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:
nums[l, ..., r]
that you haven't chosen previously.x
of nums[l, ..., r]
with the highest prime score. If multiple such elements exist, choose the one with the smallest index.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
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:
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:
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
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:
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
Case | How to Handle |
---|---|
Empty nums array | Return 0 immediately as there are no numbers to operate on. |
nums array with a single element | Return nums[0] immediately, no operations possible. |
k is 0 | Return the sum of the array, as no operations will be performed. |
All elements in nums are the same | Optimal 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 nums | Limit operations to the array size if k is excessive. |
Negative numbers in nums | Consider absolute values and sign changes depending on the defined operations, ensuring correct score calculation. |
Integer overflow during score calculation | Use long or appropriate data type to store the score if integer overflow is possible. |
nums array with extremely large numbers | The chosen data type for computation should be large enough to accommodate intermediate values. |