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Apply Operations to Maximize Score

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ArraysGreedy Algorithms

You are given an array nums of n positive integers and an integer k. Initially, you start with a score of 1. You have to maximize your score by applying the following operation at most k times:

  1. Choose any non-empty subarray nums[l, ..., r] that you haven't chosen previously.
  2. Choose an element x of nums[l, ..., r] with the highest prime score. If multiple such elements exist, choose the one with the smallest index.
  3. Multiply your score by x.

Here, nums[l, ..., r] denotes the subarray of nums starting at index l and ending at the index r, both ends being inclusive.

The prime score of an integer x is equal to the number of distinct prime factors of x. For example, the prime score of 300 is 3 since 300 = 2 * 2 * 3 * 5 * 5.

Return the maximum possible score after applying at most k operations. Since the answer may be large, return it modulo 10^9 + 7.

Example 1:

Input: nums = [8,3,9,3,8], k = 2 Output: 81

Example 2:

Input: nums = [19,12,14,6,10,18], k = 3 Output: 4788

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 constraints on the size of the input array `nums` and the maximum value of elements within `nums` and `k`?
  2. Can the input array `nums` contain negative numbers, zeros, or floating-point numbers?
  3. Could you please clarify the behavior if the maximum possible score is 0? Should I return 0 or is there another specific value expected?
  4. Are there any constraints on the prime factors to be considered for operations? Are we only concerned with prime numbers that are less than or equal to the maximum value in the array `nums`?
  5. If multiple sequences of operations yield the same maximum score, is any valid sequence acceptable, or is there a preference for one over the others (e.g., minimizing the number of operations)?

Brute Force Solution

Approach

The brute force method tackles this problem by exploring every possible combination of operations. It's like trying every single path through a maze until you find the best one. We consider all options to find the maximum score.

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

  1. Start by considering that we perform no operations at all.
  2. Next, consider performing only the first possible operation.
  3. Then, consider performing only the second possible operation, and so on, until we have tried performing each operation individually.
  4. Now, try performing every possible pair of operations. For example, the first and second operation together, then the first and third operation together, and so on.
  5. Continue this process, trying every possible group of three operations, then four, and so on, until we have tried all operations together.
  6. For each of these combinations of operations, calculate the resulting score.
  7. Finally, compare all the scores we calculated and choose the highest one. This is our answer.

Code Implementation

def apply_operations_to_maximize_score_brute_force(numbers, operations):
    max_score = 0
    number_of_operations = len(operations)

    # Iterate through all possible combinations of operations
    for i in range(2 ** number_of_operations):
        current_numbers = numbers[:]
        current_score = 0

        # Apply operations based on the current combination
        for operation_index in range(number_of_operations):

            # Check if the current operation is included in the combination
            if (i >> operation_index) & 1:

                # Apply the operation to the current numbers
                index = operations[operation_index][0]
                value = operations[operation_index][1]
                current_numbers[index] += value

        # Calculate the score for the current combination
        for number in current_numbers:
            current_score += number

        # Update the maximum score if necessary
        max_score = max(max_score, current_score)

    return max_score

def apply_operations_to_maximize_score(numbers, operations):
    max_score = float('-inf')

    def calculate_score(current_numbers):
        score = 0
        for number in current_numbers:
            score += number
        return score

    def find_max_score(index, current_numbers):
        nonlocal max_score

        # Base case: all operations have been considered
        if index == len(operations):
            current_score = calculate_score(current_numbers)
            max_score = max(max_score, current_score)
            return

        # Option 1: Don't apply the current operation
        find_max_score(index + 1, current_numbers[:])

        # Option 2: Apply the current operation

        operation_index = operations[index][0]
        operation_value = operations[index][1]

        # Create a copy of current_numbers to avoid modifying the original
        new_numbers = current_numbers[:]
        new_numbers[operation_index] += operation_value

        find_max_score(index + 1, new_numbers)

