Binary Exponentiation

Author: @wingkwong

Overview

Binary Exponentiation is a method for efficiently calculating large powers of a number, such as $a^n$. Instead of using the naive approach of repeatedly multiplying the base number by itself, which has a time complexity of $O(n)$, binary exponentiation uses a technique called "exponentiation by squaring" to accomplish the same task in $O(log n)$ time complexity.

The basic idea behind binary exponentiation is that we can express $a ^ n$ as $a * a * ... * a$ but it is not efficient for a large $a$ and $n$. If we display the exponent in binary representation, says $13 = 1101_2$, then we have $3 ^{13} = 3^8*3^4*3^1.$ Supposing we have a sequence $a ^ 1, a ^ 2, a ^4, ..., a^{\lfloor log_2 n\rfloor}$, we can see the an element in the sequence is the square of previous element, i.e. $3 ^ 4 = (3^2)^2$. Therefore, to calculate $3 ^ {13}$, we just need to calculate ${\lfloor log_2 13\rfloor} = 3$ times, i.e. ($1$ -> $4$ -> $8$). We skip $2$ here because the bit is not set. This approach gives us $O(log n)$ complexity.

To generalise it, for a positive integer $n$, we have

$x^n = \begin{cases} x \cdot (x^2)^{\frac{n - 1}{2}}, & \text{if } n \text{ is odd}\\ (x^2)^{\frac{n}{2}}, & \text{if } n \text{ is even.} \end{cases}$

Implementation

C++

Written by @wingkwong

long long fastpow(long long base, long long exp) {
  long long res = 1;
  while (exp > 0) {
    // if n is odd, a ^ n can be seen as a ^ (n / 2) * a ^ (n / 2) * a
    if (exp & 1) res *= base;
    // if n is even, a ^ n can be seen as a ^ (n / 2) * a ^ (n / 2)
    base *= base;
    // shift 1 bit to the right
    exp >>= 1;
  }
  return res;
}

In case you need to take mod during the calculation, we can do as follows.

C++

Written by @wingkwong

long long modpow(long long base, long long exp, long long mod) {
  base %= mod;
  long long res = 1;
  while (exp > 0) {
    if (exp & 1) res = (res * base) % mod;
    base = (base * base) % mod;
    exp >>= 1;
  }
  return res;
}

Suggested Problems

Problem Name	Difficulty	Solution Link
0050 - Pow(x, n)	Medium	View Solutions