Theta notation on constant time. Why we use the 1?

Question

In asymptotic notation when it is stated that if the problem size is small enough (e.g. n<c for some constant c) the solution takes constant time and is writen as Theta(1).
Why we write 1 inside the Theta?
What does the 1 mean? Why not Theta(c)?

Answer 1

This is all very hand-wavy, but there is a mathematical reason why we don't use Theta(c) and instead use Theta(1). I'll use Big O notation instead to show this.

It has to do with a property of Big Theta (as well as Big O and Big Omega) notation. If you have a function with growth rate O(g(x)) and another with growth rate O(c * g(x)) where c is some constant, you would say they have the same growth rate. That is O(c * g(x)) = O(g(x))

We can say this because the definition of Big O notation (f(x) = O(g(x))) means that we have a function f(x) and function g(x) such that |f(x)| <= k * |g(x)| for some constant k and large enough values of x. When multiplying by the constant c, we would then have:

O(c * g(x)) => k * |c * g(x)| = k * |c| * |g(x)| <= k' * g(x) where k' = k * |c|

Note that |k' * g(x)| <= k'' g(x) for some constant k'' and large enough values of x, which means k' * g(x) grows at a rate of O(g(x)) and therefore O(c * g(x)) = O(g(x))

When g(x) = 1, we have O(1) growth, saying O(c) growth for some value of c doesn't tell us anything because the constant is already factored in to the definition of Big O notation. Simplified O(c) = O(1)

Answer 2

Those notations are meant to denote the asymptotic growth. Constants do not grow and thus it's pretty equal which constant you choose. However, there's a convention that you choose 1 to indicate no growth.

I assume that this is due to the fact that you want to simplify the mathematical terms in question. When you've got a constant factor just divide by it and all that's left of it is 1. This makes comparisons easier.

Example:

O(34 * n^2) = O(1 * n^2) = O(n^2)

and

O(2567.2343 * n^2 / 5) = O(n^2)

See what I mean? As these mathematical terms get more and more complicated, you don't want to have noisy constants when they're not relevant for the information you're interested in. Why should I write O(2342.4534675767) when it can be easier expressed with O(1), which communicates the facts of the case unambiguously.

Further, the wikipedia article about time complexity also implies it's a convention:

An algorithm is said to be constant time (also written as O(1) time) ...

Answer 3

Well, of course you could write Theta(c) (or O(c)) but why does that differ from Theta(n)? n is just a variable that denotes the size of the input. You could write "The function is Theta(c) where c is a constant". The important addendum is ...where c is a constant. You have to explicitly state that an identifier is not a variable.

Consider graph theory where the bounds for an algorithm is often described as a function of |V| and |E|, or the node and edge count, respectively. Then it might be prudent to state "The function is Theta(|V| * |E|^2)".

Theta(1) however is always a constant - assuming normal mathematical practices.

Answer 4

Theta notation is about growth as a function of some variable - typically n. If it were necessary to clarify which variable is intended, the way to write it would be Theta(n^0). From there it's a simple step to apply the identity n^0 = 1 (for n != 0).

Answer 5

O(c) specifically means that the associated class of algorithms grows linearly with c, where c is the size of an input to the algorithm or a parameter to the algorithm. It isn't the same c that is used to explain O-notation, because that c is only relevant to the explanation, not the usage. O(c) contains a different c that must come from the algorithm input context.