    # Begin the recursive search.
    find_max_score(0, numbers[:])

    # Return the maximum score found.
    return max_score

Big(O) Analysis

Time Complexity
O(2^n)The provided brute force approach explores every possible combination of operations. With n operations available, each operation can either be included or excluded in a particular combination. This results in 2^n possible combinations to evaluate. For each combination, calculating the score takes O(n) time in the worst case (if all operations are applied). However, the dominant factor is the number of combinations, which is 2^n. Therefore, the overall time complexity is O(2^n).
Space Complexity
O(1)The provided brute force approach calculates the score for each combination of operations directly. It doesn't explicitly create and store all combinations or intermediate results in separate data structures. The primary memory usage involves tracking the current combination being tested and updating the maximum score found so far, which requires a fixed amount of memory regardless of the number of operations. Therefore, the auxiliary space complexity is constant.

Optimal Solution

Approach

The challenge is to find the largest possible score by strategically applying operations. We can achieve this by prioritizing prime numbers to maximize our gains and then using all the prime numbers that are present to obtain the maximum score.

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

  1. First, we need to identify all the prime numbers available to us within the range of numbers given.
  2. Next, we want to process these prime numbers in descending order, because multiplying by larger numbers is beneficial to the score more.
  3. Whenever we find a prime number, we must perform the corresponding operation. Essentially multiply by the prime number.
  4. We keep track of each operation. By ordering the prime numbers, we ensure that operations applied earlier benefit from the largest prime numbers.
  5. By applying the operations greedily, based on the prime numbers, we make optimal use of the numbers we have and obtain the largest score.

Code Implementation

def apply_operations_to_maximize_score(numbers, prime_factors, operation_count):
    prime_numbers_available = sorted(prime_factors, reverse=True)
    current_score = 1
    operations_applied = 0

    # Processing primes in descending order to maximize score.
    for prime_number in prime_numbers_available:

        # Early exit if we've used all our operations.
        if operations_applied >= operation_count:
            break

        current_score *= prime_number
        operations_applied += 1

    numbers_length = len(numbers)
    remaining_operations = operation_count - operations_applied

    # Use remaining operations with numbers array.
    for i in range(min(remaining_operations, numbers_length)):
        current_score *= numbers[i]

    return current_score

Big(O) Analysis

Time Complexity
O(n * log(log(n)))The described solution involves identifying prime numbers within a range, let's say up to n, which can be done using the Sieve of Eratosthenes. The Sieve of Eratosthenes has a time complexity of O(n * log(log(n))). The rest of the operations such as sorting the primes and applying them involve iterating through the primes identified, which is bounded by the number of primes less than n. The number of primes less than or equal to n is approximately n / ln(n). Sorting can then be O((n / ln(n)) * log(n / ln(n))). Since O(n * log(log(n))) dominates the other operations, the overall time complexity is O(n * log(log(n))).
Space Complexity
O(N)The space complexity is dominated by the need to identify prime numbers within the range of the input numbers. This typically involves creating a boolean array or a set to mark numbers as prime or not prime, which requires space proportional to the range of input numbers (let's assume the maximum number in the input determines the range, or input size N). Therefore, the auxiliary space used to store prime numbers scales linearly with N. Additionally, the operations are performed in place or involve simple scalar variables; hence, those operations contribute constant space complexity.

Edge Cases

CaseHow to Handle
Empty nums arrayReturn 0 immediately as there are no numbers to operate on.
nums array with a single elementReturn nums[0] immediately, no operations possible.
k is 0Return the sum of the array, as no operations will be performed.
All elements in nums are the sameOptimal strategy might involve repeatedly using the same index; handle based on problem constraints and operation definition, potentially requiring tracking indices used.
Large k value exceeds the possible number of operations on numsLimit operations to the array size if k is excessive.
Negative numbers in numsConsider absolute values and sign changes depending on the defined operations, ensuring correct score calculation.
Integer overflow during score calculationUse long or appropriate data type to store the score if integer overflow is possible.
nums array with extremely large numbersThe chosen data type for computation should be large enough to accommodate intermediate values.