高精度模板
完整模板 1
作者：小黑AWM + MashPlant
注：可以直接把BigInt和int一样用cin、cout都行，就是高精乘为了速度才用了FFT降低了精度，有需要可以自行更改。
#include <cstdio>
#include <iostream>
#include <cmath>
#include <string>
#include <cstring>
#include <vector>
#include <algorithm>
using namespace std;
const double PI = acos(-1.0);
struct Complex{
double x,y;
Complex(double _x = 0.0,double _y = 0.0){
x = _x;
y = _y;
}
Complex operator-(const Complex &b)const{
return Complex(x - b.x,y - b.y);
}
Complex operator+(const Complex &b)const{
return Complex(x + b.x,y + b.y);
}
Complex operator*(const Complex &b)const{
return Complex(x*b.x - y*b.y,x*b.y + y*b.x);
}
};
void change(Complex y[],int len){
int i,j,k;
for(int i = 1,j = len/2;i<len-1;i++){
if(i < j) swap(y[i],y[j]);
k = len/2;
while(j >= k){
j = j - k;
k = k/2;
}
if(j < k) j+=k;
}
}
void fft(Complex y[],int len,int on){
change(y,len);
for(int h = 2;h <= len;h<<=1){
Complex wn(cos(on*2*PI/h),sin(on*2*PI/h));
for(int j = 0;j < len;j += h){
Complex w(1,0);
for(int k = j;k < j + h/2;k++){
Complex u = y[k];
Complex t = w*y[k + h/2];
y[k] = u + t;
y[k + h/2] = u - t;
w = w*wn;
}
}
}
if(on == -1){
for(int i = 0;i < len;i++){
y[i].x /= len;
}
}
}
class BigInt
{
#define Value(x, nega) ((nega) ? -(x) : (x))
#define At(vec, index) ((index) < vec.size() ? vec[(index)] : 0)
static int absComp(const BigInt &lhs, const BigInt &rhs)
{
if (lhs.size() != rhs.size())
return lhs.size() < rhs.size() ? -1 : 1;
for (int i = lhs.size() - 1; i >= 0; --i)
if (lhs[i] != rhs[i])
return lhs[i] < rhs[i] ? -1 : 1;
return 0;
}
using Long = long long;
const static int Exp = 9;
const static Long Mod = 1000000000;
mutable std::vector<Long> val;
mutable bool nega = false;
void trim() const
{
while (val.size() && val.back() == 0)
val.pop_back();
if (val.empty())
nega = false;
}
int size() const { return val.size(); }
Long &operator[](int index) const { return val[index]; }
Long &back() const { return val.back(); }
BigInt(int size, bool nega) : val(size), nega(nega) {}
BigInt(const std::vector<Long> &val, bool nega) : val(val), nega(nega) {}
public:
friend std::ostream &operator<<(std::ostream &os, const BigInt &n)
{
if (n.size())
{
if (n.nega)
putchar('-');
for (int i = n.size() - 1; i >= 0; --i)
{
if (i == n.size() - 1)
printf("%lld", n[i]);
else
printf("%0*lld", n.Exp, n[i]);
}
}
else
putchar('0');
return os;
}
friend BigInt operator+(const BigInt &lhs, const BigInt &rhs)
{
BigInt ret(lhs);
return ret += rhs;
}
friend BigInt operator-(const BigInt &lhs, const BigInt &rhs)
{
BigInt ret(lhs);
return ret -= rhs;
}
BigInt(Long x = 0)
{
if (x < 0)
x = -x, nega = true;
while (x >= Mod)
val.push_back(x % Mod), x /= Mod;
if (x)
val.push_back(x);
}
BigInt(const char *s)
{
int bound = 0, pos;
if (s[0] == '-')
nega = true, bound = 1;
Long cur = 0, pow = 1;
for (pos = strlen(s) - 1; pos >= Exp + bound - 1; pos -= Exp, val.push_back(cur), cur = 0, pow = 1)
for (int i = pos; i > pos - Exp; --i)
