**π‘ Problem Formulation:** In many applications involving text processing, it’s essential to determine the number of distinct substrings that a given string can produce. This task becomes more challenging when addressing different queries, perhaps in a dynamic programming or database context. For example, given the input string `"aab"`

, we might want to know the number of unique substrings, which in this case would be `["a", "ab", "aab", "b", "aa"]`

, totaling to 5.

## Method 1: Brute Force Approach

Using a brute force approach involves iterating through all possible substrings of the given string and storing them in a set to ensure uniqueness. This method, while straightforward, can be computationally expensive for longer strings due to its quadratic time complexity.

Here’s an example:

def unique_substrings(s): substrings = set() for i in range(len(s)): for j in range(i + 1, len(s) + 1): substrings.add(s[i:j]) return len(substrings) unique_count = unique_substrings("aab") print(unique_count)

Output: 5

This code snippet defines a function `unique_substrings`

that takes a string `s`

and computes all possible substrings by two nested loops. Each substring is added to a set to avoid duplicates, and the number of unique substrings is returned by checking the length of the set.

## Method 2: Using itertools Combinations

Python’s `itertools`

module can be used to generate combinations of the indices of the input string and then create substrings accordingly. This method offers clean and readable code but doesn’t necessarily improve on the time complexity.

Here’s an example:

from itertools import combinations def unique_substrings_itertools(s): substrings = set([''.join(s[i:j]) for i, j in combinations(range(len(s) + 1), 2)]) return len(substrings) unique_count = unique_substrings_itertools("aab") print(unique_count)

Output: 5

The function `unique_substrings_itertools`

creates a set of substrings using a generator expression that iterates over pairs of indices created by `combinations`

. The combinations generated are used to slice the string `s`

and collect unique substrings.

## Method 3: Dynamic Programming

Dynamic programming might be utilized to reduce redundant calculations by storing already computed results. A matrix or hash map could be used to keep track of whether a substring has already been encountered. This method can be more efficient in certain cases but requires careful implementation to avoid common DP pitfalls.

Not Applicable: This task does not have an optimization structure that would benefit significantly from a standard dynamic programming approach. Therefore, we will omit an example for this method.

## Method 4: Using Trie Data Structure

A trie (prefix tree) can efficiently store and search for prefixes of strings. We can insert all suffixes of the string into the trie and count the number of nodes to find the number of unique substrings. This method is efficient in terms of lookup times but might have higher space complexity.

Here’s an example:

class TrieNode: def __init__(self): self.children = {} def insert_suffix(self, s): node = self for char in s: if char not in node.children: node.children[char] = TrieNode() node = node.children[char] def count_nodes(node): return 1 + sum(count_nodes(child) for child in node.children.values()) if node.children else 1 def unique_substrings_trie(s): root = TrieNode() for i in range(len(s)): root.insert_suffix(s[i:]) return count_nodes(root) - 1 unique_count = unique_substrings_trie("aab") print(unique_count)

Output: 5

The snippet includes a `TrieNode`

class that represents each node in the trie. The method `insert_suffix`

is used to insert all suffixes of a string into the trie. `count_nodes`

is a recursive function that counts all nodes in the trie, which corresponds to the number of unique substrings.

## Bonus One-Liner Method 5: Using Set Comprehension

A concise one-liner method to find the number of different substrings can be constructed using Python’s set comprehension. This combines the logic of looping over indices and building unique subsets in a compact form.

Here’s an example:

unique_count = len({s[i:j] for i in range(len(s)) for j in range(i + 1, len(s) + 1)}) print(unique_count)

Output: 5

This line of code creates a set of all possible substrings using a nested set comprehension. The length of this set represents the number of unique substrings, and this compact approach simplifies the code from the brute force method.

## Summary/Discussion

**Method 1: Brute Force Approach.**Easy to understand and implement. Inefficient for large strings due to its O(n^2) complexity.**Method 2: Using itertools Combinations.**Offers cleaner code, but does not improve on the brute force’s quadratic time complexity. Readability is a strength here.**Method 3: Dynamic Programming.**Though it is often used for optimization problems, dynamic programming is not particularly suited to this problem and was therefore omitted.**Method 4: Using Trie Data Structure.**Efficient look-up times and works well with incremental string queries. However, it can be space-intensive and more complex to implement.**Bonus Method 5: Using Set Comprehension.**A concise and Pythonic way to solve the problem. While it doesn’t offer a performance gain over the brute force method, the one-liner is elegant and simple.