Binary String With Substrings Representing 1 To N

Binary String With Substrings Representing 1 To N

What is this problem about?

The "Binary String With Substrings Representing 1 To N interview question" asks you to verify if a given binary string $s$ contains the binary representations of all integers from 1 to $n$ as substrings. For example, if $n=3$, you need to check if "1" (binary for 1), "10" (binary for 2), and "11" (binary for 3) are all present in $s$.

Why is this asked in interviews?

Google asks the "Binary String With Substrings Representing 1 To N coding problem" to test a candidate's ability to analyze constraints and string complexity. While the problem sounds like it might require checking many values, there's a mathematical limit: as $n$ grows, the length of the binary representations grows, making it impossible for a short string $s$ to contain all of them. It evaluates "Bit Manipulation interview pattern" and "String interview pattern" skills.

Algorithmic pattern used

This problem can be solved with String Searching or Rolling Hash.

1. Direct Approach: For each $i$ from 1 to $n$, convert $i$ to a binary string and check if it's a substring of $s$.
2. Optimization/Constraint Check: If $n$ is very large (e.g., $n > 2 imes length(s)$), it's mathematically impossible for all binary strings to be present, so you can return false immediately.
3. Iterative Search: Start from $n$ and work downwards to 1. Since larger numbers' binary forms often contain the binary forms of smaller numbers, this can sometimes be slightly faster in practice.

Example explanation

$s = "0110"$, $n = 3$

• Binary for 1: "1". Found at index 1.
• Binary for 2: "10". Found at index 2.
• Binary for 3: "11". Found at index 1. All are found. Result: true. If $n = 4$ (binary "100"), it's not in "0110", so the result would be false.

Common mistakes candidates make

• Ignoring the $n$ limit: Trying to check millions of substrings without realizing that the string $s$ is likely too short to hold them.
• Slow conversion: Using an inefficient method to convert integers to binary strings.
• Redundant checks: Not realizing that if "110" is present, "11" and "10" are also likely present (though not guaranteed to be at the same positions, they are sub-components).

Interview preparation tip

In string problems involving binary numbers, always consider the length of the strings. A binary representation of $n$ has $\log_2(n)$ bits. If $s$ is short, $n$ cannot be very large. This kind of "Complexity Analysis" is what interviewers look for in "Medium" problems.