cur += (s[i] - '0') * pow, pow *= 10;
for (cur = 0, pow = 1; pos >= bound; --pos)
cur += (s[pos] - '0') * pow, pow *= 10;
if (cur)
val.push_back(cur);
}
BigInt &operator=(const char *s){
BigInt n(s);
*this = n;
return n;
}
BigInt &operator=(const Long x){
BigInt n(x);
*this = n;
return n;
}
friend std::istream &operator>>(std::istream &is, BigInt &n){
string s;
is >> s;
n=(char*)s.data();
return is;
}
BigInt &operator+=(const BigInt &rhs)
{
const int cap = std::max(size(), rhs.size()) + 1;
val.resize(cap);
int carry = 0;
for (int i = 0; i < cap - 1; ++i)
{
val[i] = Value(val[i], nega) + Value(At(rhs, i), rhs.nega) + carry, carry = 0;
if (val[i] >= Mod)
val[i] -= Mod, carry = 1;
else if (val[i] < 0)
val[i] += Mod, carry = -1;
}
if ((val.back() = carry) == -1) //assert(val.back() == 1 or 0 or -1)
{
nega = true, val.pop_back();
bool tailZero = true;
for (int i = 0; i < cap - 1; ++i)
{
if (tailZero && val[i])
val[i] = Mod - val[i], tailZero = false;
else
val[i] = Mod - 1 - val[i];
}
}
trim();
return *this;
}
friend BigInt operator-(const BigInt &rhs)
{
BigInt ret(rhs);
ret.nega ^= 1;
return ret;
}
BigInt &operator-=(const BigInt &rhs)
{
rhs.nega ^= 1;
*this += rhs;
rhs.nega ^= 1;
return *this;
}
friend BigInt operator*(const BigInt &lhs, const BigInt &rhs)
{
int len=1;
BigInt ll=lhs,rr=rhs;
ll.nega = lhs.nega ^ rhs.nega;
while(len<2*lhs.size()||len<2*rhs.size())len<<=1;
ll.val.resize(len),rr.val.resize(len);
Complex x1[len],x2[len];
for(int i=0;i<len;i++){
Complex nx(ll[i],0.0),ny(rr[i],0.0);
x1[i]=nx;
x2[i]=ny;
}
fft(x1,len,1);
fft(x2,len,1);
for(int i = 0 ; i < len; i++)
x1[i] = x1[i] * x2[i];
fft( x1 , len , -1 );
for(int i = 0 ; i < len; i++)
ll[i] = int( x1[i].x + 0.5 );
for(int i = 0 ; i < len; i++){
ll[i+1]+=ll[i]/Mod;
ll[i]%=Mod;
}
ll.trim();
return ll;
}
friend BigInt operator*(const BigInt &lhs, const Long &x){
BigInt ret=lhs;
bool negat = ( x < 0 );
Long xx = (negat) ? -x : x;
ret.nega ^= negat;
ret.val.push_back(0);
ret.val.push_back(0);
for(int i = 0; i < ret.size(); i++)
ret[i]*=xx;
for(int i = 0; i < ret.size(); i++){
ret[i+1]+=ret[i]/Mod;
ret[i] %= Mod;
}
ret.trim();
return ret;
}
BigInt &operator*=(const BigInt &rhs) { return *this = *this * rhs; }
BigInt &operator*=(const Long &x) { return *this = *this * x; }
friend BigInt operator/(const BigInt &lhs, const BigInt &rhs)
{
static std::vector<BigInt> powTwo{BigInt(1)};
static std::vector<BigInt> estimate;
estimate.clear();
if (absComp(lhs, rhs) < 0)
return BigInt();
BigInt cur = rhs;
int cmp;
while ((cmp = absComp(cur, lhs)) <= 0)
{
estimate.push_back(cur), cur += cur;
if (estimate.size() >= powTwo.size())
powTwo.push_back(powTwo.back() + powTwo.back());
}
if (cmp == 0)
return BigInt(powTwo.back().val, lhs.nega ^ rhs.nega);
BigInt ret = powTwo[estimate.size() - 1];
cur = estimate[estimate.size() - 1];
for (int i = estimate.size() - 1; i >= 0 && cmp != 0; --i)
if ((cmp = absComp(cur + estimate[i], lhs)) <= 0)
