12tqian's Competitive Programming Library
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Big Integer
(library/numerical/big-integer.hpp)
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- Last update: 2022-07-21 16:12:33-04:00
- Include:
#include "library/numerical/big-integer.hpp"
Big Integer
A struct for big integer.
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#pragma once
// base and base_digits must be consistent
constexpr int base = 1000000000;
constexpr int base_digits = 9;
struct BigInt {
// value == 0 is represented by empty z
std::vector<int> z; // digits
// sign == 1 <==> value >= 0
// sign == -1 <==> value < 0
int sign;
BigInt() : sign(1) {}
BigInt(long long v) { *this = v; }
BigInt& operator=(long long v) {
sign = v < 0 ? -1 : 1; v *= sign;
z.clear(); for (; v > 0; v = v / base) z.push_back((int) (v % base));
return *this;
}
BigInt(const std::string& s) { read(s); }
BigInt& operator+=(const BigInt& other) {
if (sign == other.sign) {
for (int i = 0, carry = 0; i < other.z.size() || carry; ++i) {
if (i == z.size())
z.push_back(0);
z[i] += carry + (i < other.z.size() ? other.z[i] : 0);
carry = z[i] >= base;
if (carry)
z[i] -= base;
}
} else if (other != 0 /* prevent infinite loop */) {
*this -= -other;
}
return *this;
}
friend BigInt operator+(BigInt a, const BigInt& b) { return a += b; }
BigInt& operator-=(const BigInt& other) {
if (sign == other.sign) {
if (sign == 1 && *this >= other || sign == -1 && *this <= other) {
for (int i = 0, carry = 0; i < other.z.size() || carry; ++i) {
z[i] -= carry + (i < other.z.size() ? other.z[i] : 0);
carry = z[i] < 0;
if (carry)
z[i] += base;
}
trim();
} else {
*this = other - *this;
this->sign = -this->sign;
}
} else {
*this += -other;
}
return *this;
}
friend BigInt operator-(BigInt a, const BigInt& b) { return a -= b; }
BigInt& operator*=(int v) {
if (v < 0) sign = -sign, v = -v;
for (int i = 0, carry = 0; i < z.size() || carry; ++i) {
if (i == z.size())
z.push_back(0);
long long cur = (long long) z[i] * v + carry;
carry = (int) (cur / base);
z[i] = (int) (cur % base);
}
trim();
return *this;
}
BigInt operator*(int v) const { return BigInt(*this) *= v; }
friend std::pair<BigInt, BigInt> divmod(const BigInt& a1, const BigInt& b1) {
int norm = base / (b1.z.back() + 1);
BigInt a = a1.abs() * norm;
BigInt b = b1.abs() * norm;
BigInt q, r;
q.z.resize(a.z.size());
for (int i = (int) a.z.size() - 1; i >= 0; i--) {
r *= base;
r += a.z[i];
int s1 = b.z.size() < r.z.size() ? r.z[b.z.size()] : 0;
int s2 = b.z.size() - 1 < r.z.size() ? r.z[b.z.size() - 1] : 0;
int d = (int) (((long long) s1 * base + s2) / b.z.back());
r -= b * d;
while (r < 0)
r += b, --d;
q.z[i] = d;
}
q.sign = a1.sign * b1.sign;
r.sign = a1.sign;
q.trim();
r.trim();
return {q, r / norm};
}
friend BigInt sqrt(const BigInt& a1) {
BigInt a = a1;
while (a.z.empty() || a.z.size() % 2 == 1)
a.z.push_back(0);
int n = a.z.size();
int firstDigit = (int) ::sqrt((double) a.z[n - 1] * base + a.z[n - 2]);
int norm = base / (firstDigit + 1);
a *= norm;
a *= norm;
while (a.z.empty() || a.z.size() % 2 == 1)
a.z.push_back(0);
BigInt r = (long long) a.z[n - 1] * base + a.z[n - 2];
firstDigit = (int) ::sqrt((double) a.z[n - 1] * base + a.z[n - 2]);
int q = firstDigit;
BigInt res;
for (int j = n / 2 - 1; j >= 0; j--) {
for (;; --q) {
BigInt r1 = (r - (res * 2 * base + q) * q) * base * base +
(j > 0 ? (long long) a.z[2 * j - 1] * base + a.z[2 * j - 2] : 0);
if (r1 >= 0) {
r = r1;
break;
}
}
res *= base;
res += q;
if (j > 0) {
int d1 = res.z.size() + 2 < r.z.size() ? r.z[res.z.size() + 2] : 0;
