You are given a binary string s
and a positive integer k
. Return the length of the longest subsequence of s
that makes up a binary number less than or equal to k
. Note the following:
For example, given s = "1001010"
and k = 5
, the longest subsequence of s
that makes up a binary number less than or equal to 5 is "00010"
, which is equal to 2 in decimal. The length of this subsequence is 5, so you should return 5. Other possible subsequences are "00100"
(4 in decimal) and "00101"
(5 in decimal).
Another example: given s = "00101001"
and k = 1
, the longest subsequence of s
that makes up a binary number less than or equal to 1 is "000001"
, which is equal to 1 in decimal. The length of this subsequence is 6, so you should return 6.
How would you approach this problem efficiently? Consider edge cases and optimize for both time and space complexity.
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 goal is to find the longest combination of numbers from the binary sequence that sums up to a value less than or equal to a target value. The brute force approach involves checking every possible combination of numbers from the given sequence.
Here's how the algorithm would work step-by-step:
def longest_binary_subsequence_brute_force(binary_sequence, target_value):
max_length = 0
sequence_length = len(binary_sequence)
# Iterate through all possible subsequences
for i in range(1 << sequence_length):
current_subsequence = []
current_sum = 0
current_length = 0
# Construct the subsequence and calculate its sum
for j in range(sequence_length):
# Check if j-th bit is set in the current combination
if (i >> j) & 1:
current_subsequence.append(binary_sequence[j])
current_sum += binary_sequence[j]
current_length += 1
# Check if the sum is less than or equal to the target
if current_sum <= target_value:
# Update max_length if needed
if current_length > max_length:
max_length = current_length
return max_length
The goal is to find the longest possible sequence of ones and zeros from the input, but treating it as a binary number that is less than or equal to a given limit. Instead of trying every possible combination, we should prioritize using zeros because they contribute nothing to the overall value, and then strategically place ones where needed.
Here's how the algorithm would work step-by-step:
def longest_binary_subsequence(
binary_string: str, maximum_value: int
) -> int:
zero_count = 0
one_count = 0
current_value = 0
# Count the zeros
for bit in binary_string:
if bit == '0':
zero_count += 1
# Include ones from right to left
for i in range(len(binary_string) - 1, -1, -1):
if binary_string[i] == '1':
# Adding a '1' may exceed the limit
if current_value + (1 << (len(binary_string) - 1 - i)) <= maximum_value:
current_value += (1 << (len(binary_string) - 1 - i))
one_count += 1
# Return the length of the longest subsequence
return zero_count + one_count
Case | How to Handle |
---|---|
Empty binary string | Return 0 as there's no subsequence. |
k is zero and the string contains only 1s | Return 0 as no subsequence is less than or equal to k. |
k is negative | Return 0 as binary subsequences are always positive. |
String contains only zeros and k is greater than zero | Return the length of the string, as any subsequence is valid. |
String contains only ones and k is less than the smallest subsequence's value (1) | Return 0 because no subsequence satisfies the condition. |
Integer overflow when calculating the decimal value of a long subsequence | The algorithm should efficiently count '0's and a limited number of '1's to prevent integer overflow when calculating subsequences. |
k is very large (close to max int) and string has many 1s. | Limit the number of ones considered from the right to the number of bits required to represent k. |
String of all ones, k is one | Return 1, representing a single '1' subsequence. |