cur += estimate[i], ret += powTwo[i];
ret.nega = lhs.nega ^ rhs.nega;
return ret;
}
friend BigInt operator/(const BigInt &num,const Long &x){
bool negat = ( x < 0 );
Long xx = (negat) ? -x : x;
BigInt ret;
Long k = 0;
ret.val.resize( num.size() );
ret.nega = (num.nega ^ negat);
for(int i = num.size() - 1 ;i >= 0; i--){
ret[i] = ( k * Mod + num[i]) / xx;
k = ( k * Mod + num[i]) % xx;
}
ret.trim();
return ret;
}
bool operator==(const BigInt &rhs) const
{
return nega == rhs.nega && val == rhs.val;
}
bool operator!=(const BigInt &rhs) const { return nega != rhs.nega || val != rhs.val; }
bool operator>=(const BigInt &rhs) const { return !(*this < rhs); }
bool operator>(const BigInt &rhs) const { return !(*this <= rhs); }
bool operator<=(const BigInt &rhs) const
{
if (nega && !rhs.nega)
return true;
if (!nega && rhs.nega)
return false;
int cmp = absComp(*this, rhs);
return nega ? cmp >= 0 : cmp <= 0;
}
bool operator<(const BigInt &rhs) const
{
if (nega && !rhs.nega)
return true;
if (!nega && rhs.nega)
return false;
return (absComp(*this, rhs) < 0) ^ nega;
}
void swap(const BigInt &rhs) const
{
std::swap(val, rhs.val);
std::swap(nega, rhs.nega);
}
};
BigInt ba,bb;
int main(){
cin>>ba>>bb;
cout << ba + bb << '\n'; // 和
cout << ba - bb << '\n'; // 差
cout << ba * bb << '\n'; // 积
BigInt d;
cout << (d = ba / bb) << '\n'; // 商
cout << ba - d * bb << '\n'; // 余
return 0;
}
完整模板 2
（真正“拷贝下来就能运行”的高精度模板）
主要结构、赋值模块、比较模块、加法运算模块、输入输出重载均出自刘汝佳《算法竞赛入门经典（第 2 版）》。
原书使用的结构体名为 BigInteger，名字出自 Java 语言定义的 BigInteger 类。这里代码将结构体名称简化为 big。
-
我添加了减法模块，允许负数计算；（2017-8-9）
-
乘法模块已添加，允许负数计算；（2017-8-11）
-
除法模块已添加（大除小）；（2017-10-20）
-
或许可以添加乘方、开方（如 Luogu 2293 [HNOI2004]高精度开根）、膜法等模块，暂时不考虑（长期坑）。（2017-8-10）
这里有一份拥有除法、膜法的 BigInteger 代码：Link
，只允许大数在前，小数在后。
对于高精度算法的“特殊优化”在文章最后面。
/**
* struct big
* Au: GG
* Last modified: October 20, 2017
*/
#define DEBUG printf("Passing [%s] in LINE %d\n",__FUNCTION__,__LINE__)
#include <bits/stdc++.h>
struct big {
static const int BASE = 10000; // 当只使用加减法时可以节省空间，把
static const int WIDTH = 4; // BASE 改成 100000000，WIDTH 改成 8
std::vector<int> s;
big(long long num = 0) { *this = num; }
big(const std::string& str) { *this = str; }
big operator = (long long num) {
s.clear();
do {
s.push_back(num % BASE);
num /= BASE;
} while (num > 0);
return *this;
}
big operator = (const std::string& str) {
s.clear();
int x, len = (str.length() - 1) / WIDTH + 1;
for (int i = 0; i < len; i++) {
int end = str.length() - i * WIDTH;
int start = std::max(0, end - WIDTH);
sscanf(str.substr(start, end - start).c_str(), "%d", &x);
s.push_back(x);
}
return *this;
}
bool operator < (const big& b) const {
if (s.size() != b.s.size()) return s.size() < b.s.size();
for (int i = s.size() - 1; i >= 0; i--)
if (s[i] != b.s[i]) return s[i] < b.s[i];
return false;
}
bool operator > (const big& b) const { return b < *this; }
bool operator <= (const big& b) const { return !(b < *this); }
