[BZOJ2179]FFT快速傅立叶

[BZOJ2179]FFT快速傅立叶

试题描述

给出两个n位10进制整数x和y，你需要计算x*y。

输入

第一行一个正整数n。 第二行描述一个位数为n的正整数x。 第三行描述一个位数为n的正整数y。

输出

输出一行，即x*y的结果。

输入示例

1
3
4

输出示例

12

数据规模及约定

n<=60000

题解

可以把一个数看成一个多项式，乘起来之后处理一下进位即可。

#include <iostream>
#include <cstdio>
#include <algorithm>
#include <cmath>
#include <stack>
#include <vector>
#include <queue>
#include <cstring>
#include <string>
#include <map>
#include <set>
using namespace std;

const int BufferSize = 1 << 16;
char buffer[BufferSize], *Head, *Tail;
inline char Getchar() {
	if(Head == Tail) {
		int l = fread(buffer, 1, BufferSize, stdin);
		Tail = (Head = buffer) + l;
	}
	return *Head++;
}
int read() {
	int x = 0, f = 1; char c = Getchar();
	while(!isdigit(c)){ if(c == '-') f = -1; c = Getchar(); }
	while(isdigit(c)){ x = x * 10 + c - '0'; c = Getchar(); }
	return x * f;
}

#define maxn 240010
const double pi = acos(-1.0);
int n;
struct Complex {
	double a, b;
	Complex operator + (const Complex& t) {
		Complex ans;
		ans.a = a + t.a;
		ans.b = b + t.b;
		return ans;
	}
	Complex operator - (const Complex& t) {
		Complex ans;
		ans.a = a - t.a;
		ans.b = b - t.b;
		return ans;
	}
	Complex operator * (const Complex& t) {
		Complex ans;
		ans.a = a * t.a - b * t.b;
		ans.b = a * t.b + b * t.a;
		return ans;
	}
	Complex operator *= (const Complex& t) {
		*this = *this * t;
		return *this;
	}
} a[maxn], b[maxn];

char A[(maxn>>2)+10], B[(maxn>>2)+10];
int Ord[maxn], num[maxn];
void FFT(Complex* x, int n, int tp) {
	for(int i = 0; i < n; i++) if(i < Ord[i]) swap(x[i], x[Ord[i]]);
	for(int i = 1; i < n; i <<= 1) {
		Complex wn, w; wn.a = cos(pi / i); wn.b = tp * sin(pi / i);
		for(int j = 0; j < n; j += (i << 1)) {
			w.a = 1.0; w.b = 0.0;
			for(int k = 0; k < i; k++) {
				Complex t1 = x[j+k], t2 = w * x[j+k+i];
				x[j+k] = t1 + t2;
				x[j+k+i] = t1 - t2;
				w *= wn;
			}
		}
	}
	return ;
}

int main() {
	n = read();
	char tc = Getchar();
	while(!isdigit(tc)) tc = Getchar();
	for(int i = 0; i < n; i++) A[i] = tc, tc = Getchar();
	while(!isdigit(tc)) tc = Getchar();
	for(int i = 0; i < n; i++) {
		B[i] = tc;
		if(i < n - 1) tc = Getchar();
	}
	for(int i = n - 1; i >= 0; i--) {
		a[n-1-i].a = (double)(A[i] - '0'); a[i].b = 0.0;
		b[n-1-i].a = (double)(B[i] - '0'); b[i].b = 0.0;
	}
	
	int m = (n - 1) * 2, L = 0;
	for(n = 1; n <= m; n <<= 1) L++;
	for(int i = 0; i < n; i++) Ord[i] = (Ord[i>>1] >> 1) | ((i & 1) << L - 1);
	FFT(a, n, 1); FFT(b, n, 1);
	for(int i = 0; i <= n; i++) a[i] *= b[i];
	FFT(a, n, -1);
	for(int i = 0; i <= n; i++) {
		int x = (int)(a[i].a / n + .5);
		num[i] += x;
		num[i+1] += num[i] / 10;
		num[i] %= 10;
	}
	
	int i = n;
	for(; !num[i]; i--) ;
	for(; i >= 0; i--) putchar(num[i] + '0'); putchar('\n');
	
	return 0;
}