int d2 = res.z.size() + 1 < r.z.size() ? r.z[res.z.size() + 1] : 0;
int d3 = res.z.size() < r.z.size() ? r.z[res.z.size()] : 0;
q = (int) (((long long) d1 * base * base + (long long) d2 * base + d3) / (firstDigit * 2));
}
}
res.trim();
return res / norm;
}
BigInt operator/(const BigInt& v) const { return divmod(*this, v).first; }
BigInt operator%(const BigInt& v) const { return divmod(*this, v).second; }
BigInt& operator/=(int v) {
if (v < 0) sign = -sign, v = -v;
for (int i = (int) z.size() - 1, rem = 0; i >= 0; --i) {
long long cur = z[i] + rem * (long long) base;
z[i] = (int) (cur / v);
rem = (int) (cur % v);
}
trim();
return *this;
}
BigInt operator/(int v) const { return BigInt(*this) /= v; }
int operator%(int v) const {
if (v < 0) v = -v;
int m = 0;
for (int i = (int) z.size() - 1; i >= 0; --i)
m = (int) ((z[i] + m * (long long) base) % v);
return m * sign;
}
BigInt& operator*=(const BigInt& v) { return *this = *this * v; }
BigInt& operator/=(const BigInt& v) { return *this = *this / v; }
bool operator<(const BigInt& v) const {
if (sign != v.sign)
return sign < v.sign;
if (z.size() != v.z.size())
return z.size() * sign < v.z.size() * v.sign;
for (int i = (int) z.size() - 1; i >= 0; i--)
if (z[i] != v.z[i])
return z[i] * sign < v.z[i] * sign;
return false;
}
bool operator>(const BigInt& v) const { return v < *this; }
bool operator<=(const BigInt& v) const { return !(v < *this); }
bool operator>=(const BigInt& v) const { return !(*this < v); }
bool operator==(const BigInt& v) const { return !(*this < v) && !(v < *this); }
bool operator!=(const BigInt& v) const { return *this < v || v < *this; }
void trim() {
while (!z.empty() && z.back() == 0) z.pop_back();
if (z.empty()) sign = 1;
}
bool is_zero() const { return z.empty(); }
friend BigInt operator-(BigInt v) {
if (!v.z.empty()) v.sign = -v.sign;
return v;
}
BigInt abs() const { return sign == 1 ? *this : -*this; }
long long longValue() const {
long long res = 0;
for (int i = (int) z.size() - 1; i >= 0; i--)
res = res * base + z[i];
return res * sign;
}
friend BigInt gcd(const BigInt& a, const BigInt& b) {
return b.is_zero() ? a : gcd(b, a % b);
}
friend BigInt lcm(const BigInt& a, const BigInt& b) {
return a / gcd(a, b) * b;
}
void read(const std::string& s) {
sign = 1;
z.clear();
int pos = 0;
while (pos < s.size() && (s[pos] == '-' || s[pos] == '+')) {
if (s[pos] == '-')
sign = -sign;
++pos;
}
for (int i = (int) s.size() - 1; i >= pos; i -= base_digits) {
int x = 0;
for (int j = std::max(pos, i - base_digits + 1); j <= i; j++)
x = x * 10 + s[j] - '0';
z.push_back(x);
}
trim();
}
friend std::istream& operator>>(std::istream& stream, BigInt& v) {
std::string s; stream >> s;
v.read(s);
return stream;
}
friend std::ostream& operator<<(std::ostream& stream, const BigInt& v) {
if (v.sign == -1)
stream << '-';
stream << (v.z.empty() ? 0 : v.z.back());
for (int i = (int) v.z.size() - 2; i >= 0; --i)
stream << std::setw(base_digits) << std::setfill('0') << v.z[i];
return stream;
}
static std::vector<int> convert_base(const std::vector<int>& a, int old_digits, int new_digits) {
std::vector<long long> p(std::max(old_digits, new_digits) + 1);
p[0] = 1;
for (int i = 1; i < p.size(); i++)
p[i] = p[i - 1] * 10;
std::vector<int> res;
long long cur = 0;
int cur_digits = 0;
for (int v : a) {
cur += v * p[cur_digits];
cur_digits += old_digits;
while (cur_digits >= new_digits) {
res.push_back(int(cur % p[new_digits]));
cur /= p[new_digits];
cur_digits -= new_digits;
}
}
res.push_back((int) cur);
