i] - '0' + a[i] - '0' + b[i] - '0') / 10 + '0';
    ans[i] = (ans[i] - '0' + a[i] - '0' + b[i] - '0') % 10 + '0';
  }
  r = maxlen;
  while (r > 0 && ans[r] == '0') r--;
  ans = ans.substr(0, r + 1), l = 0;
  while (l < r) (swap(ans[l], ans[r]), l++, r--);
  return neg ? "-" + ans : ans;
}

Subtraction

原理

同 A + B 竖式计算，不够借位，同样倒序实现。

实现

Subtraction
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string sub(string a, string b) {
  bool neg = false;
  if (a[0] == '+') a = a.substr(1);
  if (b[0] == '+') b = b.substr(1);
  if (b[0] == '-') return add(a, b.substr(1));
  if (a[0] == '-') return "-" + add(a.substr(1), b);
  if (a.size() < b.size() || (a.size() == b.size() && a < b))
    (neg = true, swap(a, b));
  ull maxlen = max(a.size(), b.size()), l = 0, r = a.size() - 1;
  while (l < r) (swap(a[l], a[r]), l++, r--);
  l = 0, r = b.size() - 1;
  while (l < r) (swap(b[l], b[r]), l++, r--);
  a += string(maxlen - a.size(), '0');
  b += string(maxlen - b.size(), '0');
  string ans = a;
  for (ull i = 0; i < maxlen; ++i) {
    ans[i + 1] -= ans[i] >= b[i] ? 0 : 1;
    ans[i] = (ans[i] - b[i] + 10) % 10 + '0';
  }
  r = maxlen - 1;
  while (r > 0 && ans[r] == '0') r--;
  ans = ans.substr(0, r + 1), l = 0;
  while (l < r) (swap(ans[l], ans[r]), l++, r--);
  return neg ? "-" + ans : ans;
}

Multiplication

原理

竖式计算，位数不足补零，同样倒序实现。

实现

Multiplication
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string mul(string a, string b) {
  bool neg = false;
  if (a[0] == '+') a = a.substr(1);
  if (b[0] == '+') b = b.substr(1);
  if (a[0] == '-' && b[0] != '-') {
    neg = true, a = a.substr(1);
  } else if (a[0] != '-' && b[0] == '-') {
    neg = true, b = b.substr(1);
  } else if (a[0] == '-' && a[0] == b[0]) {
    a = a.substr(1), b = b.substr(1);
  }
  ull maxlen = max(a.size(), b.size());
  ull l = 0, r = a.size() - 1;
  string ans(maxlen * 2, '0');
  while (l < r) (swap(a[l], a[r]), l++, r--);
  l = 0, r = b.size() - 1;
  while (l < r) (swap(b[l], b[r]), l++, r--);
  if (a.size() < b.size()) swap(a, b);
  for (ull i = 0; i < b.size(); ++i)
    for (ull j = 0; j < a.size(); ++j) {
      ans[i + j + 1] += (ans[i + j] - '0' + (a[j] - '0') * (b[i] - '0')) /
      10; ans[i + j] = (ans[i + j] - '0' + (a[j] - '0') * (b[i] - '0')) % 10
      + '0';
    }
  r = 2 * maxlen - 1;
  while (r > 0 && ans[r] == '0') r--;
  ans = ans.substr(0, r + 1), l = 0;
  while (l < r) (swap(ans[l], ans[r]), l++, r--);
  return neg && ans != "0" ? "-" + ans : ans;
}

Division

原理

竖式计算

实现

  • 仅整除，不计算小数
Division
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string div(string a, string b) {
  bool neg = false;
  if (a[0] == '+') a = a.substr(1);
  if (b[0] == '+') b = b.substr(1);
  if (a[0] == '-' && b[0] != '-') {
    neg = true, a = a.substr(1);
  } else if (a[0] != '-' && b[0] == '-') {
    neg = true, b = b.substr(1);
  } else if (a[0] == '-' && a[0] == b[0]) {
    a = a.substr(1), b = b.substr(1);
  }
  string ans;
  if (a.size() < b.size())
    ans = "0";
  else {
    string remain = a.substr(0, b.size() - 1), temp;
    for (ull i = b.size() - 1; i < a.size(); ++i) {
      remain += a[i], remain = mul(remain, "1");
      if (remain.size() > b.size() ||
          (remain.size() == b.size() && remain >= b))
        for (char t = '9'; t >= '0'; --t) {
          temp = mul(b, string(1, t));
          if (temp.size() < remain.size() ||
              (temp.size() == remain.size() && temp <= remain)) {
            ans += t, remain = sub(remain, temp);
            break;
          }
        }
      else
        ans += '0';
    }
  }
  if (ans != "0") {
    int i = 0;
    while (ans[i] == '0') i++;
    ans = ans.substr(i);
  }
  return neg && ans != "0" ? "-" + ans : ans;
}

BigInt 类

  • 运算符重载不太会用，可能有问题，以下代码仅供参考
BigInt
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class BigInt {
 public:
  string value;
  BigInt(string s) { value = s; };
  BigInt() { value = "0"; };
  friend BigInt operator-(BigInt a, BigInt b) {
    return BigInt(sub(a.value, b.value));
  }
  friend BigInt operator+(BigInt a, BigInt b) {
    return BigInt(add(a.value, b.value));
  }
  friend BigInt operator/(BigInt a, BigInt b) {
    return BigInt(div(a.value, b.value));
  }
  friend BigInt operator*(BigInt a, BigInt b) {
    return BigInt(mul(a.value, b.value));
  }
  BigInt& operator+=(BigInt b) {
    value = add(value, b.value);
    return *this;
  }
  BigInt& operator-=(BigInt b) {
    value = sub(value, b.value);
    return *this;
  }
  BigInt& operator*=(BigInt b) {
    value = mul(value, b.value);
    return *this;
  }
  BigInt& operator/=(BigInt b) {
    value = div(value, b.value);
    return *this;
  }
};

