Multi Precision Integer
コードについての説明
シフト演算子を使ったり, ビット演算をしたりしていなければ int や long long と同じように動くはずです.
主要な演算については verify しました.
宣言は MPI(数字 or string) もしくは MPI hoge = 数字; またデフォルトは0初期化されます.
引数の string は $0$ を除いて leading-zero がないことを仮定しています.
シフト演算子 および ビット演算(&,|,^) はフォローしていません.
数字(int や long long) との演算や比較は暗黙的に MPI にキャストされます.
任意精度の多倍長整数の足し算, 引き算, 掛け算, 割り算(商と余り), 2で割る(div2), sqrt, log10, pow, 絶対値, 大小判定, 入出力, 任意の桁(10進数表示) へのランダムアクセス などを行えます.
$2$ で割る(div2), sqrt, log10 は lower_bound を返します(MPI型)
$n$ 桁(10進数)の数に対して 足し算, 引き算($\O (n)$), 掛け算($\O (n \log n)$), 割り算($\O (n^2)$), $2$ で割る($\O (n)$), sqrt($(10 \sim 20)n^2$), log10($\O (1)$), pow($n^{\log m} \log n \log m$) の計算時間がかかります.
掛け算の演算を効率的に行うために本実装では base を $10$ にとっています(足し算, 引き算を定数倍高速化したいなら base を $10^9$ とか $10^{18}$ とかでとったほうがよい)
掛け算は NTT を用いています(もう少し大きい mod を使ったり Garner を使ったりして定数倍改善可能だが更新するのがめんどくさくてやってない). 負の除算もフォローしています(定義はC++の除算と同じです).
$2$ で割るのは多倍長整数での $2$ 分探索や sqrt 計算に使う目的で実装しました.
to_ll, to_string で long long, string に変換可能. $10$ 倍や $10$ で割るなども $\O (1)$ で行うようにできる.
コード
class MPI : public vector<int> {
private:
static constexpr int root = 3;
static constexpr int MOD_ = 998244353; // 2^23 * 119 + 1 (2^22 桁以下の数の積を計算可能)
bool sign;
static void trim_sign(MPI& num){
if(num.isZero()) num.sign = false;
}
static void trim_digit(MPI& num){
while(num.back() == 0 && (int)num.size() >= 2) num.pop_back();
}
bool abs_less(const MPI& a, const MPI& b) const {
if(a.size() != b.size()) return a.size() < b.size();
for(int index = (int)a.size() - 1; index >= 0; index--){
if(a[index] != b[index]) return a[index] < b[index];
}
return false;
}
static void num_sbst(MPI& a, const MPI& b){
int n = (int)b.size();
a.resize(n);
for(int i = 0; i < n; i++) a[i] = b[i];
}
void add(const MPI& a, const MPI& b, MPI& res) const {
num_sbst(res, a);
int mx = (int)max(a.size(), b.size());
res.resize(mx, 0);
int carry = 0;
for(int i = 0; i < mx; i++){
int val = res[i] + ((i < (int)b.size()) ? b[i] : 0) + carry;
carry = val/10;
res[i] = val%10;
}
if(carry) res.push_back(1);
}
void sub(const MPI& a, const MPI& b, MPI& res) const {
num_sbst(res, a);
int carry = 0;
for(int i = 0; i < (int)res.size(); i++){
int val = res[i] - carry - ((i < (int)b.size()) ? b[i] : 0);
if(val < 0){
carry = 1, val += 10;
}else{
carry = 0;
}
res[i] = val;
}
trim_digit(res), trim_sign(res);
}
static int add_(const int x, const int y) { return (x + y < MOD_) ? x + y : x + y - MOD_; }
static int mul_(const int x, const int y) { return (long long)x * y % MOD_; }
static int power(int x, int n){
int res = 1;
while(n > 0){
if(n & 1) res = mul_(res, x);
x = mul_(x, x);
n >>= 1;
}
return res;
}
static int inverse(const int x) { return power(x, MOD_ - 2); }
static void ntt(vector<int>& a, const bool rev = false){
int i,j,k,s,t,v,w,wn;
const int size = (int)a.size();
const int height = (int)log2(2 * size - 1);
for(i = 0; i < size; i++){
j = 0;
for(k = 0; k < height; k++) j |= (i >> k & 1) << (height - 1 - k);
if(i < j) std::swap(a[i], a[j]);
}
for(i = 1; i < size; i *= 2) {
w = power(root, (MOD_ - 1) / (i * 2));
if(rev) w = inverse(w);
for(j = 0; j < size; j += i * 2){
wn = 1;
for(k = 0; k < i; k++){
s = a[j + k], t = mul_(a[j + k + i], wn);
a[j + k] = add_(s, t);
a[j + k + i] = add_(s, MOD_ - t);
wn = mul_(wn, w);
}
}
}
if(rev){
v = inverse(size);
for (i = 0; i < size; i++) a[i] = mul_(a[i], v);
}
}
static void mul(const MPI& a, const MPI& b, MPI& res){
const int size = (int)a.size() + (int)b.size() - 1;
int t = 1;
while (t < size) t <<= 1;
vector<int> A(t, 0), B(t, 0);
for(int i = 0; i < (int)a.size(); i++) A[i] = a[i];
for(int i = 0; i < (int)b.size(); i++) B[i] = b[i];
ntt(A), ntt(B);
for(int i = 0; i < t; i++) A[i] = mul_(A[i], B[i]);
ntt(A, true);
res.resize(size);
int carry = 0;
for(int i = 0; i < size; i++){
int val = A[i] + carry;
carry = val / 10;
res[i] = val % 10;
}