while (!res.empty() && res.back() == 0)
res.pop_back();
return res;
}
static std::vector<long long> karatsubaMultiply(const std::vector<long long>& a, const std::vector<long long>& b) {
int n = a.size();
std::vector<long long> res(n + n);
if (n <= 32) {
for (int i = 0; i < n; i++)
for (int j = 0; j < n; j++)
res[i + j] += a[i] * b[j];
return res;
}
int k = n >> 1;
std::vector<long long> a1(a.begin(), a.begin() + k);
std::vector<long long> a2(a.begin() + k, a.end());
std::vector<long long> b1(b.begin(), b.begin() + k);
std::vector<long long> b2(b.begin() + k, b.end());
std::vector<long long> a1b1 = karatsubaMultiply(a1, b1);
std::vector<long long> a2b2 = karatsubaMultiply(a2, b2);
for (int i = 0; i < k; i++)
a2[i] += a1[i];
for (int i = 0; i < k; i++)
b2[i] += b1[i];
std::vector<long long> r = karatsubaMultiply(a2, b2);
for (int i = 0; i < a1b1.size(); i++)
r[i] -= a1b1[i];
for (int i = 0; i < a2b2.size(); i++)
r[i] -= a2b2[i];
for (int i = 0; i < r.size(); i++)
res[i + k] += r[i];
for (int i = 0; i < a1b1.size(); i++)
res[i] += a1b1[i];
for (int i = 0; i < a2b2.size(); i++)
res[i + n] += a2b2[i];
return res;
}
BigInt operator*(const BigInt& v) const {
std::vector<int> a6 = convert_base(this->z, base_digits, 6);
std::vector<int> b6 = convert_base(v.z, base_digits, 6);
std::vector<long long> a(a6.begin(), a6.end());
std::vector<long long> b(b6.begin(), b6.end());
while (a.size() < b.size())
a.push_back(0);
while (b.size() < a.size())
b.push_back(0);
while (a.size() & (a.size() - 1))
a.push_back(0), b.push_back(0);
std::vector<long long> c = karatsubaMultiply(a, b);
BigInt res;
res.sign = sign * v.sign;
for (int i = 0, carry = 0; i < c.size(); i++) {
long long cur = c[i] + carry;
res.z.push_back((int) (cur % 1000000));
carry = (int) (cur / 1000000);
}
res.z = convert_base(res.z, 6, base_digits);
res.trim();
return res;
}
};
BigInt random_big_int(int n) {
std::string s;
for (int i = 0; i < n; i++) {
s += rand() % 10 + '0';
}
return BigInt(s);
}// base and base_digits must be consistent
constexpr int base = 1000000000;
constexpr int base_digits = 9;
struct BigInt {
// value == 0 is represented by empty z
std::vector<int> z; // digits
// sign == 1 <==> value >= 0
// sign == -1 <==> value < 0
int sign;
BigInt() : sign(1) {}
BigInt(long long v) { *this = v; }
BigInt& operator=(long long v) {
sign = v < 0 ? -1 : 1; v *= sign;
z.clear(); for (; v > 0; v = v / base) z.push_back((int) (v % base));
return *this;
}
BigInt(const std::string& s) { read(s); }
BigInt& operator+=(const BigInt& other) {
if (sign == other.sign) {
for (int i = 0, carry = 0; i < other.z.size() || carry; ++i) {
if (i == z.size())
z.push_back(0);
z[i] += carry + (i < other.z.size() ? other.z[i] : 0);
carry = z[i] >= base;
if (carry)
z[i] -= base;
}
} else if (other != 0 /* prevent infinite loop */) {
*this -= -other;
}
return *this;
}
friend BigInt operator+(BigInt a, const BigInt& b) { return a += b; }
BigInt& operator-=(const BigInt& other) {
if (sign == other.sign) {
if (sign == 1 && *this >= other || sign == -1 && *this <= other) {
for (int i = 0, carry = 0; i < other.z.size() || carry; ++i) {
z[i] -= carry + (i < other.z.size() ? other.z[i] : 0);
carry = z[i] < 0;
if (carry)
z[i] += base;
}
trim();
} else {
*this = other - *this;
this->sign = -this->sign;
}
} else {
*this += -other;
}
return *this;
}
friend BigInt operator-(BigInt a, const BigInt& b) { return a -= b; }
BigInt& operator*=(int v) {
if (v < 0) sign = -sign, v = -v;
for (int i = 0, carry = 0; i < z.size() || carry; ++i) {
if (i == z.size())
z.push_back(0);