UPDATE

  • 优化了实现，使用 reverse 和 insert 以取得更高的性能
implement
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string add(string a, string b);
string sub(string a, string b);
string mul(string a, string b);
string div(string a, string b);

string add(string a, string b) {
  bool neg = false;
  if (a[0] == '+') a = a.substr(1);
  if (b[0] == '+') b = b.substr(1);
  if (a[0] == '-' && b[0] == '-') {
    neg = true, a = a.substr(1), b = b.substr(1);
  } else if (a[0] == '-' && b[0] != '-')
    return sub(b, a.substr(1));
  else if (a[0] != '-' && b[0] == '-')
    return sub(a, b.substr(1));
  ull maxlen = max(a.size(), b.size()), r = a.size() - 1;
  string ans(maxlen + 1, '0');
  if (a.size() > b.size())
    b.insert(0, a.size() - b.size(), '0');
  else
    a.insert(0, b.size() - a.size(), '0');
  reverse(a.begin(), a.end());
  reverse(b.begin(), b.end());
  for (ull i = 0; i < maxlen; ++i) {
    ans[i + 1] = (ans[i] - '0' + a[i] - '0' + b[i] - '0') / 10 + '0';
    ans[i] = (ans[i] - '0' + a[i] - '0' + b[i] - '0') % 10 + '0';
  }
  r = maxlen;
  while (r > 0 && ans[r] == '0') r--;
  ans = ans.substr(0, r + 1);
  reverse(ans.begin(), ans.end());
  return neg ? "-" + ans : ans;
}

string sub(string a, string b) {
  bool neg = false;
  if (a[0] == '+') a = a.substr(1);
  if (b[0] == '+') b = b.substr(1);
  if (b[0] == '-') return add(a, b.substr(1));
  if (a[0] == '-') return "-" + add(a.substr(1), b);
  if (a.size() < b.size() || (a.size() == b.size() && a < b))
    (neg = true, swap(a, b));
  ull maxlen = max(a.size(), b.size()), r = a.size() - 1;
  b.insert(0, a.size() - b.size(), '0');
  reverse(a.begin(), a.end());
  reverse(b.begin(), b.end());
  string ans = a;
  for (ull i = 0; i < maxlen; ++i) {
    ans[i + 1] -= ans[i] >= b[i] ? 0 : 1;
    ans[i] = (ans[i] - b[i] + 10) % 10 + '0';
  }
  r = maxlen - 1;
  while (r > 0 && ans[r] == '0') r--;
  ans = ans.substr(0, r + 1);
  reverse(ans.begin(), ans.end());
  return neg ? "-" + ans : ans;
}

string mul(string a, string b) {
  bool neg = false;
  if (a[0] == '+') a = a.substr(1);
  if (b[0] == '+') b = b.substr(1);
  if (a[0] == '-' && b[0] != '-') {
    neg = true, a = a.substr(1);
  } else if (a[0] != '-' && b[0] == '-') {
    neg = true, b = b.substr(1);
  } else if (a[0] == '-' && a[0] == b[0]) {
    a = a.substr(1), b = b.substr(1);
  }
  ull maxlen = max(a.size(), b.size());
  ull r = a.size() - 1;
  string ans(maxlen * 2, '0');
  reverse(a.begin(), a.end());
  reverse(b.begin(), b.end());
  if (a.size() < b.size()) swap(a, b);
  for (ull i = 0; i < b.size(); ++i)
    for (ull j = 0; j < a.size(); ++j) {
      ans[i + j + 1] += (ans[i + j] - '0' + (a[j] - '0') * (b[i] - '0')) / 10;
      ans[i + j] = (ans[i + j] - '0' + (a[j] - '0') * (b[i] - '0')) % 10 + '0';
    }
  r = 2 * maxlen - 1;
  while (r > 0 && ans[r] == '0') r--;
  ans = ans.substr(0, r + 1);
  reverse(ans.begin(), ans.end());
  return neg && ans != "0" ? "-" + ans : ans;
}

string div(string a, string b) {
  bool neg = false;
  if (a[0] == '+') a = a.substr(1);
  if (b[0] == '+') b = b.substr(1);
  if (a[0] == '-' && b[0] != '-') {
    neg = true, a = a.substr(1);
  } else if (a[0] != '-' && b[0] == '-') {
    neg = true, b = b.substr(1);
  } else if (a[0] == '-' && a[0] == b[0]) {
    a = a.substr(1), b = b.substr(1);
  }
  string ans;
  if (a.size() < b.size())
    ans = "0";
  else {
    string remain = a.substr(0, b.size() - 1), temp;
    for (ull i = b.size() - 1; i < a.size(); ++i) {
      remain += a[i], remain = mul(remain, "1");
      if (remain.size() > b.size() ||
          (remain.size() == b.size() && remain >= b))
        for (char t = '9'; t >= '0'; --t) {
          temp = mul(b, string(1, t));
          if (temp.size() < remain.size() ||
              (temp.size() == remain.size() && temp <= remain)) {
            ans += t, remain = sub(remain, temp);
            break;
          }
        }
      else
        ans += '0';
    }
  }
  if (ans != "0") {
    int i = 0;
    while (ans[i] == '0') i++;
    ans = ans.substr(i);
  }
  return neg && ans != "0" ? "-" + ans : ans;
}