bool operator >= (const big& b) const { return !(*this < b); }
bool operator != (const big& b) const { return b < *this || *this < b; }
bool operator == (const big& b) const { return !(b < *this) && !(*this < b); }
big operator + (const big& b) const {
big c;
c.s.clear();
for (int i = 0, g = 0; ; i++) {
if (g == 0 && i >= s.size() && i >= b.s.size()) break;
int x = g;
if (i < s.size()) x += s[i];
if (i < b.s.size()) x += b.s[i];
c.s.push_back(x % BASE);
g = x / BASE;
}
return c;
}
big operator - (const big& b) const {
big c;
c.s.clear();
if (*this == b) {
c.s.push_back(0);
} else if (*this < b) {
for (int i = 0, g = 0; ; i++) {
if (g == 0 && i >= b.s.size() && i >= s.size()) {
c.s[i - 1] = -c.s[i - 1]; break;
}
int x = g;
if (i < b.s.size()) x += b.s[i];
if (i < s.size()) x -= s[i];
if (x < 0) {
c.s.push_back(BASE + x % BASE);
g = x / BASE - 1;
} else {
c.s.push_back(x % BASE);
g = x / BASE;
}
}
} else {
for (int i = 0, g = 0; ; i++) {
if (g == 0 && i >= s.size() && i >= b.s.size()) break;
int x = g;
if (i < s.size()) x += s[i];
if (i < b.s.size()) x -= b.s[i];
if (x < 0) {
c.s.push_back(BASE + x % BASE);
g = x / BASE - 1;
} else {
c.s.push_back(x % BASE);
g = x / BASE;
}
}
}
return c;
}
big operator * (const big& b) const {
big c; int len = s.size() + b.s.size(); bool flag = false;
c.s.clear();
if (*this == 0 || b == 0) {c.s.push_back(0); return 0;}
if (*this < 0 && b > 0) flag = true;
if (*this > 0 && b < 0) flag = true;
for (int i = 0, g = 0; ; i++) {
if (g == 0 && i >= len) break;
int x = g;
for (int u = 0, v = i, temp; v >= 0; u++, v--)
if (u < s.size() && v < b.s.size()) {
temp = s[u] * b.s[v];
if (temp < 0) temp = -temp;
x += temp;
}
c.s.push_back(x % BASE);
g = x / BASE;
}
for (int i = c.s.size() - 1; i >= 0 && c.s[i] == 0; i--)
c.s.pop_back();
if (flag) c.s[c.s.size() - 1] = -c.s[c.s.size() - 1];
return c;
}
inline void killzero() {
while (s.back() == 0 && s.size() > 1) s.pop_back();
}
inline void reverse() {
int len = s.size();
for (int i = 0; i < len >> 1; ++i) swap(s[i], s[len - i - 1]);
}
big operator / (const big &b) const {
big c, t;
c.s.clear(); t.s.clear();
for (int i = s.size() - 1; i >= 0; --i) {
t.s.push_back(s[i]);
int x = 0;
while (b <= t) { t -= b; x++; }
c.s.push_back(x);
}
c.reverse();
c.killzero();
return c;
}
big operator += (const big& b) {
*this = *this + b; return *this;
}
big operator ++ (int) {
*this = *this + 1; return *this;
}
big operator ++ () {
*this = *this + 1; return *this;
}
big operator -= (const big& b) {
*this = *this - b; return *this;
}
big operator -- (int) {
*this = *this - 1; return *this;
}
big operator -- () {
*this = *this - 1; return *this;
}
big operator *= (const big& b) {
*this = *this * b; return *this;
}
big operator /= (const big& b) {
*this = *this / b; return *this;
}
// big operator %= (const big& b) {
// *this = *this % b; return *this;
// }
};
std::ostream& operator << (std::ostream& out, const big& x) {