long long cur = (long long) z[i] * v + carry;
carry = (int) (cur / base);
z[i] = (int) (cur % base);
}
trim();
return *this;
}
BigInt operator*(int v) const { return BigInt(*this) *= v; }
friend std::pair<BigInt, BigInt> divmod(const BigInt& a1, const BigInt& b1) {
int norm = base / (b1.z.back() + 1);
BigInt a = a1.abs() * norm;
BigInt b = b1.abs() * norm;
BigInt q, r;
q.z.resize(a.z.size());
for (int i = (int) a.z.size() - 1; i >= 0; i--) {
r *= base;
r += a.z[i];
int s1 = b.z.size() < r.z.size() ? r.z[b.z.size()] : 0;
int s2 = b.z.size() - 1 < r.z.size() ? r.z[b.z.size() - 1] : 0;
int d = (int) (((long long) s1 * base + s2) / b.z.back());
r -= b * d;
while (r < 0)
r += b, --d;
q.z[i] = d;
}
q.sign = a1.sign * b1.sign;
r.sign = a1.sign;
q.trim();
r.trim();
return {q, r / norm};
}
friend BigInt sqrt(const BigInt& a1) {
BigInt a = a1;
while (a.z.empty() || a.z.size() % 2 == 1)
a.z.push_back(0);
int n = a.z.size();
int firstDigit = (int) ::sqrt((double) a.z[n - 1] * base + a.z[n - 2]);
int norm = base / (firstDigit + 1);
a *= norm;
a *= norm;
while (a.z.empty() || a.z.size() % 2 == 1)
a.z.push_back(0);
BigInt r = (long long) a.z[n - 1] * base + a.z[n - 2];
firstDigit = (int) ::sqrt((double) a.z[n - 1] * base + a.z[n - 2]);
int q = firstDigit;
BigInt res;
for (int j = n / 2 - 1; j >= 0; j--) {
for (;; --q) {
BigInt r1 = (r - (res * 2 * base + q) * q) * base * base +
(j > 0 ? (long long) a.z[2 * j - 1] * base + a.z[2 * j - 2] : 0);
if (r1 >= 0) {
r = r1;
break;
}
}
res *= base;
res += q;
if (j > 0) {
int d1 = res.z.size() + 2 < r.z.size() ? r.z[res.z.size() + 2] : 0;
int d2 = res.z.size() + 1 < r.z.size() ? r.z[res.z.size() + 1] : 0;
int d3 = res.z.size() < r.z.size() ? r.z[res.z.size()] : 0;
q = (int) (((long long) d1 * base * base + (long long) d2 * base + d3) / (firstDigit * 2));
}
}
res.trim();
return res / norm;
}
BigInt operator/(const BigInt& v) const { return divmod(*this, v).first; }
BigInt operator%(const BigInt& v) const { return divmod(*this, v).second; }
BigInt& operator/=(int v) {
if (v < 0) sign = -sign, v = -v;
for (int i = (int) z.size() - 1, rem = 0; i >= 0; --i) {
long long cur = z[i] + rem * (long long) base;
z[i] = (int) (cur / v);
rem = (int) (cur % v);
}
trim();
return *this;
}
BigInt operator/(int v) const { return BigInt(*this) /= v; }
int operator%(int v) const {
if (v < 0) v = -v;
int m = 0;
for (int i = (int) z.size() - 1; i >= 0; --i)
m = (int) ((z[i] + m * (long long) base) % v);
return m * sign;
}
BigInt& operator*=(const BigInt& v) { return *this = *this * v; }
BigInt& operator/=(const BigInt& v) { return *this = *this / v; }
bool operator<(const BigInt& v) const {
if (sign != v.sign)
return sign < v.sign;
if (z.size() != v.z.size())
return z.size() * sign < v.z.size() * v.sign;
for (int i = (int) z.size() - 1; i >= 0; i--)
if (z[i] != v.z[i])
return z[i] * sign < v.z[i] * sign;
return false;
}
bool operator>(const BigInt& v) const { return v < *this; }
bool operator<=(const BigInt& v) const { return !(v < *this); }
bool operator>=(const BigInt& v) const { return !(*this < v); }
bool operator==(const BigInt& v) const { return !(*this < v) && !(v < *this); }
bool operator!=(const BigInt& v) const { return *this < v || v < *this; }
void trim() {
while (!z.empty() && z.back() == 0) z.pop_back();
if (z.empty()) sign = 1;
}
bool is_zero() const { return z.empty(); }
friend BigInt operator-(BigInt v) {
if (!v.z.empty()) v.sign = -v.sign;
return v;
}
BigInt abs() const { return sign == 1 ? *this : -*this; }
long long longValue() const {
