Taro Logo

Maximize Subarray GCD Score

Hard
Google logo
Google
47 views
Topics:
ArraysSliding WindowsGreedy Algorithms

You are given an array of positive integers nums and an integer k. You may perform at most k operations. In each operation, you can choose one element in the array and double its value. Each element can be doubled at most once. The score of a contiguous subarray is defined as the product of its length and the greatest common divisor (GCD) of all its elements. Your task is to return the maximum score that can be achieved by selecting a contiguous subarray from the modified array. The greatest common divisor (GCD) of an array is the largest integer that evenly divides all the array elements.

Example 1: nums = [2,4], k = 1 If we double nums[0] to 4 using one operation. The modified array becomes [4, 4]. The GCD of the subarray [4, 4] is 4, and the length is 2. Thus, the maximum possible score is 2 * 4 = 8.

Example 2: nums = [3,5,7], k = 2 If we double nums[2] to 14 using one operation. The modified array becomes [3, 5, 14]. The GCD of the subarray [14] is 14, and the length is 1. Thus, the maximum possible score is 1 * 14 = 14.

Example 3: nums = [5,5,5], k = 1 The subarray [5, 5, 5] has a GCD of 5, and its length is 3. Since doubling any element doesn't improve the score, the maximum score is 3 * 5 = 15.

How would you approach this problem, keeping in mind the constraints: 1 <= n == nums.length <= 1500, 1 <= nums[i] <= 10^9, and 1 <= k <= n?

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:

  1. What are the possible ranges for the integer values within the input array?
  2. Can the input array be empty or null?
  3. If there are multiple subarrays with the same maximum GCD score, should I return any one of them, or is there a specific one I should prioritize?
  4. What should I return if all the numbers in the input array are zero?
  5. What is the expected behavior if the greatest common divisor (GCD) of a subarray is zero? Should I consider such a subarray valid and include it in score calculation?

Brute Force Solution

Approach

The brute force method for maximizing the subarray GCD score is all about trying every single possible combination. We'll explore every possible slice of the given sequence and see how each performs, eventually picking the best one.

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

  1. Consider every possible start position for a slice of the sequence.
  2. For each start position, consider every possible end position after that start.
  3. Calculate the 'score' for the slice starting at the selected start position and ending at the selected end position. The 'score' is based on the greatest common divisor (GCD) of the numbers in that slice.
  4. Keep track of the highest 'score' you've seen so far.
  5. Once you've tried every possible start and end position, the highest 'score' you've kept track of will be your answer.

Code Implementation

def greatest_common_divisor(first_number, second_number): 
    while(second_number): 
        first_number, second_number = second_number, first_number % second_number
    return first_number

def calculate_subarray_gcd(number_array, start_index, end_index):
    current_gcd = number_array[start_index]
    for index in range(start_index + 1, end_index + 1):
        current_gcd = greatest_common_divisor(current_gcd, number_array[index])
    return current_gcd

def maximize_subarray_gcd_score(number_array):
    max_score = 0
    array_length = len(number_array)

    # Iterate through all possible start indices
    for start_index in range(array_length):

        # Iterate through all possible end indices for each start index
        for end_index in range(start_index, array_length):

            # Calculate the GCD of the current subarray
            subarray_gcd = calculate_subarray_gcd(number_array, start_index, end_index)

            # Update the maximum score if necessary
            current_score = subarray_gcd * (end_index - start_index + 1)
            if current_score > max_score:
                max_score = current_score

    return max_score

Big(O) Analysis

Time Complexity
O(n³)The described brute force approach iterates through all possible subarrays. There are n possible starting positions. For each starting position, there are approximately n possible ending positions, creating a nested loop structure. Inside these loops, we calculate the GCD of the subarray, which takes O(n) time in the worst case as we may need to iterate through all the elements of the subarray to compute the GCD. Therefore, the overall time complexity is O(n * n * n) which simplifies to O(n³).
Space Complexity
O(1)The brute force method iterates through all possible subarrays using nested loops, but it only needs to store a few variables: the start and end indices of the current subarray, the current GCD, and the maximum GCD seen so far. These variables take up a constant amount of space, independent of the input array size N. No auxiliary data structures, like lists or hash maps, are created. Therefore, the space complexity is O(1).

