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Select K Disjoint Special Substrings

Medium

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Topics:

Strings

Given a string s of length n and an integer k, determine whether it is possible to select k disjoint special substrings.

A special substring is a substring where:

Any character present inside the substring should not appear outside it in the string.
The substring is not the entire string s.

Note that all k substrings must be disjoint, meaning they cannot overlap.

Return true if it is possible to select k such disjoint special substrings; otherwise, return false.

Example 1:

Input: s = "abcdbaefab", k = 2

Output: true

Explanation:

We can select two disjoint special substrings: "cd" and "ef".
"cd" contains the characters 'c' and 'd', which do not appear elsewhere in s.
"ef" contains the characters 'e' and 'f', which do not appear elsewhere in s.

Example 2:

Input: s = "cdefdc", k = 3

Output: false

Explanation:

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

Constraints:

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

Try coding it in LeetCode

Select K Disjoint Special Substrings

Medium

Asked by:

15 views

Topics:

Strings

Given a string s of length n and an integer k, determine whether it is possible to select k disjoint special substrings.

A special substring is a substring where:

Any character present inside the substring should not appear outside it in the string.
The substring is not the entire string s.

Note that all k substrings must be disjoint, meaning they cannot overlap.

Return true if it is possible to select k such disjoint special substrings; otherwise, return false.

Example 1:

Input: s = "abcdbaefab", k = 2

Output: true

Explanation:

We can select two disjoint special substrings: "cd" and "ef".
"cd" contains the characters 'c' and 'd', which do not appear elsewhere in s.
"ef" contains the characters 'e' and 'f', which do not appear elsewhere in s.

Example 2:

Input: s = "cdefdc", k = 3

Output: false

Explanation:

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

Constraints:

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

Try coding it in LeetCode

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What are the constraints on the length of the input string and the value of K? Is K always a valid value (i.e., K >= 0 and <= a reasonable upper bound)?
What defines a "special substring"? Is it a substring with some specific property or is this property also part of the input?
If there are multiple sets of K disjoint special substrings, should I return any valid set, or is there a specific criteria for choosing between them (e.g., maximizing total length, lexicographically smallest starting indices)?
If it is not possible to select K disjoint special substrings, what should the function return? Should I return null, an empty list, or throw an exception?
What characters can the input string contain? Are there any restrictions on the types of characters (e.g., only lowercase English letters, ASCII characters, Unicode characters)?

Brute Force Solution

Approach

The brute force approach to this problem means we're going to try every possible combination to find the best one. We'll check all possible substrings, and for each substring, we check all other substrings to see if they are disjoint. Finally, we choose K of them that satisfy the special condition.

Here's how the algorithm would work step-by-step:

First, we need to consider all possible pieces of the original string. Think of cutting the string in every possible place to create many different substrings.
Next, we check if each of these pieces meet the 'special' condition. Only some of these pieces will qualify.
Now, we start selecting a few pieces to see if they are disjoint (not overlapping). We begin by checking every pair of pieces to see if they overlap at any point.
If two pieces don't overlap, we continue checking if we can add a third special piece that doesn't overlap with the first two, and so on, until we have checked all possible groups of K disjoint special substrings.
We keep track of the 'best' combination of K disjoint special special substrings, according to some rule (such as, maximize the length of the special substrings).
After exhausting all combinations, we pick the 'best' combination of K disjoint special substrings we found.

Code Implementation

def select_k_disjoint_special_substrings(input_string, k_value):    all_substrings = []
    string_length = len(input_string)

    # Generate all possible substrings.
    for i in range(string_length):
        for j in range(i, string_length):
            substring = input_string[i:j+1]
            all_substrings.append((substring, i, j))

    special_substrings = []
    #In this implementation, all substrings are considered special.    special_substrings = all_substrings

    best_combination = []
    max_length = 0

    # Iterate through all combinations.
    for i in range(1 << len(special_substrings)):
        current_combination = []
        for j in range(len(special_substrings)):
            if (i >> j) & 1:
                current_combination.append(special_substrings[j])

