Maximum Number of Non-Overlapping Substrings

This problem asks for finding the maximum number of non-overlapping substrings in a string. The solution follows a greedy approach that prioritizes substrings that do not overlap. The main trick is that two valid substrings cannot overlap unless one is nested inside the other, which simplifies the problem considerably.

Problem Statement

You are given a string s of lowercase English letters. Your task is to find the maximum number of non-empty substrings of s that do not overlap. If there are multiple solutions with the same number of substrings, return the one with the minimum total length. There is guaranteed to be a unique solution with the minimum length.

For example, in the string "adefaddaccc", the optimal solution is to choose the substrings "e", "f", and "ccc", as these are non-overlapping. The problem emphasizes a greedy approach, focusing on selecting substrings that minimize overlap, with the additional constraint of selecting the shortest possible combination when multiple valid solutions exist.

Examples

Example 1

Input: s = "adefaddaccc"

Output: ["e","f","ccc"]

The following are all the possible substrings that meet the conditions: [ "adefaddaccc" "adefadda", "ef", "e", "f", "ccc", ] If we choose the first string, we cannot choose anything else and we'd get only 1. If we choose "adefadda", we are left with "ccc" which is the only one that doesn't overlap, thus obtaining 2 substrings. Notice also, that it's not optimal to choose "ef" since it can be split into two. Therefore, the optimal way is to choose ["e","f","ccc"] which gives us 3 substrings. No other solution of the same number of substrings exist.

Example 2

Input: s = "abbaccd"

Output: ["d","bb","cc"]

Notice that while the set of substrings ["d","abba","cc"] also has length 3, it's considered incorrect since it has larger total length.

Constraints

• 1 <= s.length <= 105
• s contains only lowercase English letters.

Solution Approach

Greedy Substring Selection

The key to solving this problem is to prioritize selecting substrings that do not overlap. You can use a greedy approach where you begin by choosing substrings that have no intersection with each other. Ensure to validate that no substring contains another one, as this would invalidate the choice.

Efficient Substring Validation

To ensure substrings do not overlap, maintain an invariant by keeping track of their starting and ending indices. This will help you easily check if a new substring intersects with any of the already chosen ones. Efficient validation is crucial, especially when dealing with large strings.

Final Selection with Minimum Length

After selecting the non-overlapping substrings, compare their total lengths. If multiple combinations result in the same number of substrings, choose the one with the smallest total length. This ensures you get the optimal solution with minimal length.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time and space complexity depend on the approach used to maintain substring indices and perform validation. In the worst case, the approach may require linear time and space relative to the length of the string, O(n), for substring comparison and index tracking. However, optimizations in the selection process could reduce overhead.

What Interviewers Usually Probe

• Evaluates understanding of greedy algorithms and invariant validation.
• Tests ability to optimize substring selection for minimum length.
• Checks for efficiency in handling large strings with non-overlapping substring constraints.

Common Pitfalls or Variants

Common pitfalls

• Choosing substrings that overlap, which violates the problem’s constraints.
• Failing to track the starting and ending indices of chosen substrings.
• Not considering all valid substring combinations with the same number of substrings.

Follow-up variants

• Modify the problem to allow overlapping substrings and ask how the approach changes.
• Increase the size of the string and test how efficiently the algorithm handles larger inputs.
• Change the string to include repeating characters and check how that affects substring selection.

FAQ

What is the primary pattern in the 'Maximum Number of Non-Overlapping Substrings' problem?

The problem follows a greedy approach where you select substrings based on non-overlap, ensuring the total length is minimized.

How do I know if two substrings overlap in the 'Maximum Number of Non-Overlapping Substrings' problem?

You can check if two substrings overlap by comparing their start and end indices. If one substring’s range intersects the other’s, they overlap.

What is the significance of selecting the minimum total length in the problem?

Selecting the minimum total length ensures that you return the optimal solution when multiple valid combinations of substrings are possible.

How does the greedy algorithm apply to 'Maximum Number of Non-Overlapping Substrings'?

The greedy algorithm helps prioritize selecting non-overlapping substrings that are the smallest possible, minimizing the total length while maximizing the count.

What happens if substrings are allowed to overlap in this problem?

Allowing substrings to overlap would change the problem's constraints, potentially leading to multiple valid solutions, but it would complicate substring selection.