Optimal Solution

Approach

The goal is to pick groups of numbers such that the greatest common divisor (GCD) of each group is as large as possible, and then sum up those GCDs. We want to avoid checking every single possible group by smartly selecting groups based on the properties of GCDs.

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

  1. First, consider that the GCD of a single number is just the number itself. This is the largest possible GCD for any group containing that number.
  2. Start by sorting the input numbers from largest to smallest. This is because larger numbers have a greater potential to contribute a higher GCD score.
  3. Iterate through the sorted numbers. For each number, consider it as a potential GCD for a subarray.
  4. For each number, check all subarrays it is a part of. Calculate the GCD of that subarray.
  5. If the GCD of that subarray is equal to the number being considered, this is the best possible GCD from that sub array, use this number as a GCD.
  6. Make sure to only count each number once, meaning skip this number if its been already used as a gcd

Code Implementation

def maximize_subarray_gcd_score(numbers):
    numbers.sort(reverse=True)
    used_numbers = set()
    total_gcd_score = 0

    for number_index in range(len(numbers)):
        current_number = numbers[number_index]
        if current_number in used_numbers:
            continue

        best_gcd = 0
        
        for subarray_start_index in range(len(numbers)):
            for subarray_end_index in range(subarray_start_index, len(numbers)):
                
                current_subarray = numbers[subarray_start_index:subarray_end_index+1]
                
                #Consider only subarrays containing the current number
                if current_number not in current_subarray:
                    continue

                subarray_gcd = current_subarray[0]
                for index in range(1, len(current_subarray)):
                    subarray_gcd = find_gcd(subarray_gcd, current_subarray[index])

                # It's optimal to pick the largest GCD
                if subarray_gcd == current_number:
                  best_gcd = current_number
                  break
            if best_gcd == current_number:
              break
                  
        #Only add the gcd score if we found a gcd == the number
        if best_gcd > 0:
            total_gcd_score += best_gcd

            #Track used numbers to avoid re-counting.
            used_numbers.add(current_number)

    return total_gcd_score

def find_gcd(first_number, second_number):
    while(second_number):
        first_number, second_number = second_number, first_number % second_number
    return first_number

Big(O) Analysis

Time Complexity
O(n² * log(max(arr)))Sorting the input array of size n takes O(n log n) time. The outer loop iterates through each of the n elements in the sorted array. The inner loop checks all possible subarrays containing the current element, which in the worst case, requires iterating through up to n elements. Calculating the GCD of a subarray takes O(log(max(arr))) time, where max(arr) is the largest element in the array. Therefore, the dominant cost is iterating through all possible subarrays and computing their GCDs. The nested loops lead to a time complexity of approximately n * n * log(max(arr)), which simplifies to O(n² * log(max(arr))). Since the sorting can be included in this complexity, the overall time complexity becomes O(n² * log(max(arr))).
Space Complexity
O(N)The algorithm sorts the input array of size N, which may require O(N) auxiliary space depending on the sorting algorithm used. Furthermore, the algorithm utilizes a set or boolean array of size N to keep track of numbers that have already been used as GCDs, which consumes O(N) space. The iteration and GCD calculations themselves only use constant space. Therefore, the overall auxiliary space complexity is O(N).

Edge Cases

CaseHow to Handle
Empty input arrayReturn 0 immediately, as no subarray can be formed.
Array with a single elementReturn the value of that single element as a single element subarray's GCD is itself.
Array with all elements being 0Return 0, since GCD(0,0,0...) is technically undefined, but 0 is the common practice.
Array with very large numbers that could lead to integer overflow in GCD calculationsUse a data type with sufficient range (e.g., long long in C++, long in Java, or Python's arbitrary precision integers) to avoid overflow in GCD computations.
Array with negative numbers (if the problem states only positive integers are accepted)Throw an IllegalArgumentException or return an error code/message indicating invalid input if the problem explicitly restricts input to positive integers.
Array with a mix of very small and very large numbersThe GCD algorithm should handle this correctly without special casing, but it's worth considering for potential performance implications, and ensure correct behavior when the GCD is trivially 1.
Array where all numbers are the sameThe GCD of any subarray will be equal to the element's value; select the largest possible subarray (the entire array) to maximize the score.
Maximum array size exceeding available memoryConsider using iterative approaches and, if possible, process the array in chunks to avoid exceeding memory limits, especially if the GCD calculation is memory-intensive.