        # Check for disjointedness and number of substrings.
        if len(current_combination) == k_value:
            is_disjoint = True
            for index_one in range(len(current_combination)):
                for index_two in range(index_one + 1, len(current_combination)):
                    substring_one = current_combination[index_one]
                    substring_two = current_combination[index_two]
                    if not (substring_one[2] < substring_two[1] or substring_two[2] < substring_one[1]):
                        is_disjoint = False
                        break
                if not is_disjoint:
                    break

            # If disjoint, compute total length and update best combination
            if is_disjoint:
                total_length = sum(len(substring[0]) for substring in current_combination)
                if total_length > max_length:
                    max_length = total_length
                    best_combination = current_combination

    return best_combination

Big(O) Analysis

Time Complexity

O(n^(2k)) – Generating all possible substrings from a string of length n takes O(n^2) time. Checking if each substring is 'special' takes O(1) time per substring so it doesn't affect the overall time complexity. We need to consider all combinations of K disjoint substrings, and since there are O(n^2) substrings, selecting K of them requires considering (n^2 choose K) combinations, which is O((n^2)^K). Checking if K substrings are disjoint requires comparing each pair of substrings, which takes O(k^2) time, but since k is constant compared to n, this simplifies to O(1). Therefore, the overall time complexity is O(n^(2k)).

Space Complexity

O(N^2) – The algorithm considers all possible substrings, which can be up to N^2 where N is the length of the input string. It then needs to store these 'special' substrings for disjoint checking which results in an auxiliary space of O(N^2). Additionally, keeping track of the 'best' combination among all possible combinations of K disjoint special substrings requires potentially storing all special substrings again, hence O(N^2). Therefore, the auxiliary space complexity is O(N^2).

Optimal Solution

Approach

The best way to solve this is to use a dynamic programming approach. We try to find the best solution for smaller parts of the problem and then use those to build the bigger, final answer. This avoids checking every possible combination, making it much faster.

Here's how the algorithm would work step-by-step:

Think of each possible place where a substring could end as a potential stopping point.
For each stopping point, figure out the best score you can get by including a substring that ends there, considering you need to select 'K' non-overlapping substrings.
To figure out the best score for each stopping point, consider the best score from the previous stopping points where it's possible to start a substring without overlapping.
We can calculate this using information we have from previous positions to figure out how to get to that point.
Keep track of these scores as you go, always storing the best solution for each endpoint.
Once you've reached the end of the entire string, the best score you've saved is your answer.

Code Implementation

def select_k_disjoint_substrings(input_string, k_value, special_substrings):
    string_length = len(input_string)
    dp_table = [[(0) for _ in range(string_length + 1)] for _ in range(k_value + 1)]

    for substrings_selected in range(1, k_value + 1):
        for ending_index in range(1, string_length + 1):
            dp_table[substrings_selected][ending_index] = dp_table[substrings_selected][ending_index - 1]
            for start_index in range(1, ending_index + 1):
                substring = input_string[start_index - 1:ending_index]

                # Check if substring is special
                if substring in special_substrings:
                    # To avoid overlapping, update dp_table
                    if start_index == 1:
                        dp_table[substrings_selected][ending_index] = max(
                            dp_table[substrings_selected][ending_index], len(substring)
                        )
                    else:
                        # Add length to previously found best score
                        dp_table[substrings_selected][ending_index] = max(
                            dp_table[substrings_selected][ending_index],
                            dp_table[substrings_selected - 1][start_index - 1] + len(substring),
                        )

    # The value at the end is the maximum length.
    return dp_table[k_value][string_length]

Big(O) Analysis

Time Complexity

O(n*k) – The algorithm iterates through all possible ending positions of substrings, which is O(n). For each ending position, it considers the best possible score attainable by including a valid substring ending at that position, given that we need to select k disjoint substrings. Determining the best score at each ending position requires checking previous non-overlapping positions and maintaining k scores, resulting in an O(k) operation within each of the n iterations. Thus, the overall time complexity is O(n*k).

Space Complexity

O(N*K) – The dynamic programming approach outlined keeps track of scores for each possible stopping point (up to N) and for each selected substring (up to K). This implies storing intermediate results in a data structure, likely a 2D array or table of size N*K, to store the best scores calculated so far. Therefore, the auxiliary space used grows linearly with both the length of the string (N) and the number of substrings to select (K). Consequently, the space complexity is O(N*K).

