Jobs

Interviews

Discussions

Find the Maximum Sequence Value of Array

Hard

Google

2 views

Topics:

ArraysBit ManipulationDynamic Programming

You are given an integer array nums and a positive integer k. The value of a sequence seq of size 2 * x is defined as: (seq[0] OR seq[1] OR ... OR seq[x - 1]) XOR (seq[x] OR seq[x + 1] OR ... OR seq[2 * x - 1]). Return the maximum value of any subsequence of nums having size 2 * k.

For example:

nums = [2,6,7], k = 1. The subsequence [2, 7] has the maximum value of 2 XOR 7 = 5.
nums = [4,2,5,6,7], k = 2. The subsequence [4, 5, 6, 7] has the maximum value of (4 OR 5) XOR (6 OR 7) = 1 XOR 7 = 6. Note that you can select any subsequence, the elements do not have to be contiguous.

What is the most efficient algorithm to solve this problem? What are the time and space complexities? Discuss edge cases and how they are handled in your solution.

Find the Maximum Sequence Value of Array

Hard

Google

2 views

Topics:

ArraysBit ManipulationDynamic Programming

For example:

nums = [2,6,7], k = 1. The subsequence [2, 7] has the maximum value of 2 XOR 7 = 5.
nums = [4,2,5,6,7], k = 2. The subsequence [4, 5, 6, 7] has the maximum value of (4 OR 5) XOR (6 OR 7) = 1 XOR 7 = 6. Note that you can select any subsequence, the elements do not have to be contiguous.

What is the most efficient algorithm to solve this problem? What are the time and space complexities? Discuss edge cases and how they are handled in your solution.

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 range of integer values within the input array? Can I expect negative numbers, zero, or only positive integers?
What constitutes a 'sequence' in this problem? Are we looking for contiguous subsequences or non-contiguous subsequences?
If the input array is empty or null, what should the function return?
Are there any constraints on the length of the input array? What is the maximum possible size of the array?
If multiple sequences have the same maximum value, is it acceptable to return any one of them, or is there a specific criteria to choose one?

Brute Force Solution

Approach

The brute force strategy for finding the maximum sequence value considers every possible sub-sequence within the given sequence. It calculates the sequence value for each and every sub-sequence. Finally, it compares all these values and returns the highest one found.

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

First, consider the very first number in the sequence as a sub-sequence by itself and calculate its sequence value.
Next, consider the first two numbers together as a sub-sequence and calculate their sequence value.
Continue this process, adding one more number at a time to the sub-sequence from the beginning of the sequence, and calculating the sequence value each time until you've used all numbers in the sequence.
Now, do the same thing again, but start with the second number in the sequence as a sub-sequence.
Repeat this process, each time starting with the next number in the sequence as the start of a new sub-sequence, until you have considered all starting points.
Each time you calculate a sequence value, remember to store it.
After going through all the possible sub-sequences, compare all the stored sequence values to find the largest one.
Return the largest sequence value you found.

Code Implementation

def find_maximum_sequence_value_brute_force(sequence):
    maximum_sequence_value = float('-inf')

    # Iterate through all possible starting positions
    for start_index in range(len(sequence)):

        # Iterate through all possible ending positions for each start position
        for end_index in range(start_index, len(sequence)):
            current_sub_sequence = sequence[start_index:end_index + 1]
            current_sequence_value = calculate_sequence_value(current_sub_sequence)

            # Store the max sequence value found
            if current_sequence_value > maximum_sequence_value:

                maximum_sequence_value = current_sequence_value

    return maximum_sequence_value

def calculate_sequence_value(sub_sequence):
    if not sub_sequence:
        return 0

    sequence_value = 0
    for number in sub_sequence:
        sequence_value += number
    return sequence_value

Big(O) Analysis

Time Complexity

O(n²) – The described brute force algorithm iterates through all possible sub-sequences of the input array. For each starting index i, the algorithm iterates from i to the end of the array to form sub-sequences. The outer loop runs n times, and the inner loop (for creating sub-sequences) also runs up to n times in the worst case. Therefore, the total number of operations is proportional to n + (n-1) + (n-2) + ... + 1, which sums to approximately n*(n+1)/2. This simplifies to O(n²).