long long res = 0;
for (int i = (int) z.size() - 1; i >= 0; i--)
res = res * base + z[i];
return res * sign;
}
friend BigInt gcd(const BigInt& a, const BigInt& b) {
return b.is_zero() ? a : gcd(b, a % b);
}
friend BigInt lcm(const BigInt& a, const BigInt& b) {
return a / gcd(a, b) * b;
}
void read(const std::string& s) {
sign = 1;
z.clear();
int pos = 0;
while (pos < s.size() && (s[pos] == '-' || s[pos] == '+')) {
if (s[pos] == '-')
sign = -sign;
++pos;
}
for (int i = (int) s.size() - 1; i >= pos; i -= base_digits) {
int x = 0;
for (int j = std::max(pos, i - base_digits + 1); j <= i; j++)
x = x * 10 + s[j] - '0';
z.push_back(x);
}
trim();
}
friend std::istream& operator>>(std::istream& stream, BigInt& v) {
std::string s; stream >> s;
v.read(s);
return stream;
}
friend std::ostream& operator<<(std::ostream& stream, const BigInt& v) {
if (v.sign == -1)
stream << '-';
stream << (v.z.empty() ? 0 : v.z.back());
for (int i = (int) v.z.size() - 2; i >= 0; --i)
stream << std::setw(base_digits) << std::setfill('0') << v.z[i];
return stream;
}
static std::vector<int> convert_base(const std::vector<int>& a, int old_digits, int new_digits) {
std::vector<long long> p(std::max(old_digits, new_digits) + 1);
p[0] = 1;
for (int i = 1; i < p.size(); i++)
p[i] = p[i - 1] * 10;
std::vector<int> res;
long long cur = 0;
int cur_digits = 0;
for (int v : a) {
cur += v * p[cur_digits];
cur_digits += old_digits;
while (cur_digits >= new_digits) {
res.push_back(int(cur % p[new_digits]));
cur /= p[new_digits];
cur_digits -= new_digits;
}
}
res.push_back((int) cur);
while (!res.empty() && res.back() == 0)
res.pop_back();
return res;
}
static std::vector<long long> karatsubaMultiply(const std::vector<long long>& a, const std::vector<long long>& b) {
int n = a.size();
std::vector<long long> res(n + n);
if (n <= 32) {
for (int i = 0; i < n; i++)
for (int j = 0; j < n; j++)
res[i + j] += a[i] * b[j];
return res;
}
int k = n >> 1;
std::vector<long long> a1(a.begin(), a.begin() + k);
std::vector<long long> a2(a.begin() + k, a.end());
std::vector<long long> b1(b.begin(), b.begin() + k);
std::vector<long long> b2(b.begin() + k, b.end());
std::vector<long long> a1b1 = karatsubaMultiply(a1, b1);
std::vector<long long> a2b2 = karatsubaMultiply(a2, b2);
for (int i = 0; i < k; i++)
a2[i] += a1[i];
for (int i = 0; i < k; i++)
b2[i] += b1[i];
std::vector<long long> r = karatsubaMultiply(a2, b2);
for (int i = 0; i < a1b1.size(); i++)
r[i] -= a1b1[i];
for (int i = 0; i < a2b2.size(); i++)
r[i] -= a2b2[i];
for (int i = 0; i < r.size(); i++)
res[i + k] += r[i];
for (int i = 0; i < a1b1.size(); i++)
res[i] += a1b1[i];
for (int i = 0; i < a2b2.size(); i++)
res[i + n] += a2b2[i];
return res;
}
BigInt operator*(const BigInt& v) const {
std::vector<int> a6 = convert_base(this->z, base_digits, 6);
std::vector<int> b6 = convert_base(v.z, base_digits, 6);
std::vector<long long> a(a6.begin(), a6.end());
std::vector<long long> b(b6.begin(), b6.end());
while (a.size() < b.size())
a.push_back(0);
while (b.size() < a.size())
b.push_back(0);
while (a.size() & (a.size() - 1))
a.push_back(0), b.push_back(0);
std::vector<long long> c = karatsubaMultiply(a, b);
BigInt res;
res.sign = sign * v.sign;
for (int i = 0, carry = 0; i < c.size(); i++) {
long long cur = c[i] + carry;
res.z.push_back((int) (cur % 1000000));
carry = (int) (cur / 1000000);
}
res.z = convert_base(res.z, 6, base_digits);
res.trim();
return res;
}
};
BigInt random_big_int(int n) {
std::string s;
for (int i = 0; i < n; i++) {
s += rand() % 10 + '0';
}
return BigInt(s);
}