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
?
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 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:
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
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:
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
Case | How to Handle |
---|---|
Empty input array | Return 0 immediately, as no subarray can be formed. |
Array with a single element | Return the value of that single element as a single element subarray's GCD is itself. |
Array with all elements being 0 | Return 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 calculations | Use 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 numbers | The 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 same | The 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 memory | Consider using iterative approaches and, if possible, process the array in chunks to avoid exceeding memory limits, especially if the GCD calculation is memory-intensive. |