Problem Description

The problem requires calculating the total number of distinct substrings that can be formed from a given string s. A substring is defined as any sequence of characters that can be derived from the string by deleting zero or more characters from the beginning and zero or more characters from the end. For example, in the string "abc", some of the substrings include "a", "ab", "abc", and "b". The key here is to ensure that each substring is counted only once, even if it occurs multiple times in different positions of the string.

Intuition

The intuitive approach to solving this problem relies on generating all possible substrings of the given string and then counting the unique ones. To achieve this, two nested loops can be used where the outer loop iterates over the start position of a substring and the inner loop iterates over the end position. For each pair of start and end positions, a substring is extracted and stored in a set, which inherently eliminates duplicates due to its unordered collection of unique elements. After all possible substrings are added to the set, the number of distinct substrings can be determined by simply taking the size of the set.

This approach is easy to implement and understand, and although not the most efficient in terms of time complexity, it works well for strings of reasonable length.

Learn more about Trie patterns.

Solution Approach

The solution's implementation relies on the concept of hash sets and comprehensions in Python. Here's a step-by-step explanation of the code provided:

1. Initialize the length of the string n = len(s). This is required to control the loops' range.
2. A set is created using a set comprehension ({}) that iterates through all possible start (i) and end (j) positions of the substrings. In Python, sets store only unique elements which means any duplicate substrings generated will not be added to the set.
3. The outer loop begins at the first character (i starts at 0) and continues until the end of the string (i < n).
4. The inner loop begins at the character immediately following the start position (j starts at i + 1) and continues to the end of the string (j <= n). This ensures that we consider all possible end positions for substrings starting at position i.
5. During each iteration of the inner loop, a new substring s[i:j] is sliced from the original string s, with i inclusive and j exclusive.
6. The sliced substring is then added to the set. If the substring already exists in the set, it's ignored, as sets do not store duplicates.
7. After the looping, the set will contain all possible unique substrings of the original string.
8. The total number of distinct substrings is the size of the set, which is obtained using the len() function on the set.

To summarize, the algorithm leverages a brute-force approach to generate all substrings and a data structure (set) that naturally handles the uniqueness constraint. It's excellent for its simplicity and ease of understanding. However, for large strings, this approach may lead to time-consuming operations because the number of substrings grows quadratically with the length of the input string.

Example Walkthrough

Let us consider a simple example using the string "abc" to illustrate the solution approach.

1. First, we initialize the length of the string n = len(s). In our case, n would be 3, since "abc" is three characters long.
2. We then create a set to store our substrings. This set will only keep unique elements.
3. We start our outer loop, where i is our starting point of the substring. For our string "abc", i will take values 0, 1, and 2.
4. For each value of i, the inner loop starts from i + 1 and continues until the end of the string. This is to ensure we cover all possible substrings starting at index i.
5. As the loops execute, we generate substrings using s[i:j] and add them to our set. For i = 0, j will go from 1 to 3, generating "a", "ab", and "abc".
6. We then move to i = 1, where j will go from 2 to 3, giving us substrings "b" and "bc".
7. Lastly, for i = 2, j will only be 3, which gives us the substring "c".
8. All these substrings are added to our set: {"a", "ab", "abc", "b", "bc", "c"}.
9. After both loops have completed, we end up with a set containing all unique substrings.
10. The total number of distinct substrings is simply the size of this set. Here, we have 6 unique substrings. Thus, the answer is 6.

Through this example, we can see how the proposed solution methodically enumerates all the substrings while enforcing uniqueness by utilizing the properties of a set.

Solution Implementation

Time and Space Complexity

The given Python code snippet defines a function that counts the distinct substrings of a given string s. To analyze the complexity, let's consider n to be the length of the input string s.

Time Complexity: O(n^3)

The time complexity of the given function is O(n^3) for the following reasons:

1. Two nested loops iterate over the string to generate all possible substrings, which leads to O(n^2) complexity.

2. For each iteration, a substring is created using slicing s[i:j]. In Python, slicing a string is O(k) where k is the number of characters in the substring. In the worst case, k can be n leading to O(n) for the slicing operation.

3. These two combined give us a total of O(n^2) * O(n) = O(n^3) complexity since for each pair (i, j), a new substring of the string s can be created with O(n) complexity.

Space Complexity: O(n^2)

The space complexity of the given function is O(n^2) for the following reasons:

1. A set is used to store the distinct substrings. In the worst case, the number of distinct substrings of a string of length n is the sum of the series 1 + 2 + 3 + ... + n, which results in n(n + 1)/2 substrings, equating to an O(n^2) space complexity.

2. Even though string slicing itself does not take additional space since it's creating new strings that reference the same characters within the original string, in Python, slicing creates new objects and the total space in the set would be the sum of the length of all unique substrings.

Considering these points, the complexities are as follows:

• Time Complexity: O(n^3)
• Space Complexity: O(n^2)

Learn more about how to find time and space complexity quickly using problem constraints.