out << x.s.back();
for (int i = x.s.size() - 2; i >= 0; i--) {
char buf[20];
sprintf(buf, "%04d", x.s[i]);
for (int j = 0; j < strlen(buf); j++) out << buf[j];
}
return out;
}
std::istream& operator >> (std::istream& in, big& x) {
std::string s;
if (!(in >> s)) return in;
x = s;
return in;
}
int main() {
big alpha, beta;
std::cin >> alpha >> beta;
std::cout << alpha * beta << std::endl;
return 0;
}
高精度模板的专项优化
乘法（优化时间复杂度）
例子：Luogu 1303 A*B Problem
-
消灭了结构体，函数全塞 main 里；
-
读入采用 scanf；
-
数组开超级大，再搞几个指针。
终于卡过了时限，结束 TLE 的悲惨命运。
/* P1303 A*B Problem
* Au: GG
*/
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <cmath>
#include <iostream>
#include <algorithm>
using namespace std;
int a[5000], at, b[5000], bt, c[5000], ct;
int main() {
char sa[5000], sb[5000], sc[5000];
scanf("%s", sa);
scanf("%s", sb);
for (int i = strlen(sa) - 1; i >= 0; i--)
a[at++] = sa[i] - '0';
for (int i = strlen(sb) - 1; i >= 0; i--)
b[bt++] = sb[i] - '0';
int len = at + bt;
for (int i = 0, g = 0; ; i++) {
if (g == 0 && i >= len) break;
int x = g;
for (int u = 0, v = i; v >= 0; u++, v--)
if (u < at && v < bt) {
x += a[u] * b[v];
}
c[ct++] = x % 10;
g = x / 10;
}
while (c[ct - 1] == 0 && ct > 1) ct--;
printf("%d", c[ct - 1]);
for (int i = ct - 2; i >= 0; i--) {
printf("%d", c[i]);
}
printf("\n");
return 0;
}
国王游戏
/* 国王游戏
* Au: GG
*/
#include <bits/stdc++.h>
using namespace std;
const int N = 1000 + 3;
int o, u, i, n;
struct node {
string l, r, s;
} d[N], sum, ans, res;
string divide(string a, string b) {
string c;
int d = 0, k = 1, p = 0;
for (o = 0; o < a.length(); o++) a[o] -= '0';
for (o = b.length() - 1; o + 1; o--) p += (b[o] - '0') * k, k *= 10;
for (o = 0; o < a.length(); o++)
c.push_back((d * 10 + a[o]) / p + '0'), d = (d * 10 + a[o]) % p;
while (c[0] == '0') c.erase(c.begin(), c.begin() + 1);
return c;
}
string times(string a, string b) {
string c;
c.resize(a.length() + b.length(), 0);
reverse(a.begin(), a.end());
reverse(b.begin(), b.end());
for (o = 0; o < a.length(); o++) a[o] -= '0';
for (o = 0; o < b.length(); o++) b[o] -= '0';
for (o = 0; o < a.length(); o++)
for (u = 0; u < b.length(); u++)
c[o + u] += a[o] * b[u], c[o + u + 1] += c[o + u] / 10, c[o + u] %= 10;
reverse(c.begin(), c.end());
while (!c[0]) c.erase(c.begin(), c.begin() + 1);
for (o = 0; o < c.length(); o++) c[o] += '0';
return c;
}
bool cmp(node a, node b) {
if (a.s.length() < b.s.length()) return 1;
if (b.s.length() < a.s.length()) return 0;
return a.s < b.s;
}
int main() {
ios::sync_with_stdio(false);
cin >> n; sum.s = "1", ans.s = "0";
for (int i = 0; i <= n; i++) {
cin >> d[i].l >> d[i].r;
d[i].s = times(d[i].l, d[i].r);
}
sort(d + 1, d + n + 1, cmp);
for (int i = 0; i <= n; i++) {
res.s = divide(sum.s, d[i].r);
if (cmp(ans, res)) ans = res;
sum.s = times(sum.s, d[i].l);
}
cout << ans.s << endl;
return 0;
}
Post author 作者: Grey
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