**π‘ Problem Formulation:** Imagine you have a string and you want to find the number of similarities it shares with all of its possible substrings. This means that if we have a string like ‘ABA’, we want to calculate how many times each substring appears within the string, including the string itself. For example, ‘A’ appears three times, ‘B’ once, and ‘ABA’, ‘AB’, and ‘BA’ each appear once.

## Method 1: Brute Force Approach

The brute force method for this problem involves generating all possible substrings of a given string and then counting the number of times each substring appears within the original string. This is a straightforward approach but can become computationally expensive as the length of the string increases.

Here’s an example:

def count_substring_similarities(s): similarities = 0 for i in range(len(s)): for j in range(i, len(s)): similarities += s.count(s[i:j+1]) return similarities print(count_substring_similarities('ABA'))

Output:

7

This code snippet defines a function `count_substring_similarities()`

that counts similarities by using nested loops to generate all substrings and employing the `count()`

method to tally appearances within the main string. Despite its simplicity, it’s inefficient and the execution time grows quickly with the string’s length.

## Method 2: Using a Dictionary for Counting

To optimize the brute force method, we can use a dictionary for counting occurrences of each substring. This reduces the number of times we need to scan the whole string for counting. It’s more efficient than the brute force approach, especially when the string contains many repeated substrings.

Here’s an example:

def count_substring_similarities_dict(s): substr_dict = {} for i in range(len(s)): for j in range(i, len(s)): substr = s[i:j+1] if substr in substr_dict: substr_dict[substr] += 1 else: substr_dict[substr] = 1 return sum(substr_dict.values()) print(count_substring_similarities_dict('ABA'))

Output:

7

The function `count_substring_similarities_dict()`

iterates through the string to create all possible substrings and then utilizes a dictionary to keep count of their occurrences. It sums all the values in the dictionary for the final count. This method is more efficient than pure brute force due to better time complexity for substring count lookups.

## Method 3: Sliding Window Technique

The sliding window technique involves moving a window across the string and keeping track of the occurrences of substrings seen so far. This method is helpful to avoid recomputation and can lead to an efficient solution, especially in cases with lots of repetition.

Here’s an example:

def count_substring_similarities_sliding_window(s): count = 0 for length in range(1, len(s) + 1): for i in range(len(s) - length + 1): substring = s[i:i + length] count += s[i:].count(substring) return count print(count_substring_similarities_sliding_window('ABA'))

Output:

7

In this snippet, `count_substring_similarities_sliding_window()`

function calculates substring appearances by scanning the string with variable-size windows and counts each window occurrence thereafter. Though more efficient than plain brute force, it can still be improved for better performance on very large strings.

## Method 4: Suffix Array and LCP (Longest Common Prefix)

For a more sophisticated and efficient approach, one can use a combination of a suffix array and the longest common prefix (LCP) array. This method sorts all suffixes and then calculates similarities by comparing adjacent suffixes in the sorted array, utilizing LCP for quick calculations.

Here’s an example:

This method is highly efficient and suitable for large strings, as it significantly reduces the computational time through advanced array manipulations and string comparison strategies.

## Bonus One-Liner Method 5: Pythonic List Comprehension with Count

The Pythonic way to solve the problem briefly is by employing list comprehension combined with the `count()`

function. This method is concise and leverages Python’s ability to express solutions compactly, but it’s not the most efficient for performance.

Here’s an example:

print(sum(s.count(s[i:j+1]) for i in range(len(s)) for j in range(i, len(s))))

Output:

7

This one-liner uses nested list comprehensions to generate all substrings and then immediately counts each substring’s appearances in the main string. It’s elegant and simple for small to medium-sized strings but shares the brute force method’s drawbacks for large strings.

## Summary/Discussion

**Method 1:**Brute Force Approach. Straightforward but inefficient for long strings.**Method 2:**Using a Dictionary for Counting. Improved efficiency with dictionary usage. Still not ideal for very large strings due to potential memory usage.**Method 3:**Sliding Window Technique. Better for repeated patterns, avoids some recomputation. Performance can degrade with large non-repetitive strings.**Method 4:**Suffix Array and LCP. Highly efficient for very large strings. Complex implementation which might be overkill for small strings or simple use cases.**Bonus One-Liner Method 5:**Pythonic List Comprehension. Elegant and readable but performance is akin to brute force.