快速幂

概述

快速幂，二进制取幂（Binary Exponentiation，也称平方法），是一个在 $\Theta(\log n)$ 的时间内计算 $a^n$ 的小技巧，而暴力的计算需要 $\Theta(n)$ 的时间。

解释

计算 $a$ 的 $n$ 次方表示将 $n$ 个 $a$ 乘在一起: $a^{n} = \underbrace{a \times a \cdots \times a}_{n\text{ 个 a}}$ 。然而当 $a,n$ 太大的时侯，这种方法就不太适用了。不过我们知道: $a^{b+c} = a^b \cdot a^c,\,\,a^{2b} = a^b \cdot a^b = (a^b)^2$ 。二进制取幂的想法是，我们将取幂的任务按照指数的 二进制表示 来分割成更小的任务。

简单版本

首先我们将 n 表示为 2 进制，举一个例子： $3^{13} = 3^{(1101)_2} = 3^8 \cdot 3^4 \cdot 3^1$ 因为 $n$ 有 $\lfloor \log_2 n \rfloor + 1$ 个二进制位，因此当我们知道了 $a^1, a^2, a^4, a^8, \dots, a^{2^{\lfloor \log_2 n \rfloor}}$ 后，我们只用计算 $\Theta(\log n)$ 次乘法就可以计算出 $a^n$ 。

迭代代码

double binpow(double a, long long b) {
    if (b < 0)
          return 1.0 / binpow(a, -b); // x^{-a} = 1/(x^{a})
    double res = 1;
    while (b > 0) {
        if (b & 1)
            res *= a;
        a *= a;
        b >>= 1;
    }
    return res;
}
C++

模意义下取幂

##green## 问: 计算 $x^n\bmod m$ 。

我们需要知道: 取模的运算不会干涉乘法运算

#include <cstdio>

long long hxPow(long long a, long long b, long long mod) {
    a %= mod;
    long long res = 1;
    while (b) {
        if (b & 1)
            res = res * a % mod;
        a = a * a % mod;
        b >>= 1;
    }
    return res;
}

int main() {
    long long a, b, mod;
    scanf("%lld %lld %lld", &a, &b, &mod);
    printf("%lld^%lld mod %lld=%lld\n", a, b, mod, hxPow(a, b, mod));
    return 0;
}
C++

注意: 不能优化的时候对幂(指数)进行取模, 因为费马小定理 (我不会, 见 1969. 数组元素的最小非零乘积的题解区!^[2])

题目:

指数过大

题目: 蓝桥杯-周赛-萌新题-Q3

给定 $2$ 个整数 $x$ , $p$ 你需要输出 $x^p \mod 10$ 的结果

输入格式:

一个整数 $T(1 \le T \le 10^3)$
对于每组数据
- 包含两个正整数 $x$ , $p (1 \le x \le 10^3 , 1 \le p \le 10^{2 \times 10^5})$
- 数据保证 $1 \leq \sum_{i=1}^{T} \log _{10} p_{i} \leq 2 \times 10^{5}$ 。换句话说便是保证所有数据指数之和在 $1 \sim 2 \times 10^{5}$ 。

示例:

1
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0
C

此处涉及到 字符串 输入了!

即 $p$ 应该使用 $string$ 来存储

先上码, 再讲解:

#include <iostream>
using namespace std;

long long mod = 10;

// 普通快速幂
long long hxpow(long long x, long long p) {
    x %= mod;
    long long res = 1;
    while (p) {
        if (p & 1)
          res = (res * x) % mod;
        x = (x * x) % mod;
        p >>= 1;
    }
    return res;
}

int main()
{
    int t;
    cin >> t;
  
    while (t--) {
      long long x;
      string p;
      cin >> x >> p;
  
      long long res = 1;
      for (int i = p.size() - 1; i >= 0; --i) { // 拆分
        res = res * hxpow(x, p[i] - '0') % mod; // 为什么我忘记 - '0' 也能过?!
        x = hxpow(x, 10) % mod;
      }
  
      cout << res % mod << "\n";
    }
  
    return 0;
}
C++

对于 $x^{345}$ 我们可以拆分为 $(x^1)^5 \times (x^{10})^4 \times (x^{100})^3$ (从后往前拆分) , 所以就有上面代码部分.