Space Complexity

O(1) – The provided algorithm only uses a few variables to store the current sequence value and the maximum sequence value found so far. No auxiliary data structures like arrays, lists, or hashmaps are used to store intermediate subsequences or results. The amount of extra memory needed remains constant irrespective of the input array size N, so the space complexity is O(1).

Optimal Solution

Approach

The key is to avoid checking every single possible combination of numbers. Instead, focus on making choices that incrementally build the best possible result by comparing two options at each step and keeping track of what maximizes our value so far.

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

Begin by considering the first number in the sequence.
Next, consider including the second number to the existing sequence or starting a new sequence with the second number alone. Calculate which option results in a higher sequence value and keep the better one. This decision determines if we continue with the existing sequence or begin a new one.
Repeat the decision-making process for each subsequent number. At each stage, decide whether to extend the current sequence by including the number or start a new sequence with that number. Base the decision on which choice results in a higher value.
Keep track of the highest sequence value found so far during this process. As you proceed, if a new sequence value exceeds the current maximum, update the maximum value.
After examining all the numbers, the maximum sequence value found will be your final answer.

Code Implementation

def find_maximum_sequence_value(numbers):
    maximum_sequence_value = 0
    current_sequence_value = 0

    for number in numbers:
        # Decide whether to extend the current sequence or start a new one
        if current_sequence_value + number > number:
            current_sequence_value += number

        # Starting new sequence from current number if its greater
        else:
            current_sequence_value = number

        # Keep track of the highest sequence value found so far
        if current_sequence_value > maximum_sequence_value:

            maximum_sequence_value = current_sequence_value

    return maximum_sequence_value

Big(O) Analysis

Time Complexity

O(n) – The algorithm iterates through each of the n numbers in the input array once. For each number, it performs a constant amount of work by comparing two options (extend the sequence or start a new one) and updating the maximum value found so far. Since the amount of work done for each number is constant and independent of n, the overall time complexity is directly proportional to n, which means the time complexity is O(n).

Space Complexity

O(1) – The algorithm, as described, maintains a few scalar variables to track the current maximum sequence value and potentially the value of the current sequence being built. No auxiliary data structures like arrays, hash maps, or lists are created that scale with the input size N (the number of elements in the array). The memory usage remains constant irrespective of the input array's size. Therefore, the space complexity is O(1).

Edge Cases

Case	How to Handle
Null or empty input array	Return 0 (or throw an exception, depending on requirements) since no sequence can be formed.
Array with one element	Return that element itself since it's the only possible sequence of length 1.
Array with all elements equal to zero	Handle zeros appropriately; if the algorithm multiplies, make sure it doesn't cause unexpected zeros; consider zero as a valid element.
Array with all negative numbers	The algorithm should correctly handle negative numbers and find the maximum sequence even within negative numbers.
Large input array causing potential memory or time issues	Optimize the algorithm for time complexity and consider using data structures that minimize memory usage (e.g., in-place operations if possible).
Integer overflow during calculations of sequence value	Use appropriate data types (e.g., long) or modulo operations to prevent integer overflow.
Array containing extremely large or extremely small numbers	Ensure that calculations involving these numbers do not lead to overflow or underflow, potentially using larger data types or scaling techniques.
Array with many duplicate elements that form a significant part of a possible sequence	Ensure algorithm is robust and counts each unique element only once in valid sequence or applies a count appropriately.

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 range of integer values within the input array? Can I expect negative numbers, zero, or only positive integers?
What constitutes a 'sequence' in this problem? Are we looking for contiguous subsequences or non-contiguous subsequences?
If the input array is empty or null, what should the function return?
Are there any constraints on the length of the input array? What is the maximum possible size of the array?
If multiple sequences have the same maximum value, is it acceptable to return any one of them, or is there a specific criteria to choose one?