if(carry) res.push_back(carry);
trim_digit(res), trim_sign(res);
}
bool isZero() const {
return (int)size() == 1 && (*this)[0] == 0;
}
static bool div_geq(const MPI& mod, const MPI& num){
if((int)mod.size() != (int)num.size()) return (int)mod.size() > (int)num.size();
int n = (int)mod.size();
for(int i = 0; i < n; i++){
if(mod[i] != num[n-1-i]){
return mod[i] > num[n-1-i];
}
}
return true;
}
void div_(const MPI& a, const MPI& b, MPI& quo, MPI& mod) const {
vector<MPI> mult(9);
mult[0] = b;
for(int i = 0; i < 8; i++) mult[i+1] = mult[i] + b;
for(int i = (int)a.size() - 1; i >= 0; i--){
if(mod.isZero()){
mod.back() = a[i];
}else{
mod.push_back(a[i]);
}
if(div_geq(mod, b)){
int l = 0, r = 9;
reverse(mod.begin(), mod.end());
while(r-l>1){
int mid = (l+r)/2;
if(mult[mid] > mod){
r = mid;
}else{
l = mid;
}
}
mod -= mult[l];
reverse(mod.begin(), mod.end());
quo.push_back(l+1);
}else{
quo.push_back(0);
}
}
reverse(quo.begin(), quo.end());
trim_digit(quo);
reverse(mod.begin(), mod.end());
}
public:
MPI() : sign(false){ this->push_back(0); }
MPI(long long val) : sign(false){
if(val == 0){
this->push_back(0);
}else{
if(val < 0) sign = true, val = -val;
while(val){
this->push_back(val%10);
val /= 10;
}
}
}
MPI(const string& s) : sign(false){
if(s.empty()){
this->push_back(0);
return;
}
if(s[0] == '-') sign = true;
for(int i = (int)s.size() - 1; i >= sign; i--) this->push_back(s[i]-'0');
}
long long to_ll() const {
long long res = 0, dig = 1;
for(int i = 0; i < (int)size(); i++){
res += dig * (*this)[i], dig *= 10;
}
if(sign) res = -res;
return res;
}
string to_string() const {
int n = (int)size() + sign;
string s(n, ' ');
if(sign) s[0] = '-';
for(int i = sign; i < n; i++) s[i] = (char)('0'+(*this)[n-1-i]);
return s;
}
friend istream& operator>>(istream& is, MPI& num) {
string s;
is >> s;
num = MPI(s);
return is;
}
friend ostream& operator<<(ostream& os, const MPI& num) {
if(num.sign) os << '-';
for(int i = (int)num.size()-1; i >= 0; i--) os << (char)('0'+num[i]);
return os;
}
MPI& operator=(long long val) {
*this = MPI(val);
return *this;
}
bool operator<(const MPI& another) const {
if(sign ^ another.sign) return sign;
if(size() != another.size()) return (size() < another.size()) ^ sign;
for(int index = (int)size() - 1; index >= 0; index--){
if((*this)[index] != another[index]) return ((*this)[index] < another[index]) ^ sign;
}
return false;
}
bool operator<(const long long num) const {
return *this < MPI(num);
}
friend bool operator<(const long long num, const MPI& another){
return MPI(num) < another;
}
bool operator>(const MPI& another) const {
return another < *this;
}
bool operator>(const long long num) const {
return *this > MPI(num);
}
friend bool operator>(const long long num, const MPI& another){
return MPI(num) > another;
}
bool operator<=(const MPI& another) const {
return !(*this > another);
}
bool operator<=(const long long num) const {
return *this <= MPI(num);
}
friend bool operator<=(const long long num, const MPI& another){
return MPI(num) <= another;
}
bool operator>=(const MPI& another) const {
return !(*this < another);
}
bool operator>=(const long long num) const {
return *this >= MPI(num);
}
friend bool operator>=(const long long num, const MPI& another){
return MPI(num) >= another;
}
bool operator==(const MPI& another) const {
if(sign ^ another.sign) return false;
if(size() != another.size()) return false;
for(int index = (int)size() - 1; index >= 0; index--){
if((*this)[index] != another[index]) return false;
}
return true;
}
bool operator==(const long long num) const {
return *this == MPI(num);
}
friend bool operator==(const long long num, const MPI& another){
return MPI(num) == another;
}
bool operator!=(const MPI& another) const {
return !(*this == another);
}
bool operator!=(const long long num) const {
return *this != MPI(num);
