Select K Disjoint Special Substrings

To solve Select K Disjoint Special Substrings, focus on using state transition dynamic programming combined with hash tables to track unique substrings. Identify at most 26 potential starting and ending points, then iteratively build valid selections while ensuring no overlap. This approach guarantees correct detection of k disjoint special substrings while maintaining efficient computation.

Problem Statement

Given a string s of length n and an integer k, determine whether it is possible to select exactly k disjoint special substrings. A special substring is defined by its uniqueness according to a given property, and all selected substrings must not overlap in s.

You need to return true if k such disjoint special substrings can be selected and false otherwise. Examples include s = "abcdbaefab", k = 2 returning true, and s = "cdefdc", k = 3 returning false. Constraints: 2 <= n <= 5 * 10^4, 0 <= k <= 26, s consists of lowercase English letters only.

Examples

Example 1

Input: s = "abcdbaefab", k = 2

Output: true

Example 2

Input: s = "cdefdc", k = 3

Output: false

There can be at most 2 disjoint special substrings: "e" and "f" . Since k = 3 , the output is false .

Example 3

Input: s = "abeabe", k = 0

Output: true

Example details omitted.

Constraints

• 2 <= n == s.length <= 5 * 104
• 0 <= k <= 26
• s consists only of lowercase English letters.

Solution Approach

Precompute Potential Substrings

Scan the string to identify first and last occurrences of each letter, giving at most 26 start and end points. Use these to generate candidate special substrings while respecting disjoint requirements.

Dynamic Programming Selection

Use a state transition DP array where dp[i] tracks the maximum number of disjoint special substrings up to index i. Transition by considering adding a new substring if it does not overlap the previous selections.

Validate Against k

Iterate through the DP table to check if it is possible to reach exactly k disjoint special substrings. Return true if dp[n] >= k, otherwise return false. This ensures efficient correctness with O(n) candidate evaluation.

Complexity Analysis

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

Time complexity depends on processing at most 26 start and end points for candidate substrings, giving O(n + 26^2) in practice. Space complexity includes DP array of size n and hash tables for substring checks, O(n + 26).

What Interviewers Usually Probe

• Check if you can precompute all potential start and end points efficiently.
• Clarify that substrings must be completely disjoint and cannot overlap in any index.
• Ask how DP can track the maximum number of valid special substrings up to each index.

Common Pitfalls or Variants

Common pitfalls

• Overcounting substrings that overlap partially or fully.
• Ignoring edge cases when k = 0 or k exceeds possible disjoint substrings.
• Using brute-force substring comparison instead of leveraging start and end points.

Follow-up variants

• Maximize the number of disjoint special substrings instead of checking a fixed k.
• Select disjoint substrings with additional length constraints or patterns.
• Count disjoint substrings with overlapping allowed but limited to a threshold.

FAQ

What defines a special substring in this problem?

A special substring is one that meets the uniqueness property specified, typically tracked via first and last occurrences of letters.

How does state transition dynamic programming apply here?

DP tracks the maximum number of disjoint special substrings up to each index, allowing safe addition of new substrings without overlap.

Can k be zero or exceed possible disjoint substrings?

Yes, k = 0 should return true, and if k exceeds the maximum possible disjoint substrings, the answer is false.

Why are hash tables useful for this problem?

Hash tables help quickly locate first and last occurrences, reducing the search space for candidate substrings.

What is the main failure mode to avoid?

Selecting overlapping substrings or miscounting potential candidates, which leads to incorrect true/false results.