大数模板

  1 const int base = 1000000000;
  2 const int base_digits = 9;
  3 struct bigint {
  4     vector<int> z; 
  5     int sign;
  6     bigint() : sign(1) {}
  7     bigint(long long v) { *this = v; }
  8     bigint &operator=(long long v) {
  9         sign = v < 0 ? -1 : 1;
 10         v *= sign;
 11         z.clear();
 12         for (; v > 0; v = v / base) z.push_back((int)(v % base));
 13         return *this;
 14     }
 15 
 16     bigint(const string &s) { read(s); }
 17 
 18     bigint &operator+=(const bigint &other) {
 19         if (sign == other.sign) {
 20             for (int i = 0, carry = 0; i < other.z.size() || carry; ++i) {
 21                 if (i == z.size())
 22                     z.push_back(0);
 23                 z[i] += carry + (i < other.z.size() ? other.z[i] : 0);
 24                 carry = z[i] >= base;
 25                 if (carry)
 26                     z[i] -= base;
 27             }
 28         }
 29         else if (other != 0 /* prevent infinite loop */) {
 30             *this -= -other;
 31         }
 32         return *this;
 33     }
 34 
 35     friend bigint operator+(bigint a, const bigint &b) { return a += b; }
 36 
 37     bigint &operator-=(const bigint &other) {
 38         if (sign == other.sign) {
 39             if (sign == 1 && *this >= other || sign == -1 && *this <= other) {
 40                 for (int i = 0, carry = 0; i < other.z.size() || carry; ++i) {
 41                     z[i] -= carry + (i < other.z.size() ? other.z[i] : 0);
 42                     carry = z[i] < 0;
 43                     if (carry)
 44                         z[i] += base;
 45                 }
 46                 trim();
 47             }
 48             else {
 49                 *this = other - *this;
 50                 this->sign = -this->sign;
 51             }
 52         }
 53         else {
 54             *this += -other;
 55         }
 56         return *this;
 57     }
 58 
 59     friend bigint operator-(bigint a, const bigint &b) { return a -= b; }
 60 
 61     bigint &operator*=(int v) {
 62         if (v < 0) sign = -sign, v = -v;
 63         for (int i = 0, carry = 0; i < z.size() || carry; ++i) {
 64             if (i == z.size())
 65                 z.push_back(0);
 66             long long cur = (long long)z[i] * v + carry;
 67             carry = (int)(cur / base);
 68             z[i] = (int)(cur % base);
 69         }
 70         trim();
 71         return *this;
 72     }
 73 
 74     bigint operator*(int v) const { return bigint(*this) *= v; }
 75 
 76     friend pair<bigint, bigint> divmod(const bigint &a1, const bigint &b1) {
 77         int norm = base / (b1.z.back() + 1);
 78         bigint a = a1.abs() * norm;
 79         bigint b = b1.abs() * norm;
 80         bigint q, r;
 81         q.z.resize(a.z.size());
 82 
 83         for (int i = (int)a.z.size() - 1; i >= 0; i--) {
 84             r *= base;
 85             r += a.z[i];
 86             int s1 = b.z.size() < r.z.size() ? r.z[b.z.size()] : 0;
 87             int s2 = b.z.size() - 1 < r.z.size() ? r.z[b.z.size() - 1] : 0;
 88             int d = (int)(((long long)s1 * base + s2) / b.z.back());
 89             r -= b * d;
 90             while (r < 0)
 91                 r += b, --d;
 92             q.z[i] = d;
 93         }
 94 
 95         q.sign = a1.sign * b1.sign;
 96         r.sign = a1.sign;
 97         q.trim();
 98         r.trim();
 99         return { q, r / norm };
100     }
101 
102     friend bigint sqrt(const bigint &a1) {
103         bigint a = a1;
104         while (a.z.empty() || a.z.size() % 2 == 1)
105             a.z.push_back(0);
106 
107         int n = a.z.size();
108 
109         int firstDigit = (int) ::sqrt((double)a.z[n - 1] * base + a.z[n - 2]);
110         int norm = base / (firstDigit + 1);
111         a *= norm;
112         a *= norm;
113         while (a.z.empty() || a.z.size() % 2 == 1)
114             a.z.push_back(0);
115 
116         bigint r = (long long)a.z[n - 1] * base + a.z[n - 2];