当然从前往后拆分也可以 $x^{345} = [(x^3)^{10} \times x^4]^{10} \times x^5$ 即

long long x;
string p;
cin >> x >> p;
long long res = 1;
for (int i = 0; i < p.size(); ++i) {
    res = (hxpow(res, 10) * hxpow(x, p[i] - '0')) % mod;
}
C++

注解

[1]

学习: oi-wiki 快速幂

[2] 留下疑问

为什么注释部分代码是错的qwq?!

class Solution {
    const int MOD = 1e9 + 7; // MOD
    long long hxPow(long long a, long long b) {
        a %= MOD;
        long long res = 1;
        while (b) {
            if (b & 1)
                res = (res * a) % MOD;
            a = (a * a) % MOD;
            b >>= 1;
        }
        return res;
    }
public:

    int minNonZeroProduct(int p) {
/*
贪心选择 ... 构造 

p = 3

如果 x 的 k 位 是 0, y 的 k 位 是 1
记 y = y' + 2^k
有 x*y = x(y' + 2^k) = xy' + x*2^k
移位交换有
   (x + 2^k) * y' = xy' + y'*2^k

如果需要满足 xy > 移动后 那么 x > y' (总之就是合理)

100 和 011 > 110 和 001 的积

(001, 010, 011, 100, 101, 110, 111) (2^p - 1) / 2 => (2^{p - 1} - 1)

[001, 110] * [010, 101] * [001, 100] * 111
=> [001, 110] * [001, 110] * [001, 110] * 111
=> 001 * 3 * 110 * 3 * 111
=> 此处的3实际上是长度 (2^p - 1) / 2
=> 即 2^{p - 2} - 1
故 res = (2^p - 1) * (2^p - 2)^(2^{p - 1} - 1)
*/
        long long fk = (1LL << p); // 为什么我把 pow(2, p) 换成 1LL << p 就ok?
        long long res_1 = fk - 1;
        long long res_pow = (fk >> 1) - 1;
        long long disu_pow = fk - 2;
        long long res_2 = hxPow(disu_pow, res_pow);
        int res = (((res_1 % MOD) * (res_2 % MOD)) % MOD);

        // res_pow(指数部分) 不能使用快速幂??!
        // long long res_1 = hxPow(2, p) - 1;
        // long long res_pow = hxPow(2, p - 1) - 1; // 不能取余 mod, 而应该取余 mod - 1
        // long long disu_pow = hxPow(2, p) - 2;
        // long long res_2 = hxPow(disu_pow, res_pow);
        // int res = (((res_1 % MOD) * (res_2 % MOD)) % MOD);
        return res;
    }
};

// 2^60 - 1 >> 1e8 // 不可能遍历判断/模拟!
// 需要数学大佬!
// 选择 1 ~ 2^p - 1 个数
/*
换 0 0 => 没换
换 1 1 => 没换
换 0 1 => a + 2^x, b - a^x
换 1 0 => 等价上面!
然后 可以对 其中一个数a -2^x 同时另一个数b +2^x (a >= 2^x)
求 他们的 最小非零 乘积

((a) - 2^x) * ((b) + 2^x) <= (a) * (b) 什么情况下?

(3 - 2) (3 + 2) <= 3 * 3

1001    0011
(9 - 8) (3 + 8) <= 9 * 3

// 使得元素凑凑出一吗? <贪心>
*/

/*
题目给的数是p 即 [0..01, 1..11] // 长度是p位
// 所以每一位配的 0 1 ...?
*/
C++

概述​

解释​

简单版本​

迭代代码​

模意义下取幂​

指数过大​

注解​

[1]​

[2] 留下疑问​

概述

解释

简单版本

迭代代码

模意义下取幂

指数过大

注解

[1]

[2] 留下疑问