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Maximum Number of Non-overlapping Palindrome Substrings #8 Most Asked

Hard

SoFi

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Topics:

StringsDynamic ProgrammingGreedy Algorithms

You are given a string s and a positive integer k.

Select a set of non-overlapping substrings from the string s that satisfy the following conditions:

The length of each substring is at least k.
Each substring is a palindrome.

Return the maximum number of substrings in an optimal selection.

A substring is a contiguous sequence of characters within a string.

Example 1:

Input: s = "abaccdbbd", k = 3
Output: 2
Explanation: We can select the substrings underlined in s = "abaccdbbd". Both "aba" and "dbbd" are palindromes and have a length of at least k = 3.
It can be shown that we cannot find a selection with more than two valid substrings.

Example 2:

Input: s = "adbcda", k = 2
Output: 0
Explanation: There is no palindrome substring of length at least 2 in the string.

Constraints:

1 <= k <= s.length <= 2000
s consists of lowercase English letters.

Try coding it in LeetCode

Maximum Number of Non-overlapping Palindrome Substrings #8 Most Asked

Hard

SoFi

2 views

Topics:

StringsDynamic ProgrammingGreedy Algorithms

You are given a string s and a positive integer k.

Select a set of non-overlapping substrings from the string s that satisfy the following conditions:

The length of each substring is at least k.
Each substring is a palindrome.

Return the maximum number of substrings in an optimal selection.

A substring is a contiguous sequence of characters within a string.

Example 1:

Input: s = "abaccdbbd", k = 3
Output: 2
Explanation: We can select the substrings underlined in s = "abaccdbbd". Both "aba" and "dbbd" are palindromes and have a length of at least k = 3.
It can be shown that we cannot find a selection with more than two valid substrings.

Example 2:

Input: s = "adbcda", k = 2
Output: 0
Explanation: There is no palindrome substring of length at least 2 in the string.

Constraints:

1 <= k <= s.length <= 2000
s consists 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 is the maximum length of the input string `s`?
Can the input string `s` be empty or null?
If no palindrome substrings can be found, what should I return?
Are we looking for any palindrome substring, or only those containing at least two characters?
If multiple valid partitions with the same maximum number of palindrome substrings exist, can I return any one of them?

Brute Force Solution

Approach

The brute force method for finding the most non-overlapping palindrome substrings involves checking every possible combination of substrings within the main string. We generate all possible substrings and test if they are palindromes. After identifying all the palindrome substrings, we find the maximum number that do not overlap.

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

First, we need to find every possible substring of the given string. Think of it like cutting a rope into every possible combination of smaller pieces.
For each substring we created, we check if it is a palindrome. A palindrome is a word that reads the same forward and backward. Like 'madam'.
We keep only the substrings that are palindromes and throw away the rest.
Now we have a list of all the palindrome substrings that exist in the original string.
Our goal is to choose as many non-overlapping palindromes as possible. 'Non-overlapping' means they do not share any letters from the original string.
We try every possible combination of the palindrome substrings we found. For each combination, we check if the palindromes overlap.
If a combination has overlapping palindromes, we discard it.
We keep track of the size of each combination of non-overlapping palindrome substrings.
Finally, after checking every possible combination, we choose the one with the most palindromes. This is the maximum number of non-overlapping palindrome substrings.

Code Implementation

def max_non_overlapping_palindrome_substrings_brute_force(input_string):
    string_length = len(input_string)
    max_palindromes = 0

    def is_palindrome(substring):
        return substring == substring[::-1]

    def solve(current_index, current_palindromes):
        nonlocal max_palindromes

        if current_index == string_length:
            max_palindromes = max(max_palindromes, len(current_palindromes))
            return

        # Explore splitting the string at the current index
        for next_index in range(current_index + 1, string_length + 1):
            substring = input_string[current_index:next_index]
            
            # Prune branches where the substring is not a palindrome.
            if is_palindrome(substring):

                # Recursively explore the remaining string
                solve(next_index, current_palindromes + [substring])

    # Explore not splitting at current index.
    solve(0, [])
    return max_palindromes

Big(O) Analysis

Time Complexity

O(2^n) – Generating all possible substrings from a string of length n takes O(n^2) time. Checking if each substring is a palindrome also takes O(n) time, resulting in O(n^3) for finding all palindrome substrings. The most expensive operation is iterating through all possible combinations of these palindrome substrings to find non-overlapping sets. In the worst-case scenario, we may have to examine 2^k combinations where k is the number of palindrome substrings, and k can be proportional to n, so this part takes O(2^n) time. Therefore, the overall time complexity is dominated by the combination check, which is O(2^n).

Space Complexity

O(N^2) – The algorithm first generates all possible substrings, which can be up to N^2 in number, where N is the length of the input string. These substrings are stored in a list, and their palindrome status is also checked and potentially stored. Therefore, the space needed to store these substrings and palindrome flags results in O(N^2) space complexity.

Optimal Solution

Approach

The goal is to find as many palindrome substrings as possible in a given string, making sure they don't overlap. The most effective way is to go through the string from left to right, greedily picking the shortest palindrome we can find at each step.

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

Start at the beginning of the string.
Look for the smallest palindrome that starts at the current position.
If you find a palindrome, count it and move to the position right after that palindrome ends.
If you don't find a palindrome starting at the current position, just move to the next letter.
Keep repeating steps two through four until you reach the end of the string.
The number of palindromes you've counted is the maximum number of non-overlapping palindrome substrings.

Code Implementation

def maximum_number_of_nonoverlapping_palindrome_substrings(input_string):
    string_length = len(input_string)
    palindrome_count = 0
    current_index = 0

    while current_index < string_length:
        longest_palindrome_length = 0

        # Find the longest palindrome starting at current_index
        for length in range(1, string_length - current_index + 1):
            substring = input_string[current_index:current_index + length]
            if substring == substring[::-1]:
                longest_palindrome_length = length

        # If we found a palindrome
        if longest_palindrome_length > 0:
            palindrome_count += 1
            # Advance the current index past it
            current_index += longest_palindrome_length

        # No palindrome found, move to the next char
        else:
            current_index += 1

    return palindrome_count

Big(O) Analysis

Time Complexity

O(n^2) – The algorithm iterates through the string of length n. For each character, it attempts to find the shortest palindrome starting at that position. In the worst case, checking for palindromes from a given position involves comparing pairs of characters, potentially up to the end of the string. This palindrome check itself can take up to O(n) time in the worst case for each of the n starting positions. Therefore, the overall time complexity approximates n * n operations. Thus the time complexity is O(n^2).

Space Complexity

O(1) – The algorithm iterates through the string, keeping track of the current position. No auxiliary data structures like arrays, hashmaps, or stacks are created. The space used is limited to a few integer variables for indexing, irrespective of the input string length N. Therefore, the space complexity is constant.

Edge Cases

Case	How to Handle
Empty string input	Return 0, as there are no substrings in an empty string.
Single-character string	Return 1, as a single character is always a palindrome.
String with all identical characters (e.g., 'aaaa')	The algorithm should correctly identify each character as a palindrome and return the string length.
Very long string (scaling performance)	Ensure the chosen algorithm (e.g., dynamic programming or greedy) is efficient enough to handle large input strings without exceeding time limits.
String with no palindromic substrings	If no palindrome is found return 0 as the valid solution.
String is already a palindrome	Return 1, since the entire string itself is one palindrome.
String containing special characters or unicode	Ensure the palindrome check correctly handles special characters and unicode values, typically through case-insensitive comparison or explicit character filtering if needed.
Overlapping palindrome substrings	The chosen algorithm should greedily select the earliest ending palindrome substring to maximize the number of non-overlapping palindromes.

Empty string input

How to Handle:

Return 0, as there are no substrings in an empty string.

Single-character string

How to Handle:

Return 1, as a single character is always a palindrome.

String with all identical characters (e.g., 'aaaa')

How to Handle:

The algorithm should correctly identify each character as a palindrome and return the string length.

Very long string (scaling performance)

How to Handle:

Ensure the chosen algorithm (e.g., dynamic programming or greedy) is efficient enough to handle large input strings without exceeding time limits.

String with no palindromic substrings

How to Handle:

If no palindrome is found return 0 as the valid solution.

String is already a palindrome

How to Handle:

Return 1, since the entire string itself is one palindrome.

String containing special characters or unicode

How to Handle:

Ensure the palindrome check correctly handles special characters and unicode values, typically through case-insensitive comparison or explicit character filtering if needed.

Overlapping palindrome substrings

How to Handle:

The chosen algorithm should greedily select the earliest ending palindrome substring to maximize the number of non-overlapping palindromes.

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 is the maximum length of the input string `s`?
Can the input string `s` be empty or null?
If no palindrome substrings can be found, what should I return?
Are we looking for any palindrome substring, or only those containing at least two characters?
If multiple valid partitions with the same maximum number of palindrome substrings exist, can I return any one of them?