117         firstDigit = (int) ::sqrt((double)a.z[n - 1] * base + a.z[n - 2]);
118         int q = firstDigit;
119         bigint res;
120 
121         for (int j = n / 2 - 1; j >= 0; j--) {
122             for (;; --q) {
123                 bigint r1 = (r - (res * 2 * base + q) * q) * base * base +
124                     (j > 0 ? (long long)a.z[2 * j - 1] * base + a.z[2 * j - 2] : 0);
125                 if (r1 >= 0) {
126                     r = r1;
127                     break;
128                 }
129             }
130             res *= base;
131             res += q;
132 
133             if (j > 0) {
134                 int d1 = res.z.size() + 2 < r.z.size() ? r.z[res.z.size() + 2] : 0;
135                 int d2 = res.z.size() + 1 < r.z.size() ? r.z[res.z.size() + 1] : 0;
136                 int d3 = res.z.size() < r.z.size() ? r.z[res.z.size()] : 0;
137                 q = (int)(((long long)d1 * base * base + (long long)d2 * base + d3) / (firstDigit * 2));
138             }
139         }
140 
141         res.trim();
142         return res / norm;
143     }
144 
145     bigint operator/(const bigint &v) const { return divmod(*this, v).first; }
146 
147     bigint operator%(const bigint &v) const { return divmod(*this, v).second; }
148 
149     bigint &operator/=(int v) {
150         if (v < 0) sign = -sign, v = -v;
151         for (int i = (int)z.size() - 1, rem = 0; i >= 0; --i) {
152             long long cur = z[i] + rem * (long long)base;
153             z[i] = (int)(cur / v);
154             rem = (int)(cur % v);
155         }
156         trim();
157         return *this;
158     }
159 
160     bigint operator/(int v) const { return bigint(*this) /= v; }
161 
162     int operator%(int v) const {
163         if (v < 0) v = -v;
164         int m = 0;
165         for (int i = (int)z.size() - 1; i >= 0; --i)
166             m = (int)((z[i] + m * (long long)base) % v);
167         return m * sign;
168     }
169 
170     bigint &operator*=(const bigint &v) { return *this = *this * v; }
171 
172     bigint &operator/=(const bigint &v) { return *this = *this / v; }
173 
174     bool operator<(const bigint &v) const {
175         if (sign != v.sign)
176             return sign < v.sign;
177         if (z.size() != v.z.size())
178             return z.size() * sign < v.z.size() * v.sign;
179         for (int i = (int)z.size() - 1; i >= 0; i--)
180             if (z[i] != v.z[i])
181                 return z[i] * sign < v.z[i] * sign;
182         return false;
183     }
184 
185     bool operator>(const bigint &v) const { return v < *this; }
186 
187     bool operator<=(const bigint &v) const { return !(v < *this); }
188 
189     bool operator>=(const bigint &v) const { return !(*this < v); }
190 
191     bool operator==(const bigint &v) const { return !(*this < v) && !(v < *this); }
192 
193     bool operator!=(const bigint &v) const { return *this < v || v < *this; }
194 
195     void trim() {
196         while (!z.empty() && z.back() == 0) z.pop_back();
197         if (z.empty()) sign = 1;
198     }
199 
200     bool isZero() const { return z.empty(); }
201 
202     friend bigint operator-(bigint v) {
203         if (!v.z.empty()) v.sign = -v.sign;
204         return v;
205     }
206 
207     bigint abs() const {
208         return sign == 1 ? *this : -*this;
209     }
210 
211     long long longValue() const {
212         long long res = 0;
213         for (int i = (int)z.size() - 1; i >= 0; i--)
214             res = res * base + z[i];
215         return res * sign;
216     }
217 
218     friend bigint gcd(const bigint &a, const bigint &b) {
219         return b.isZero() ? a : gcd(b, a % b);
220     }
221 
222     friend bigint lcm(const bigint &a, const bigint &b) {
223         return a / gcd(a, b) * b;
224     }
225 
226     void read(const string &s) {
227         sign = 1;
228         z.clear();
229         int pos = 0;
230         while (pos < s.size() && (s[pos] == ‘-‘ || s[pos] == ‘+‘)) {
231             if (s[pos] == ‘-‘)
232                 sign = -sign;
233             ++pos;
