How would you code a shuffle() function that randomizes an array uniformly using roll()?

Question Analysis

The question asks you to implement a shuffle() function that randomizes an array uniformly. The key requirement is to use a roll() function to achieve this. The roll() function can be assumed to generate a random number in a specific range, similar to rolling a die where the range is defined. Understanding this, it is important to ensure that the shuffle algorithm is unbiased, meaning every permutation of the array is equally likely. A well-known algorithm for this purpose is the Fisher-Yates shuffle (also known as the Knuth shuffle), which is effective in generating a uniform distribution of the array elements.

Answer

To implement the shuffle() function using the roll() function, we can use the Fisher-Yates shuffle algorithm. Below is a step-by-step guide and implementation:

1. Initialize: Start from the end of the array and iterate backward.
2. Generate Random Index: For each position i in the array, use roll() to generate a random index j such that 0 <= j <= i.
3. Swap Elements: Swap the elements at indices i and j.
4. Repeat: Continue the process until the beginning of the array is reached.

Here's how you can implement this:

import random

def roll(max_value):
    # Simulates a dice roll that returns a number between 0 and max_value
    return random.randint(0, max_value)

def shuffle(array):
    n = len(array)
    for i in range(n-1, 0, -1):
        j = roll(i)  # Get a random index such that 0 <= j <= i
        array[i], array[j] = array[j], array[i]  # Swap elements
    return array

Key Points:

• Uniform Distribution: The Fisher-Yates algorithm ensures that each permutation of the array is equally likely, achieving uniform randomness.
• Efficiency: The algorithm runs in O(n) time complexity, making it efficient for shuffling large arrays.
• Randomness Source: The roll() function is crucial as it provides the randomness needed to determine which elements to swap.

By following these steps, you can successfully implement a shuffle function that randomizes an array uniformly, meeting the requirements of the question.