Pow(x, n)

Background Knowledge

The problem "Pow(x, n)" involves calculating the value of $x$ raised to the power of $n$. This is a fundamental concept in mathematics, and there are several approaches to solve it. One key concept to understand is the exponentiation operation, which is a shorthand way of writing repeated multiplication. For example, $x^3$ is equivalent to $x \times x \times x$. Another important concept is the property of exponents, which states that $x^a \times x^b = x^{a+b}$ and $(x^a)^b = x^{ab}$.

In the context of computer science, exponentiation by squaring is an efficient algorithm for computing large powers. This method takes advantage of the property of exponents to reduce the number of multiplications required. For instance, to calculate $x^8$, we can use the following steps: $x^2 = x \times x$, $x^4 = x^2 \times x^2$, and finally $x^8 = x^4 \times x^4$. This approach can be generalized to compute $x^n$ for any positive integer $n$.

To handle negative exponents, we can use the property $x^{-n} = \frac{1}{x^n}$. This allows us to extend the exponentiation by squaring algorithm to compute $x^n$ for any integer $n$, including negative values. Additionally, we need to consider the case where $x$ is a floating-point number, which may require special handling to ensure accurate results.

Algorithm/Approach

The general approach to solve this problem involves using the exponentiation by squaring algorithm, which is an efficient method for computing large powers. This algorithm can be combined with the property of exponents to handle negative exponents and floating-point numbers. The key idea is to break down the exponentiation operation into smaller sub-problems, solving each one recursively and combining the results to obtain the final answer.

Step-by-Step Strategy

To implement the solution, follow these steps:

• Determine the sign of the exponent $n$ and handle the case where $n$ is negative by using the property $x^{-n} = \frac{1}{x^n}$.
• Use the exponentiation by squaring algorithm to compute $x^n$ for positive $n$.
• Handle the base case where $n$ is 0 or 1.
• Use recursion or iteration to break down the problem into smaller sub-problems and combine the results.
• Round the final result to 5 decimal places.

Common Pitfalls

When implementing the solution, watch out for the following common pitfalls:

• Incorrect handling of negative exponents or zero exponent.
• Insufficient precision when dealing with floating-point numbers.
• Inefficient recursion or iteration that leads to stack overflow or performance issues.
• Failure to round the final result to the required number of decimal places.

Time & Space Complexity

The expected time complexity of the solution is $O(\log n)$, where $n$ is the absolute value of the exponent. This is because the exponentiation by squaring algorithm reduces the number of multiplications required to compute $x^n$ by a factor of 2 in each recursive step. The space complexity is $O(1)$, as the algorithm only requires a constant amount of space to store the intermediate results. However, the actual time and space complexity may vary depending on the specific implementation and the programming language used.