Fast Exponentiation Algorithm

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Questions and Answers

Why is `long long` used for the exponent `p` instead of `int`?

To optimize memory usage for exponent storage.
To handle larger exponents that exceed the range of `int`.
To improve the precision of the power calculation.
To avoid potential overflow when negating the exponent if `n` is `INT_MIN`. (correct)

The provided `myPow` function has a time complexity of $O(n)$, where $n$ is the exponent.

False (B)

Explain how the line `if (p & 1)` contributes to the efficiency of the `myPow` function.

This line checks if the current least significant bit of the exponent p is 1. If it is, the current value of x (which represents a power of the base) is multiplied into the result. This allows the function to only perform multiplication when necessary, based on the binary representation of the exponent.

In the `myPow` function, the line `x *= x;` serves to __________ the base in each iteration.

square Signup and view all the answers

If the input `n` is negative, what transformation is applied to `x` and `p` in the `myPow` function?

<code>x</code> is inverted (<code>x = 1 / x</code>), and <code>p</code> is negated (<code>p = -p</code>). (C) Signup and view all the answers

The `main` function in the provided code includes comprehensive error handling for invalid inputs of `x` and `n`.

False (B) Signup and view all the answers

Explain the purpose of the `result` variable in the `myPow` function and how it accumulates the final power value.

The <code>result</code> variable is initialized to 1.0 and serves as the accumulator for the final power. It is multiplied by <code>x</code> only when the current bit of the exponent <code>p</code> is 1, effectively including the appropriate powers of <code>x</code> in the final result. Signup and view all the answers

The bitwise operator `&` in the expression `p & 1` is used to perform a(n) __________ operation.

AND Signup and view all the answers

What is the primary advantage of using fast exponentiation over a naive iterative approach for calculating powers?

Fast exponentiation significantly reduces the number of multiplication operations. (A) Signup and view all the answers

The `myPow` function will not work correctly if `x` is zero and `n` is negative.

True (A) Signup and view all the answers

Describe a potential scenario where the use of `double` for the base `x` could lead to inaccuracies in the `myPow` function.

Using <code>double</code> can lead to inaccuracies due to the limitations of floating-point representation. Repeated multiplication, especially with numbers close to 1, can accumulate rounding errors, leading to a result that deviates from the true value, particularly for large exponents. Signup and view all the answers

The operation `p /= 2` in the `myPow` function is equivalent to a right bit shift by __________ position(s).

one Signup and view all the answers

Which of the following best describes the purpose of the line `if (p < 0) { p = -p; x = 1 / x; }` in the provided code?

It ensures that the exponent <code>p</code> is always positive by inverting the base <code>x</code>. (A) Signup and view all the answers

Using recursion instead of iteration would generally improve the performance of the `myPow` function due to reduced overhead.

False (B) Signup and view all the answers

Explain how the `myPow` function could be adapted to handle complex numbers as the base `x`.

To handle complex numbers, the <code>double</code> type for <code>x</code> and <code>result</code> would need to be replaced with a complex number type (e.g., <code>std::complex</code>). The multiplication and division operations would then need to be performed using complex number arithmetic rules. The overall logic of the fast exponentiation algorithm would remain the same. Signup and view all the answers

The time complexity of the provided `myPow` algorithm is __________.

O(log |n|) Signup and view all the answers

What would happen if the `long long` type were replaced with `int` for the variable `p` and the input `n` is `INT_MIN`?

The program would experience integer overflow when negating <code>n</code>, leading to undefined behavior. (D) Signup and view all the answers

The `myPow` function is guaranteed to provide perfectly accurate results for all possible floating-point inputs due to the use of fast exponentiation.

False (B) Signup and view all the answers

Suggest a modification to the `myPow` function that could improve its handling of extremely large exponents without sacrificing significant performance.

One possible modification is to include checks for potential overflow during the squaring step (<code>x *= x</code>). If <code>x * x</code> exceeds the maximum representable value for <code>double</code>, the function could return positive or negative infinity, depending on the sign of <code>x</code>, or throw an exception to indicate the overflow. Signup and view all the answers

In the context of the fast exponentiation algorithm implemented in `myPow`, `exponentiation by __________` is another common name for the technique used.

squaring Signup and view all the answers

Flashcards

Fast Exponentiation

A method to efficiently calculate x to the power of n, significantly faster than multiplying x by itself n times.

Bitwise AND in Power Function

A bitwise AND operation is used to check if the least significant bit of 'p' is 1, determining if the current power of x should be included in the result.

Using 'long long' for Exponent

Converting the exponent to 'long long' prevents potential overflow issues that can occur when negating the smallest integer ('INT_MIN').

Handling Negative Exponents

If the exponent is negative, take the reciprocal of the base (x) and make the exponent positive (p).

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Squaring and Halving

Repeatedly squaring the base (x) and halving the exponent (p) allows the function to compute large powers with fewer calculations.

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Study Notes

The implementation calculates x raised to the power of n, denoted as x^n
It employs fast exponentiation, achieving a time complexity of O(log |n|).
Fast exponentiation is also known as exponentiation by squaring

Implementation Details

It uses long long for the exponent to prevent overflow, especially when n is INT_MIN.
If the exponent n is negative, it's converted to positive, and x is replaced by its reciprocal (1/x).

Algorithm Steps

Initialize a result variable to 1.0.
While the exponent p is greater than 0:
- If the current bit of p is 1 (checked using p & 1), multiply result by x.
- Square x and halve p in each iteration (x *= x and p /= 2).
Return the final result.

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Podcast

Questions and Answers

Why is `long long` used for the exponent `p` instead of `int`?

The provided `myPow` function has a time complexity of $O(n)$, where $n$ is the exponent.

Explain how the line `if (p & 1)` contributes to the efficiency of the `myPow` function.

In the `myPow` function, the line `x *= x;` serves to __________ the base in each iteration.

If the input `n` is negative, what transformation is applied to `x` and `p` in the `myPow` function?

The `main` function in the provided code includes comprehensive error handling for invalid inputs of `x` and `n`.

Explain the purpose of the `result` variable in the `myPow` function and how it accumulates the final power value.

The bitwise operator `&` in the expression `p & 1` is used to perform a(n) __________ operation.

What is the primary advantage of using fast exponentiation over a naive iterative approach for calculating powers?

The `myPow` function will not work correctly if `x` is zero and `n` is negative.

Describe a potential scenario where the use of `double` for the base `x` could lead to inaccuracies in the `myPow` function.

The operation `p /= 2` in the `myPow` function is equivalent to a right bit shift by __________ position(s).

Which of the following best describes the purpose of the line `if (p < 0) { p = -p; x = 1 / x; }` in the provided code?

Using recursion instead of iteration would generally improve the performance of the `myPow` function due to reduced overhead.

Explain how the `myPow` function could be adapted to handle complex numbers as the base `x`.

The time complexity of the provided `myPow` algorithm is __________.

What would happen if the `long long` type were replaced with `int` for the variable `p` and the input `n` is `INT_MIN`?

The `myPow` function is guaranteed to provide perfectly accurate results for all possible floating-point inputs due to the use of fast exponentiation.

Suggest a modification to the `myPow` function that could improve its handling of extremely large exponents without sacrificing significant performance.

In the context of the fast exponentiation algorithm implemented in `myPow`, `exponentiation by __________` is another common name for the technique used.