Edge Cases

Case	How to Handle
Empty input string	Return an empty list, as no substrings can be formed.
k is 0	Return an empty list as no substrings need to be selected.
k is greater than the maximum possible number of disjoint special substrings	Return the maximum possible number of disjoint special substrings instead of throwing an error.
String contains no special characters	Return an empty list, since no 'special' substring can be formed.
Overlapping special substrings are present	The solution should prioritize non-overlapping substrings, potentially using a greedy or dynamic programming approach.
Input string with very long length, close to memory limits	Ensure the algorithm's memory usage is optimized to avoid exceeding memory limits, possibly by processing substrings in chunks.
Multiple valid sets of k disjoint special substrings exist	The algorithm should either return any valid set or specify a criterion for selecting a particular set (e.g., lexicographically smallest, longest substrings).
k is a very large number	Check for integer overflow if k is used in calculations related to string indices or substring lengths; use appropriate data types.

Empty input string

How to Handle:

Return an empty list, as no substrings can be formed.

k is 0

How to Handle:

Return an empty list as no substrings need to be selected.

k is greater than the maximum possible number of disjoint special substrings

How to Handle:

Return the maximum possible number of disjoint special substrings instead of throwing an error.

String contains no special characters

How to Handle:

Return an empty list, since no 'special' substring can be formed.

Overlapping special substrings are present

How to Handle:

The solution should prioritize non-overlapping substrings, potentially using a greedy or dynamic programming approach.

Input string with very long length, close to memory limits

How to Handle:

Ensure the algorithm's memory usage is optimized to avoid exceeding memory limits, possibly by processing substrings in chunks.

Multiple valid sets of k disjoint special substrings exist

How to Handle:

The algorithm should either return any valid set or specify a criterion for selecting a particular set (e.g., lexicographically smallest, longest substrings).

k is a very large number

How to Handle:

Check for integer overflow if k is used in calculations related to string indices or substring lengths; use appropriate data types.

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What are the constraints on the length of the input string and the value of K? Is K always a valid value (i.e., K >= 0 and <= a reasonable upper bound)?
What defines a "special substring"? Is it a substring with some specific property or is this property also part of the input?
If there are multiple sets of K disjoint special substrings, should I return any valid set, or is there a specific criteria for choosing between them (e.g., maximizing total length, lexicographically smallest starting indices)?
If it is not possible to select K disjoint special substrings, what should the function return? Should I return null, an empty list, or throw an exception?
What characters can the input string contain? Are there any restrictions on the types of characters (e.g., only lowercase English letters, ASCII characters, Unicode characters)?

Brute Force Solution

Approach

Here's how the algorithm would work step-by-step:

First, we need to consider all possible pieces of the original string. Think of cutting the string in every possible place to create many different substrings.
Next, we check if each of these pieces meet the 'special' condition. Only some of these pieces will qualify.
Now, we start selecting a few pieces to see if they are disjoint (not overlapping). We begin by checking every pair of pieces to see if they overlap at any point.
If two pieces don't overlap, we continue checking if we can add a third special piece that doesn't overlap with the first two, and so on, until we have checked all possible groups of K disjoint special substrings.
We keep track of the 'best' combination of K disjoint special special substrings, according to some rule (such as, maximize the length of the special substrings).
After exhausting all combinations, we pick the 'best' combination of K disjoint special substrings we found.

Code Implementation

def select_k_disjoint_special_substrings(input_string, k_value):    all_substrings = []
    string_length = len(input_string)

    # Generate all possible substrings.
    for i in range(string_length):
        for j in range(i, string_length):
            substring = input_string[i:j+1]
            all_substrings.append((substring, i, j))

    special_substrings = []
    #In this implementation, all substrings are considered special.    special_substrings = all_substrings

    best_combination = []
    max_length = 0

    # Iterate through all combinations.
    for i in range(1 << len(special_substrings)):
        current_combination = []
        for j in range(len(special_substrings)):
            if (i >> j) & 1:
                current_combination.append(special_substrings[j])