234         }
235         for (int i = (int)s.size() - 1; i >= pos; i -= base_digits) {
236             int x = 0;
237             for (int j = max(pos, i - base_digits + 1); j <= i; j++)
238                 x = x * 10 + s[j] - ‘0‘;
239             z.push_back(x);
240         }
241         trim();
242     }
243 
244     friend istream &operator>>(istream &stream, bigint &v) {
245         string s;
246         stream >> s;
247         v.read(s);
248         return stream;
249     }
250 
251     friend ostream &operator<<(ostream &stream, const bigint &v) {
252         if (v.sign == -1)
253             stream << ‘-‘;
254         stream << (v.z.empty() ? 0 : v.z.back());
255         for (int i = (int)v.z.size() - 2; i >= 0; --i)
256             stream << setw(base_digits) << setfill(‘0‘) << v.z[i];
257         return stream;
258     }
259 
260     static vector<int> convert_base(const vector<int> &a, int old_digits, int new_digits) {
261         vector<long long> p(max(old_digits, new_digits) + 1);
262         p[0] = 1;
263         for (int i = 1; i < p.size(); i++)
264             p[i] = p[i - 1] * 10;
265         vector<int> res;
266         long long cur = 0;
267         int cur_digits = 0;
268         for (int v : a) {
269             cur += v * p[cur_digits];
270             cur_digits += old_digits;
271             while (cur_digits >= new_digits) {
272                 res.push_back(int(cur % p[new_digits]));
273                 cur /= p[new_digits];
274                 cur_digits -= new_digits;
275             }
276         }
277         res.push_back((int)cur);
278         while (!res.empty() && res.back() == 0)
279             res.pop_back();
280         return res;
281     }
282 
283     typedef vector<long long> vll;
284 
285     static vll karatsubaMultiply(const vll &a, const vll &b) {
286         int n = a.size();
287         vll res(n + n);
288         if (n <= 32) {
289             for (int i = 0; i < n; i++)
290                 for (int j = 0; j < n; j++)
291                     res[i + j] += a[i] * b[j];
292             return res;
293         }
294 
295         int k = n >> 1;
296         vll a1(a.begin(), a.begin() + k);
297         vll a2(a.begin() + k, a.end());
298         vll b1(b.begin(), b.begin() + k);
299         vll b2(b.begin() + k, b.end());
300 
301         vll a1b1 = karatsubaMultiply(a1, b1);
302         vll a2b2 = karatsubaMultiply(a2, b2);
303 
304         for (int i = 0; i < k; i++)
305             a2[i] += a1[i];
306         for (int i = 0; i < k; i++)
307             b2[i] += b1[i];
308 
309         vll r = karatsubaMultiply(a2, b2);
310         for (int i = 0; i < a1b1.size(); i++)
311             r[i] -= a1b1[i];
312         for (int i = 0; i < a2b2.size(); i++)
313             r[i] -= a2b2[i];
314 
315         for (int i = 0; i < r.size(); i++)
316             res[i + k] += r[i];
317         for (int i = 0; i < a1b1.size(); i++)
318             res[i] += a1b1[i];
319         for (int i = 0; i < a2b2.size(); i++)
320             res[i + n] += a2b2[i];
321         return res;
322     }
323 
324     bigint operator*(const bigint &v) const {
325         vector<int> a6 = convert_base(this->z, base_digits, 6);
326         vector<int> b6 = convert_base(v.z, base_digits, 6);
327         vll a(a6.begin(), a6.end());
328         vll b(b6.begin(), b6.end());
329         while (a.size() < b.size())
330             a.push_back(0);
331         while (b.size() < a.size())
332             b.push_back(0);
333         while (a.size() & (a.size() - 1))
334             a.push_back(0), b.push_back(0);
335         vll c = karatsubaMultiply(a, b);
336         bigint res;
337         res.sign = sign * v.sign;
338         for (int i = 0, carry = 0; i < c.size(); i++) {
339             long long cur = c[i] + carry;
340             res.z.push_back((int)(cur % 1000000));
341             carry = (int)(cur / 1000000);
342         }
343         res.z = convert_base(res.z, 6, base_digits);
344         res.trim();
345         return res;
346     }
347 
348 };