Generating all unique substrings of a given string is a fundamental problem in computer science with applications in text processing, data compression, and bioinformatics. This operation involves extracting every possible contiguous sequence of characters from a string and ensuring that each substring appears only once in the final list.

Concepts and Definitions

Before diving into algorithms, let’s define some key concepts:

• String: A sequence of characters, e.g., `"abc"`.
• Substring: A contiguous sequence of characters within a string. For example, `"ab"`, `"bc"`, and `"abc"` are substrings of `"abc"`.
• Unique Substrings: Substrings that appear only once in the list of all possible substrings.

Problem Explanation

The task is to generate a list of all possible unique substrings for any given string. Consider the string `s = "abc"`. The substrings are:

• Length 1: `a`, `b`, `c`
• Length 2: `ab`, `bc`
• Length 3: `abc`

Therefore, the unique substrings of `"abc"` are: `["a", "b", "c", "ab", "bc", "abc"]`.

Algorithm Overview

A straightforward approach to generate all unique substrings involves iterating over every possible starting and ending position within the string:

1. Initialize: Create a set to hold substrings and ensure uniqueness.
2. Nested Loop:
   • Use an outer loop to fix the starting position of the substring.
   • Use an inner loop to define the ending position of the substring and extract the substring.
   • Add each substring to the set.
3. Convert Set to List: Convert the set to a list to obtain the result.

Example Implementation

Here is a Python example implementing the above logic:

• Empty String: If the string is empty, the set of substrings is also empty.
• Single Character: For a single character string, the only substring is itself.
• Suffix Array: An array of all starting positions of suffixes of a string, sorted lexicographically.
• Suffix Tree: A compressed trie of all the suffixes of a string, allowing for faster substring identification and manipulation.