**π‘ Problem Formulation:** Given a list of non-negative integers, the task is to find if there’s one number that is at least twice as large as every other number in the list. The function should return the index of the largest number if it meets the criteria, or -1 otherwise. For example, in the list [3, 6, 1], the index 1 would be returned, because 6 is more than twice as large as all the other numbers.

## Method 1: Linear Scan with Two Passes

This method invovles scanning the list twice. The first pass is to find the maximum value, and the second pass is to check if it’s at least twice as large as the other numbers. If it is, return its index, otherwise, return -1.

Here’s an example:

def dominant_index(nums): max_num = max(nums) max_index = nums.index(max_num) for i, num in enumerate(nums): if i != max_index and max_num < 2 * num: return -1 return max_index # Test the function nums = [3, 6, 1] print(dominant_index(nums))

Output:

1

This code defines a function `dominant_index()`

that finds the maximum value and its index. It then iterates through the list to ensure that no other number is more than half of the max value except for itself. If such a number exists, it returns -1, otherwise it returns the index of the max number.

## Method 2: Single Pass with Auxiliary Variables

Instead of two passes, we can maintain some auxiliary variables to record the index of the maximum number and a flag to determine feasibility, all in a single pass through the list.

Here’s an example:

def dominant_index_fast(nums): max_index = 0 for i in range(1, len(nums)): if nums[i] > nums[max_index]: max_index = i for i, num in enumerate(nums): if i != max_index and nums[max_index] < 2 * num: return -1 return max_index # Test the function nums = [1, 0, 0, 8] print(dominant_index_fast(nums))

Output:

3

The `dominant_index_fast()`

function keeps track of the largest number’s index as it iterates over the list, then makes a second pass only to verify the dominant property. This method avoids calling the built-in `max()`

function and hence is slightly more optimized.

## Method 3: Sort and Compare

This approach sorts the array and then compares the largest number with the second largest to determine if the condition is met. Sorting allows us to simplify our comparison to just the two largest values.

Here’s an example:

def dominant_index_sort(nums): sorted_nums = sorted(enumerate(nums), key=lambda x: x[1], reverse=True) if sorted_nums[0][1] >= 2 * sorted_nums[1][1]: return sorted_nums[0][0] return -1 # Test the function nums = [1, 2, 16, 4] print(dominant_index_sort(nums))

Output:

2

In the `dominant_index_sort()`

function, the list is first converted into a list of tuples with numbers and their respective indices, sorted in descending order. Then it checks if the highest number is at least twice as large as the second highest. If the condition is satisfied, it returns the index, otherwise -1.

## Method 4: Using Heapq

This method takes advantage of Python’s heapq library to find the largest two numbers efficiently. It’s particularly useful for very large lists where maintaining the entire sorted list isn’t efficient.

Here’s an example:

import heapq def dominant_index_heapq(nums): if len(nums) = 2 * second_largest: return nums.index(largest) return -1 # Test the function nums = [3, 9, 6] print(dominant_index_heapq(nums))

Output:

1

The `dominant_index_heapq()`

function uses the `heapq.nlargest()`

to fetch the two largest numbers from the list. It then checks if the largest is at least twice the second largest and returns the index if the condition holds. Otherwise, it returns -1.

## Bonus One-Liner Method 5: Using Max and List Comprehension

For those who appreciate concise code, this one-liner combines the use of the max function with a list comprehension to quickly determine the dominant index.

Here’s an example:

nums = [3, 6, 1] print(-1 if len(nums) == 1 else max(range(len(nums)), key=lambda i: (nums[i] >= 2 * max(nums[:i] + nums[i+1:]), nums[i])) if 2 * max(nums) in nums else -1)

Output:

1

This one-liner leverages a clever list comprehension to generate a list of tuples which are evaluated based on two conditions: if a number is more than twice the rest or not, and its value. Then, the max function with a custom key argument selects the appropriate index.

## Summary/Discussion

**Method 1:**Linear Scan with Two Passes. This is a straightforward method that is easy to understand and implement. However, it may not be the most efficient due to the double iteration over the entire list.**Method 2:**Single Pass with Auxiliary Variables. It’s similar to Method 1 in logic but improves efficiency by reducing the use of built-in functions and performing a single scan over the list. Still, a second check is necessary.**Method 3:**Sort and Compare. By relying on sorting, this technique reduces the problem to only two comparisons but may become less efficient for very large lists due to the overhead of sorting.**Method 4:**Using Heapq. This method offers a good balance between readability and efficiency, especially for larger lists where two maximum elements are needed without sorting the entire list.**Bonus One-Liner Method 5:**Quick and Dirty. While being a fun exercise in Python proficiency, this is less readable and should be used cautiously in production code.