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

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)`?

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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) ...

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I see.But why not just Theta(c) to cover any constant?It is just a convention? –  user10326 Sep 24 '11 at 13:56
@user10326: I think it's because "c" could be misinterpreted, you clearly have to state that it is a constant while "1" does the same job unambiguously. –  Falcon Sep 24 '11 at 13:58
So the actual number is irrelevant?We use 1 instead of 5 as a convention? –  user10326 Sep 24 '11 at 14:54
@user10326: Yes, because it doesn't make a difference. But I'd refrain from using "0" because that could lead to a lot of confusion. –  Falcon Sep 24 '11 at 15:09
@user10326: Falcon his answer made perfect sense didn't it? if it would be 5 instead of 1, dropping 5 in `O(5 * n^2)` would feel less natural, while dropping `* 1` is basic math. –  Steven Jeuris Sep 24 '11 at 15:50

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)`

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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.

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`Theta(1) however is always a constant`.This is the part I do not get.Theta(c) is always a constant as well.Right?So I was wondering if the `1` has a special meaning –  user10326 Sep 24 '11 at 13:58
@user10326: no, `c` is not always a constant, since `c` is a variable if you do not explicitly state that it should be in fact interpreted as a constant... Which is exactly the subtle difference that is avoided by `1`. –  blubb Sep 24 '11 at 14:09
Ok, but it represents a constant time. –  user10326 Sep 24 '11 at 14:19
@user10326: No, no, it does not represent a constant time. It represents time that grows linearly with c. Those are different, because you need something additional to force the value of c to never change, whereas 1 never changes. –  jprete Sep 24 '11 at 17:13
@user10326: Or, put more simply: `c` isn't a constant; `c` is a letter. Other letters represent variables, how do you expect the reader to know this one doesn't as well? –  Random832 Sep 24 '11 at 18:38

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).

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But why do you say `n^0` to denote constant time and not `n^1` in your example? –  user10326 Sep 24 '11 at 17:02
@user10326, because n^1 = n is not constant. It grows linearly. –  Peter Taylor Sep 24 '11 at 19:45

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.

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