Given a string, s, determine the maximum number of unique substringsA substring is defined as a continuous sequence of characters within a string. into which s can be split. You can divide s into any sequence of nonempty substrings, ensuring their concatenation reconstructs the original string. However, each substring in the split must be distinct.

Constraints:

• $1 \leq$ s.length $\leq 16$

• s contains only lowercase English letters.

The problem requires splitting a given string into the maximum number of unique substrings. This involves exploring different ways to segment the string while ensuring each segment is distinct. For this purpose, we can use the backtracking approach, where different partition possibilities are tried recursively. The key idea is to maintain a set of seen or visited substrings and try adding new unique substrings at each step, backtracking when necessary.

We start by calling a recursive function that explores all possible ways to split the string into unique parts. We extract a substring at each step and check if it has already been used. If not, we add it to a set and recursively process the remaining part of the string. After exploring all possibilities for a particular split, we remove the substring from the set, allowing us to backtrack and try different segmentations. In this way, we consider all valid ways to split the string while keeping track of the maximum number of unique substrings found.

Here are the detailed steps of the algorithm that we have just discussed:

1. We start by initializing an empty set, seen, to keep track of unique substrings, which calls the recursive function backtrack starting from index 0. This recursive function is responsible for finding the maximum number of unique substrings starting from a given index.

2. The recursive function backtrack takes three parameters: s the input string, a start variable that keeps track of the position in the string from which we are trying to extract a new unique substring, and the set seen.

   1. If the start reaches the length of s, the entire string has been processed; we return 0 as no further splits can be made.

   2. We also initialize a variable max_count to store the maximum unique split count found during the recursive exploration.

   3. Next, we iterate over all possible substrings starting from the start.

      1. At each iteration, we extract the substring s[start:end] and store it in sub_string, representing the portion of s we consider at the current iteration. This substring extends from index start to end - 1.

      2. If sub_string is in seen, we skip it to ensure uniqueness in our partitioning. Otherwise, we add it to the set. This prevents future recursive calls from using the same substring within this partitioning attempt.

         1. Next, we recursively call backtrack(s, end, seen), moving from start to the end. This allows us to explore new substrings starting from the end onward.

         2. We add 1 to the result of backtrack because sub_string itself counts as one valid unique substring and update max_count to store the maximum number of unique substrings obtained from different partitioning paths.

         3. After exploring all possibilities for the current substring, we remove it from seen to allow backtracking and exploration of alternative splits.

   4. After exploring all possible substrings starting from the start, we return max_count.

The following illustration shows these steps in detail: