Longest Ideal Subsequence

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Amazon

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StringsDynamic Programming

You are given a string s consisting of lowercase letters and an integer k. We call a string t ideal if the following conditions are satisfied:

t is a subsequence of the string s.
The absolute difference in the alphabet order of every two adjacent letters in t is less than or equal to k.

Return the length of the longest ideal string.

A subsequence is a string that can be derived from another string by deleting some or no characters without changing the order of the remaining characters.

Note that the alphabet order is not cyclic. For example, the absolute difference in the alphabet order of 'a' and 'z' is 25, not 1.

Example 1:

Input: s = "acfgbd", k = 2
Output: 4
Explanation: The longest ideal string is "acbd". The length of this string is 4, so 4 is returned.
Note that "acfgbd" is not ideal because 'c' and 'f' have a difference of 3 in alphabet order.

Example 2:

Input: s = "abcd", k = 3
Output: 4
Explanation: The longest ideal string is "abcd". The length of this string is 4, so 4 is returned.

Constraints:

1 <= s.length <= 10^5
0 <= k <= 25
s consists of lowercase English letters.

Longest Ideal Subsequence

Medium

Amazon

3 views

Topics:

StringsDynamic Programming

You are given a string s consisting of lowercase letters and an integer k. We call a string t ideal if the following conditions are satisfied:

t is a subsequence of the string s.
The absolute difference in the alphabet order of every two adjacent letters in t is less than or equal to k.

Return the length of the longest ideal string.

A subsequence is a string that can be derived from another string by deleting some or no characters without changing the order of the remaining characters.

Note that the alphabet order is not cyclic. For example, the absolute difference in the alphabet order of 'a' and 'z' is 25, not 1.

Example 1:

Input: s = "acfgbd", k = 2
Output: 4
Explanation: The longest ideal string is "acbd". The length of this string is 4, so 4 is returned.
Note that "acfgbd" is not ideal because 'c' and 'f' have a difference of 3 in alphabet order.

Example 2:

Input: s = "abcd", k = 3
Output: 4
Explanation: The longest ideal string is "abcd". The length of this string is 4, so 4 is returned.

Constraints:

1 <= s.length <= 10^5
0 <= k <= 25
s consists of lowercase English letters.

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 `s` and the value of `k`?
Can the input string `s` contain characters other than lowercase English letters?
If no ideal subsequence exists, should I return 0?
Is an empty string considered an ideal subsequence, and if so, what should the value of `k` be to allow that?
If multiple longest ideal subsequences exist, is it sufficient to return the length of any one of them?

Brute Force Solution

Approach

The brute force approach to finding the longest ideal subsequence means we'll look at every possible subsequence. We'll then check if each of these subsequences is 'ideal' according to the rules. Finally, we'll choose the longest one we find.

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

Consider every possible group of characters you can make from the original sequence. This includes groups of one character, two characters, three characters, and so on, up to using all the characters.
For each group of characters, check if it's an 'ideal' subsequence. This means checking if the absolute difference between the alphabetical positions of any two adjacent characters in the group is less than or equal to the given limit.
If a group of characters is not an 'ideal' subsequence, discard it and move on to the next group.
If a group of characters *is* an 'ideal' subsequence, remember its length.
After checking all possible groups of characters, compare the lengths of all the 'ideal' subsequences you found.
The longest 'ideal' subsequence is your answer.

Code Implementation

def longest_ideal_subsequence_brute_force(text, limit):
    maximum_length = 0
    number_of_characters = len(text)

    # Iterate through all possible subsequences
    for i in range(1 << number_of_characters):
        subsequence = ""
        for j in range(number_of_characters):
            if (i >> j) & 1:
                subsequence += text[j]

        # Check if the subsequence is ideal
        is_ideal = True
        if len(subsequence) > 1:
            for k in range(len(subsequence) - 1):
                absolute_difference = abs(ord(subsequence[k]) - ord(subsequence[k + 1]))
                if absolute_difference > limit:
                    is_ideal = False
                    break

        # If the subsequence is ideal, update the maximum length
        if is_ideal:
            maximum_length = max(maximum_length, len(subsequence))

    return maximum_length

Big(O) Analysis

Time Complexity

O(2^n * n) – The brute force approach iterates through all possible subsequences of the input string of length n. There are 2^n possible subsequences. For each subsequence, we iterate through its elements (in the worst case, up to n elements) to check if it is an ideal subsequence by comparing adjacent characters. Thus, the time complexity is O(2^n * n), where 2^n represents the number of subsequences and n represents the maximum length of any subsequence that needs to be checked for the ideal property. Therefore the runtime grows exponentially with the input size.

Space Complexity

O(1) – The brute force approach, as described, doesn't explicitly use any auxiliary data structures whose size depends on the input string's length N. It iterates through all possible subsequences and checks if they are ideal. While generating subsequences implicitly involves some overhead, the provided explanation doesn't suggest the creation of lists or other data structures to store these subsequences. Thus, the auxiliary space used remains constant, independent of the input size N.

Optimal Solution

Approach

The best way to find the longest ideal sequence is to build it character by character. Instead of checking all possible sequences, we focus on efficiently extending the best sequences we've seen so far for each letter.

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

Think of each letter of the alphabet as the end of a potential ideal sequence.
Start looking at the letters in the input one at a time.
For each letter in the input, check if it can extend an existing ideal sequence ending with a letter 'close' to it (as defined by the 'ideal' rule).
If it can extend a sequence, choose the longest such sequence to extend. If multiple sequences are the same length, pick one.
If the current letter can't extend any existing sequence, it starts a new ideal sequence of length 1.
Keep track of the length of the longest ideal sequence ending with each letter of the alphabet. We only need to remember the lengths, not the sequences themselves.
After going through all the letters in the input, the longest of all the sequence lengths tracked for each letter of the alphabet is the answer.

Code Implementation

def longest_ideal_string(s, k):
    longest_sequence_ending_here = [0] * 26

    for char in s:
        current_char_index = ord(char) - ord('a')
        max_length = 0

        # Iterate through all possible last characters.
        for previous_char_index in range(26):
            # Check if the current char extends an existing ideal sequence
            if abs(current_char_index - previous_char_index) <= k:

                max_length = max(max_length,
                                 longest_sequence_ending_here[previous_char_index])

        # Update the length of the longest ideal sequence ending with
        # the current character.
        longest_sequence_ending_here[current_char_index] = max_length + 1

    # Find the maximum length among all ideal sequences.
    return max(longest_sequence_ending_here)

Big(O) Analysis

Time Complexity

O(n) – The algorithm iterates through the input string s of length n once. Inside the loop, for each character, it checks at most 26 other characters (the entire alphabet) to find the longest suitable sequence to extend, according to the 'ideal' rule determined by k. Since the alphabet size is constant, the inner check takes constant time. Thus, the overall time complexity is O(n * 26), which simplifies to O(n).

Space Complexity

O(1) – The algorithm maintains an array to store the length of the longest ideal subsequence ending with each letter of the alphabet. This array has a fixed size of 26, as there are 26 letters in the English alphabet. Therefore, the space used by this array is constant and independent of the input string's length N. No other auxiliary data structures are created that depend on the input size, resulting in constant space complexity.

Edge Cases

Case	How to Handle
Empty string input	Return 0, as an empty string has no subsequence.
String with length 1	Return 1, as a single character is a subsequence of length 1.
String with all identical characters	Return 1, as any single character forms a subsequence of length 1.
Large string input (performance)	The DP approach should have acceptable performance, but the character comparison could be optimized using bit manipulation if necessary.
k=0 (every character is ideal)	Return the length of the input string, as every character is ideal.
k is large enough that all characters are ideal	Return the length of the input string.
String contains only characters near the edges of ASCII range (e.g., 'a', 'b', 'z', 'y')	Ensure the absolute difference calculation handles both ends of the ASCII table correctly.
Integer overflow in length calculation (unlikely, but consider extremely large string lengths)	Although improbable given typical input constraints, be mindful of potential overflows when managing the length variables and consider using a larger datatype if required.