        # Check for disjointedness and number of substrings.
        if len(current_combination) == k_value:
            is_disjoint = True
            for index_one in range(len(current_combination)):
                for index_two in range(index_one + 1, len(current_combination)):
                    substring_one = current_combination[index_one]
                    substring_two = current_combination[index_two]
                    if not (substring_one[2] < substring_two[1] or substring_two[2] < substring_one[1]):
                        is_disjoint = False
                        break
                if not is_disjoint:
                    break

            # If disjoint, compute total length and update best combination
            if is_disjoint:
                total_length = sum(len(substring[0]) for substring in current_combination)
                if total_length > max_length:
                    max_length = total_length
                    best_combination = current_combination

    return best_combination

Big(O) Analysis

Time Complexity

Space Complexity

Optimal Solution

Approach

Here's how the algorithm would work step-by-step:

Think of each possible place where a substring could end as a potential stopping point.
For each stopping point, figure out the best score you can get by including a substring that ends there, considering you need to select 'K' non-overlapping substrings.
To figure out the best score for each stopping point, consider the best score from the previous stopping points where it's possible to start a substring without overlapping.
We can calculate this using information we have from previous positions to figure out how to get to that point.
Keep track of these scores as you go, always storing the best solution for each endpoint.
Once you've reached the end of the entire string, the best score you've saved is your answer.

Code Implementation

def select_k_disjoint_substrings(input_string, k_value, special_substrings):
    string_length = len(input_string)
    dp_table = [[(0) for _ in range(string_length + 1)] for _ in range(k_value + 1)]

    for substrings_selected in range(1, k_value + 1):
        for ending_index in range(1, string_length + 1):
            dp_table[substrings_selected][ending_index] = dp_table[substrings_selected][ending_index - 1]
            for start_index in range(1, ending_index + 1):
                substring = input_string[start_index - 1:ending_index]

                # Check if substring is special
                if substring in special_substrings:
                    # To avoid overlapping, update dp_table
                    if start_index == 1:
                        dp_table[substrings_selected][ending_index] = max(
                            dp_table[substrings_selected][ending_index], len(substring)
                        )
                    else:
                        # Add length to previously found best score
                        dp_table[substrings_selected][ending_index] = max(
                            dp_table[substrings_selected][ending_index],
                            dp_table[substrings_selected - 1][start_index - 1] + len(substring),
                        )

    # The value at the end is the maximum length.
    return dp_table[k_value][string_length]

Big(O) Analysis

Time Complexity

Space Complexity

Edge Cases

Case	How to Handle
Empty input string	Return an empty list, as no substrings can be formed.
k is 0	Return an empty list as no substrings need to be selected.
k is greater than the maximum possible number of disjoint special substrings	Return the maximum possible number of disjoint special substrings instead of throwing an error.
String contains no special characters	Return an empty list, since no 'special' substring can be formed.
Overlapping special substrings are present	The solution should prioritize non-overlapping substrings, potentially using a greedy or dynamic programming approach.
Input string with very long length, close to memory limits	Ensure the algorithm's memory usage is optimized to avoid exceeding memory limits, possibly by processing substrings in chunks.
Multiple valid sets of k disjoint special substrings exist	The algorithm should either return any valid set or specify a criterion for selecting a particular set (e.g., lexicographically smallest, longest substrings).
k is a very large number	Check for integer overflow if k is used in calculations related to string indices or substring lengths; use appropriate data types.

Empty input string

How to Handle:

Return an empty list, as no substrings can be formed.

k is 0

How to Handle:

Return an empty list as no substrings need to be selected.

k is greater than the maximum possible number of disjoint special substrings

How to Handle:

Return the maximum possible number of disjoint special substrings instead of throwing an error.

String contains no special characters

How to Handle:

Return an empty list, since no 'special' substring can be formed.

Overlapping special substrings are present

How to Handle:

The solution should prioritize non-overlapping substrings, potentially using a greedy or dynamic programming approach.

Input string with very long length, close to memory limits

How to Handle:

Ensure the algorithm's memory usage is optimized to avoid exceeding memory limits, possibly by processing substrings in chunks.

Multiple valid sets of k disjoint special substrings exist

How to Handle:

The algorithm should either return any valid set or specify a criterion for selecting a particular set (e.g., lexicographically smallest, longest substrings).

k is a very large number

How to Handle:

Check for integer overflow if k is used in calculations related to string indices or substring lengths; use appropriate data types.