# 5 Best Ways to Find Maximum Difference With Next Smaller Number in Python

π‘ Problem Formulation: The task is to write a Python program that finds the maximum difference between a number in a list and its next smaller successor. For instance, given the list [2, 3, 10, 6, 4, 8, 1], the maximum difference should be between 10 and 4 (resulting in 6), as 4 is the next smaller number after 10.

## Method 1: Brute Force Approach

This method iterates through each element in the list and compares it with each of the following elements to find the next smaller number and calculate the differences, consequently keeping track of the maximum difference found.

Here’s an example:

```def max_difference_brute(nums):
max_diff = 0
for i in range(len(nums) - 1):
for j in range(i+1, len(nums)):
if nums[i] > nums[j]:
max_diff = max(max_diff, nums[i] - nums[j])
break
return max_diff

# Example list
numbers = [2, 3, 10, 6, 4, 8, 1]
print(max_difference_brute(numbers))```

Output:

`6`

This code snippet iterates over the list using two nested loops. The outer loop picks each number, while the inner loop finds the next smaller number after the picked number and calculates the difference. If this difference is greater than the previous differences, it updates the maximum difference found.

## Method 2: Using a Stack

A stack can be used to keep track of the elements and efficiently find the next smaller element for every number in the list. This way, we can make a single pass over the list, rather than using nested iterations.

Here’s an example:

```def max_difference_stack(nums):
stack = []
max_diff = 0
for num in nums:
while stack and stack[-1] > num:
max_diff = max(max_diff, stack.pop() - num)
stack.append(num)
return max_diff

# Example list
numbers = [2, 3, 10, 6, 4, 8, 1]
print(max_difference_stack(numbers))```

Output:

`6`

The code snippet works by pushing each number onto a stack. When the next number is smaller than the stack’s top element, it calculates the difference, updates the maximum, and pops from the stack. By doing this, each element is compared only with its next smaller number.

## Method 3: Optimized Linear Traversal

This method improves upon the brute force approach by traversing the list only once. It keeps track of the maximum element encountered so far and calculates the difference with each subsequent smaller element to find the maximum difference.

Here’s an example:

```def max_difference_optimized(nums):
max_num = nums[0]
max_diff = 0
for num in nums[1:]:
if num < max_num:
max_diff = max(max_diff, max_num - num)
else:
max_num = num
return max_diff

# Example list
numbers = [2, 3, 10, 6, 4, 8, 1]
print(max_difference_optimized(numbers))```

Output:

`6`

This code snippet initializes the maximum number as the first element and iterates through the rest of the list. If the current number is smaller than the maximum number, it calculates the difference. If the current number is larger, it updates the maximum number. This process ensures we only compare numbers with their previous larger values.

## Method 4: Using Python Libraries

Python’s built-in functions can be combined to create an elegant solution. This method may use functions like `enumerate()`, `max()`, and list comprehensions to achieve the same result as the previous methods but with less explicit looping.

Here’s an example:

```def max_difference_libraries(nums):
return max([max(nums[:i]) - nums[i] for i in range(1, len(nums)) if nums[i] < max(nums[:i])], default=0)

# Example list
numbers = [2, 3, 10, 6, 4, 8, 1]
print(max_difference_libraries(numbers))```

Output:

`6`

The snippet generates a list of differences between the maximum value up to the current index and the current element if the current element is smaller. Then, it uses Python’s `max()` function to find the highest value within this list of differences.

## Bonus One-Liner Method 5: Using a Lambda Function

For those who appreciate Python’s ability to condense logic into a single line, a lambda function along with the `reduce()` function can be used to compute the maximum difference in a compact form.

Here’s an example:

```from functools import reduce

numbers = [2, 3, 10, 6, 4, 8, 1]
max_diff = reduce(lambda acc, x: max(acc[0], acc[1]-x if x<acc[1] else 0), numbers, (0, float('inf')))[0]
print(max_diff)```

Output:

`6`

This one-liner uses the `reduce()` function to carry two values through the iteration: the accumulated largest difference and the current maximum value. By setting the initial maximum to infinity, we ensure that we can calculate the difference with the first actual number in the list. The lambda function is then applied to each element, updating these values appropriately.

## Summary/Discussion

• Method 1: Brute Force Approach. Simple to understand. Can be slow for large lists as it has a quadratic time complexity.
• Method 2: Using a Stack. More efficient with linear time complexity. Utilizes extra space for the stack.
• Method 3: Optimized Linear Traversal. Efficient with linear time complexity and no extra space. Compares only with previous larger elements, potentially skipping some computations.
• Method 4: Using Python Libraries. Elegant and uses powerful Python features for concise code. Potentially less efficient due to multiple calls to `max()` function.
• Bonus Method 5: One-Liner Lambda Function. Very concise and uses advanced Python features. Can be difficult to read and understand for less experienced programmers.