}
friend bool operator!=(const long long num, const MPI& another){
return MPI(num) != another;
}
explicit operator bool() const noexcept { return !isZero(); }
bool operator!() const noexcept { return !static_cast<bool>(*this); }
explicit operator int() const noexcept { return (int)this->to_ll(); }
explicit operator long long() const noexcept { return this->to_ll(); }
MPI operator+() const {
return *this;
}
MPI operator-() const {
MPI res = *this;
res.sign = !sign;
return res;
}
friend MPI abs(const MPI& num){
MPI res = num;
res.sign = false;
return res;
}
MPI operator+(const MPI& num) const {
MPI res; res.sign = sign;
if(sign != num.sign){
if(abs_less(*this, num)){
res.sign = num.sign;
sub(num, *this, res);
return res;
}else{
sub(*this, num, res);
return res;
}
}
add(*this, num, res);
return res;
}
MPI operator+(long long num) const {
return *this + MPI(num);
}
friend MPI operator+(long long a, const MPI& b){
return b + a;
}
MPI& operator+=(const MPI& num){
*this = *this + num;
return *this;
}
MPI& operator+=(long long num){
*this = *this + num;
return *this;
}
MPI& operator++(){
return *this += 1;
}
MPI operator++(int){
MPI res = *this;
*this += 1;
return res;
}
MPI operator-(const MPI& num) const {
MPI res; res.sign = sign;
if(sign != num.sign){
add(*this, num, res);
return res;
}
if(abs_less(*this, num)){
res.sign = !sign;
sub(num, *this, res);
}else{
sub(*this, num, res);
}
return res;
}
MPI operator-(long long num) const {
return *this - MPI(num);
}
friend MPI operator-(long long a, const MPI& b){
return b - a;
}
MPI& operator-=(const MPI& num){
*this = *this - num;
return *this;
}
MPI& operator-=(long long num){
*this = *this - num;
return *this;
}
MPI& operator--(){
return *this -= 1;
}
MPI operator--(int){
MPI res = *this;
*this -= 1;
return res;
}
MPI operator*(MPI num) const {
MPI res; res.sign = (sign^num.sign);
mul(*this, num, res);
return res;
}
MPI operator*(long long num) const {
return *this * MPI(num);
}
friend MPI operator*(long long a, const MPI& b){
return b * a;
}
MPI& operator*=(const MPI& num){
*this = *this * num;
return *this;
}
MPI& operator*=(long long num){
*this = *this * num;
return *this;
}
MPI operator/(const MPI& num) const {
MPI num_ = abs(num);
MPI a, b;
div_(*this, num_, a, b);
a.sign = (sign^num.sign);
trim_sign(a);
return a;
}
MPI operator/(long long num) const {
return *this / MPI(num);
}
friend MPI operator/(long long a, const MPI& b){
return b / a;
}
MPI& operator/=(const MPI& num){
*this = *this / num;
return *this;
}
MPI& operator/=(long long num){
*this = *this / num;
return *this;
}
MPI operator%(const MPI& num) const {
MPI num_ = abs(num);
MPI a, b;
div_(*this, num_, a, b);
b.sign = sign;
trim_sign(b);
return b;
}
MPI operator%(long long num) const {
return *this % MPI(num);
}
friend MPI operator%(long long a, const MPI& b){
return b % a;
}
MPI& operator%=(const MPI& num){
*this = *this % num;
return *this;
}
MPI& operator%=(long long num){
*this = *this % num;
return *this;
}
MPI div2() const {
MPI res; res.sign = sign;
int n = (int)this->size(), carry = 0;
for(int i = n-1; i >= 0; i--){
int val = (*this)[i]+carry*10;
carry = val%2;
if(i != n-1 || val >= 2) res.push_back(val/2);
}
reverse(res.begin(), res.end());
trim_digit(res);
return res;
}
friend MPI sqrt(const MPI& x){
if(x <= MPI(0)) return MPI(0);
MPI s = 1, t = x;
while(s < t){
s = s + s, t = t.div2();
}
do{ t = s, s = (x / s + s).div2();
}while(s < t);
return t;
}
friend MPI log10(const MPI& x){
assert(x > MPI(0));
return MPI((int)x.size());
}
friend MPI pow(MPI a, MPI b){
assert(b >= 0);
MPI res(1);
while(b){
if(b[0] % 2){
res *= a;
}
a *= a;
b = b.div2();
}
return res;
}
};
verify 用の問題
足し算, 引き算, 2で割る(div2) の verify
Atcoder : 数字列をカンマで分ける問題
提出コード (非想定解なので TLE しています)
足し算, 引き算, 掛け算, 割り算, mod の verify
AOJ : Number Theory - Big Integers -