Brute Force Solution

Approach

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

First, consider the very first number in the sequence as a sub-sequence by itself and calculate its sequence value.
Next, consider the first two numbers together as a sub-sequence and calculate their sequence value.
Continue this process, adding one more number at a time to the sub-sequence from the beginning of the sequence, and calculating the sequence value each time until you've used all numbers in the sequence.
Now, do the same thing again, but start with the second number in the sequence as a sub-sequence.
Repeat this process, each time starting with the next number in the sequence as the start of a new sub-sequence, until you have considered all starting points.
Each time you calculate a sequence value, remember to store it.
After going through all the possible sub-sequences, compare all the stored sequence values to find the largest one.
Return the largest sequence value you found.

Code Implementation

def find_maximum_sequence_value_brute_force(sequence):
    maximum_sequence_value = float('-inf')

    # Iterate through all possible starting positions
    for start_index in range(len(sequence)):

        # Iterate through all possible ending positions for each start position
        for end_index in range(start_index, len(sequence)):
            current_sub_sequence = sequence[start_index:end_index + 1]
            current_sequence_value = calculate_sequence_value(current_sub_sequence)

            # Store the max sequence value found
            if current_sequence_value > maximum_sequence_value:

                maximum_sequence_value = current_sequence_value

    return maximum_sequence_value

def calculate_sequence_value(sub_sequence):
    if not sub_sequence:
        return 0

    sequence_value = 0
    for number in sub_sequence:
        sequence_value += number
    return sequence_value

Big(O) Analysis

Time Complexity

Space Complexity

Optimal Solution

Approach

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

Begin by considering the first number in the sequence.
Next, consider including the second number to the existing sequence or starting a new sequence with the second number alone. Calculate which option results in a higher sequence value and keep the better one. This decision determines if we continue with the existing sequence or begin a new one.
Repeat the decision-making process for each subsequent number. At each stage, decide whether to extend the current sequence by including the number or start a new sequence with that number. Base the decision on which choice results in a higher value.
Keep track of the highest sequence value found so far during this process. As you proceed, if a new sequence value exceeds the current maximum, update the maximum value.
After examining all the numbers, the maximum sequence value found will be your final answer.

Code Implementation

def find_maximum_sequence_value(numbers):
    maximum_sequence_value = 0
    current_sequence_value = 0

    for number in numbers:
        # Decide whether to extend the current sequence or start a new one
        if current_sequence_value + number > number:
            current_sequence_value += number

        # Starting new sequence from current number if its greater
        else:
            current_sequence_value = number

        # Keep track of the highest sequence value found so far
        if current_sequence_value > maximum_sequence_value:

            maximum_sequence_value = current_sequence_value

    return maximum_sequence_value

Big(O) Analysis

Time Complexity

Space Complexity

Edge Cases

Case	How to Handle
Null or empty input array	Return 0 (or throw an exception, depending on requirements) since no sequence can be formed.
Array with one element	Return that element itself since it's the only possible sequence of length 1.
Array with all elements equal to zero	Handle zeros appropriately; if the algorithm multiplies, make sure it doesn't cause unexpected zeros; consider zero as a valid element.
Array with all negative numbers	The algorithm should correctly handle negative numbers and find the maximum sequence even within negative numbers.
Large input array causing potential memory or time issues	Optimize the algorithm for time complexity and consider using data structures that minimize memory usage (e.g., in-place operations if possible).
Integer overflow during calculations of sequence value	Use appropriate data types (e.g., long) or modulo operations to prevent integer overflow.
Array containing extremely large or extremely small numbers	Ensure that calculations involving these numbers do not lead to overflow or underflow, potentially using larger data types or scaling techniques.
Array with many duplicate elements that form a significant part of a possible sequence	Ensure algorithm is robust and counts each unique element only once in valid sequence or applies a count appropriately.