If I have an algorithm that simply converts an input integer to its binary representation and stores the result, what is the space complexity of this algorithm?
I initially assumed that the space required would be proportional to the number of bits in the binary representation of the number. This would give an O(logn) space complexity, since the number of bits required to represent an integer n in binary is roughly log(2)n.
However, I’ve recently considered the fact that the input would be constrained to a fixed range, such as 2^31 - 1 for a 32-bit signed integer. In this case, the length of the binary representation is capped at a fixed length of 32 bits, regardless of the input size, since the input integer is constrained to fit within 32 bits.
So, is it correct to conclude that the space complexity of storing the binary representation of the integer is thus constant / O(1), given the fixed upper bound on the length of the binary string?
Thanks for the replies. I was going to provide an edit but decided to add my own answer. There seems to be a few different ideas and opinions here, and after looking into this question further, I've realized that the core of the discussion revolves around how we define the scope of our analysis—what we care about when we talk about algorithmic complexity. Does the underlying representation of the input matter in our analysis, or should we only focus on the abstract, logical behavior of the algorithm?
Here’s an adjacent question that helps us explore this idea:
If I have a program that copies an input integer to another variable, this is typically assumed to be a constant space operation. But since the input integer is likely represented as a binary number under the hood, why is this not considered an O(log n) operation, since we are copying all log(n) bits from one variable to another?
One possible answer is that we are dealing with a fixed-size input, such as a 32-bit integer, which would make the operation constant in terms of space because the input is capped at a maximum size (regardless of the actual value of the integer). But is this truly the reason we consider the operation constant space, or does the space complexity analysis depend more on what we measure the complexity with respect to?
The key insight, I believe, is that the answer depends on our model of computation—on what we are willing to count as "space" or "time" in our complexity analysis. Complexity analysis is inherently tied to the assumptions we make about the nature of the input and the environment. As one comment puts it, "All the computations of complexity depend on your model of computation, in other words on what you would like to count." Are we counting only the space that our algorithm explicitly allocates, or are we including the underlying machine representation (e.g., the binary encoding of integers)?
In this context, the RAM model of algorithmic analysis is usually what we are concerned with. In this model, the focus is on high-level operations, such as additions and comparisons, and assumes a fixed-size machine word (e.g., 32 or 64 bits). The time and space complexity are measured based on the number of operations required to solve a problem, ignoring the details of machine word size or arbitrary precision. For most algorithmic problems and competitive programming algorithms, this model is used, as it abstracts away the details of the underlying hardware and focuses on the algorithm’s efficiency with respect to input size.
In short, the real question boils down to what you’re measuring your algorithmic complexity with respect to—the input value itself or the representation of the input value on the machine. If we consider the input value in its raw form, with a fixed bit size (say 32 or 64 bits), then copying it would be an O(1) operation. But if we delve into the details of how the input is represented on the machine (in binary, with a potentially varying bit length), we might argue that the operation could take O(log n) space, reflecting the number of bits involved.
Ultimately, it’s less about whether the input is “arbitrary precision” versus “fixed size” and more about what assumptions we make when measuring algorithmic complexity. These assumptions about the model of computation—what we count and how we count it—determine whether we consider the operation constant space or logarithmic.
In Short: O(1) time with respect to what we are generally concerned about. We can do complexity analysis with respect to other lower-level details, but this generally is outside the scope of our model of analysis.
Note that also assumes that when we store the binary representation of the integer, we are not storing it as a string, as this would definitely require O(logn) space.