Longest Binary Subsequence Less Than or Equal to K

Longest Binary Subsequence Less Than or Equal to K

What is this problem about?

The Longest Binary Subsequence Less Than or Equal to K interview question asks you to analyze a binary string (composed of '0's and '1's) and an integer K. Your goal is to find the maximum possible length of a subsequence from the string such that, when this subsequence is treated as a base-2 binary number, its decimal value is less than or equal to K.

Why is this asked in interviews?

This coding problem evaluates a candidate's understanding of binary number properties combined with greedy algorithms. Interviewers love it because it looks like a complex dynamic programming problem at first glance, but it actually has a much more elegant, intuitive solution once you grasp how binary representations work. It tests whether you can avoid over-engineering and apply mathematical insights to string manipulation.

Algorithmic pattern used

The best way to solve this is using a Greedy Strategy with string traversal. Because you want the longest subsequence, every '0' in the string is strictly beneficial—a '0' never increases the binary value if placed at the front, but it adds to the length. Therefore, you should greedily take all the '0's. For the '1's, you only want to take them if they don't cause the total value to exceed K. The optimal approach is to iterate from right to left (least significant bit to most significant bit), picking '1's only while the accumulated binary value remains $\le K$.

Example explanation

Suppose your string is "1001010" and K = 5. The binary value of 5 is 101. We scan from right to left:

• Index 6 ('0'): Value = 0. We take it. Length = 1.
• Index 5 ('1'): Represents $2^1 = 2$. $0 + 2 \le 5$. We take it. Length = 2.
• Index 4 ('0'): Represents $2^2$ but it's a 0. We take it. Length = 3.
• Index 3 ('1'): Represents $2^3 = 8$. $2 + 8 = 10 > 5$. We cannot take this '1'.
• Index 2 ('0'): We greedily take all remaining '0's. Length = 4.
• Index 1 ('0'): Take it. Length = 5.
• Index 0 ('1'): We skip it because taking any more '1's will exceed K. The longest subsequence is "00010" (taking the 0s and the valid 1s), which has a length of 5.

Common mistakes candidates make

A very common mistake is trying to solve this using dynamic programming or memoization. While DP works, it is often too slow ($O(N \times K)$ or similar) and consumes unnecessary memory, leading to Time Limit Exceeded errors on larger test cases. Another mistake is calculating powers of 2 using floats (Math.pow), which can cause precision errors in many languages when dealing with large binary strings. Use bitwise shifts instead.

Interview preparation tip

To master this interview pattern, always remember that in binary, length is dictated by the highest power of 2. A '0' essentially comes for "free" when evaluating the magnitude of the final number if you just keep it in its original relative position. Practice parsing strings backwards to easily calculate running binary values without needing full string-to-integer conversions.