Brute Force Solution

Approach

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

First, we need to find every possible substring of the given string. Think of it like cutting a rope into every possible combination of smaller pieces.
For each substring we created, we check if it is a palindrome. A palindrome is a word that reads the same forward and backward. Like 'madam'.
We keep only the substrings that are palindromes and throw away the rest.
Now we have a list of all the palindrome substrings that exist in the original string.
Our goal is to choose as many non-overlapping palindromes as possible. 'Non-overlapping' means they do not share any letters from the original string.
We try every possible combination of the palindrome substrings we found. For each combination, we check if the palindromes overlap.
If a combination has overlapping palindromes, we discard it.
We keep track of the size of each combination of non-overlapping palindrome substrings.
Finally, after checking every possible combination, we choose the one with the most palindromes. This is the maximum number of non-overlapping palindrome substrings.

Code Implementation

def max_non_overlapping_palindrome_substrings_brute_force(input_string):
    string_length = len(input_string)
    max_palindromes = 0

    def is_palindrome(substring):
        return substring == substring[::-1]

    def solve(current_index, current_palindromes):
        nonlocal max_palindromes

        if current_index == string_length:
            max_palindromes = max(max_palindromes, len(current_palindromes))
            return

        # Explore splitting the string at the current index
        for next_index in range(current_index + 1, string_length + 1):
            substring = input_string[current_index:next_index]
            
            # Prune branches where the substring is not a palindrome.
            if is_palindrome(substring):

                # Recursively explore the remaining string
                solve(next_index, current_palindromes + [substring])

    # Explore not splitting at current index.
    solve(0, [])
    return max_palindromes

Big(O) Analysis

Time Complexity

Space Complexity

Optimal Solution

Approach

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

Start at the beginning of the string.
Look for the smallest palindrome that starts at the current position.
If you find a palindrome, count it and move to the position right after that palindrome ends.
If you don't find a palindrome starting at the current position, just move to the next letter.
Keep repeating steps two through four until you reach the end of the string.
The number of palindromes you've counted is the maximum number of non-overlapping palindrome substrings.

Code Implementation

def maximum_number_of_nonoverlapping_palindrome_substrings(input_string):
    string_length = len(input_string)
    palindrome_count = 0
    current_index = 0

    while current_index < string_length:
        longest_palindrome_length = 0

        # Find the longest palindrome starting at current_index
        for length in range(1, string_length - current_index + 1):
            substring = input_string[current_index:current_index + length]
            if substring == substring[::-1]:
                longest_palindrome_length = length

        # If we found a palindrome
        if longest_palindrome_length > 0:
            palindrome_count += 1
            # Advance the current index past it
            current_index += longest_palindrome_length

        # No palindrome found, move to the next char
        else:
            current_index += 1

    return palindrome_count

Big(O) Analysis

Time Complexity

Space Complexity

Edge Cases

Case	How to Handle
Empty string input	Return 0, as there are no substrings in an empty string.
Single-character string	Return 1, as a single character is always a palindrome.
String with all identical characters (e.g., 'aaaa')	The algorithm should correctly identify each character as a palindrome and return the string length.
Very long string (scaling performance)	Ensure the chosen algorithm (e.g., dynamic programming or greedy) is efficient enough to handle large input strings without exceeding time limits.
String with no palindromic substrings	If no palindrome is found return 0 as the valid solution.
String is already a palindrome	Return 1, since the entire string itself is one palindrome.
String containing special characters or unicode	Ensure the palindrome check correctly handles special characters and unicode values, typically through case-insensitive comparison or explicit character filtering if needed.
Overlapping palindrome substrings	The chosen algorithm should greedily select the earliest ending palindrome substring to maximize the number of non-overlapping palindromes.

Empty string input

How to Handle:

Return 0, as there are no substrings in an empty string.

Single-character string

How to Handle:

Return 1, as a single character is always a palindrome.

String with all identical characters (e.g., 'aaaa')

How to Handle:

The algorithm should correctly identify each character as a palindrome and return the string length.

Very long string (scaling performance)

How to Handle:

Ensure the chosen algorithm (e.g., dynamic programming or greedy) is efficient enough to handle large input strings without exceeding time limits.

String with no palindromic substrings

How to Handle:

If no palindrome is found return 0 as the valid solution.

String is already a palindrome

How to Handle:

Return 1, since the entire string itself is one palindrome.

String containing special characters or unicode

How to Handle:

Ensure the palindrome check correctly handles special characters and unicode values, typically through case-insensitive comparison or explicit character filtering if needed.

Overlapping palindrome substrings

How to Handle:

The chosen algorithm should greedily select the earliest ending palindrome substring to maximize the number of non-overlapping palindromes.

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