You are given an integer array nums
and a positive integer k
. A subsequence sub
of nums
with length x
is called valid if it satisfies:
(sub[0] + sub[1]) % k == (sub[1] + sub[2]) % k == ... == (sub[x - 2] + sub[x - 1]) % k
.Return the length of the longest valid subsequence of nums
.
For example:
nums = [1,2,3,4,5], k = 2
The longest valid subsequence is [1, 2, 3, 4, 5]
. The function should return 5.
Another example:
nums = [1,4,2,3,1,4], k = 3
The longest valid subsequence is [1, 4, 1, 4]
. The function should return 4.
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 strategy for this problem involves examining every single possible subsequence to find the longest valid one. We generate all possible combinations and then check if each one meets the problem's criteria for being valid. Finally, we compare the lengths of all the valid subsequences and select the maximum.
Here's how the algorithm would work step-by-step:
def find_max_length_valid_subsequence_brute_force(sequence):
max_subsequence_length = 0
# Iterate through all possible subsequences
for i in range(1 << len(sequence)):
subsequence = []
for j in range(len(sequence)):
# Check if the j-th element is included in the current subsequence
if (i >> j) & 1:
subsequence.append(sequence[j])
# Check if the current subsequence is valid.
if is_valid_subsequence(subsequence):
# Update the maximum length if the current subsequence is longer.
max_subsequence_length = max(max_subsequence_length, len(subsequence))
return max_subsequence_length
def is_valid_subsequence(subsequence):
if not subsequence:
return True
for i in range(len(subsequence) - 1):
if subsequence[i+1] <= subsequence[i]:
return False
return True
The key is to use a technique that efficiently skips over subsequences that are guaranteed to be invalid. We keep track of the best possible length we have seen so far to help us decide whether or not to explore a specific subsequence. This technique helps significantly reduce the search space.
Here's how the algorithm would work step-by-step:
def find_maximum_length_of_valid_subsequence(sequence):
sequence.sort()
longest_subsequence_length = 0
for starting_index in range(len(sequence)):
current_subsequence_length = 0
last_element = -1
def extend_subsequence(current_index):
nonlocal current_subsequence_length
nonlocal longest_subsequence_length
nonlocal last_element
# Early cutoff: Not enough elements remaining.
if current_subsequence_length + (len(sequence) - current_index) < longest_subsequence_length:
return
if current_index == len(sequence):
longest_subsequence_length = max(longest_subsequence_length, current_subsequence_length)
return
# Check if we can extend the subsequence.
if sequence[current_index] >= 2 * last_element:
current_length_before = current_subsequence_length
last_element = sequence[current_index]
current_subsequence_length += 1
extend_subsequence(current_index + 1)
current_subsequence_length = current_length_before
last_element = sequence[starting_index] if current_subsequence_length > 0 else -1
# Explore the possibility of skipping this element.
extend_subsequence(current_index + 1)
#Consider each element as the potential start.
current_subsequence_length = 1
last_element = sequence[starting_index]
extend_subsequence(starting_index + 1)
return longest_subsequence_length
Case | How to Handle |
---|---|
Empty input array | Return 0 immediately as an empty array cannot contain a valid subsequence. |
Input array with only one element | Return 0 because a subsequence requires at least two elements, thus one element will always be invalid. |
Array with all identical values. | Iterate through the array, for each element count the duplicates of the elements that are exactly twice the current element and those that are exactly half and choose the subsequence with maximum length. |
Array contains negative numbers. | The solution should handle negative numbers correctly by considering both multiplication and division by 2 for subsequence verification, ensure no integer overflow occurs during multiplication. |
Array contains zero. | Handle zero as special case; any zero multiplied by 2 is zero again so the subsequence should not include more than 1 zero, or division by zero errors will occur. |
Integer overflow during multiplication by 2 | Use long data type or perform checks to prevent overflow during multiplication, potentially capping the maximum value before multiplication. |
Input array with very large number of elements (performance) | Optimize the solution to reduce time complexity, potentially using memoization or dynamic programming depending on the specific algorithm used to determine subsequence validity. |
No valid subsequence exists in the input | The algorithm should return 0 when no valid subsequence is found, such as if all numbers are prime or are unrelated by factors of 2. |