[austincheney]

Before I begin just let me say I am not a mathematician. I program so that I don't have to do complex math.

I know in reality the frequency of iterations varies considerably but for simplicity of discussion let's remove variability.

Say we have a loop with 1000 iterations. That is at minimum 1000 statements in the loop body plus expression overhead from the loop itself. If this loop is nested once with a same sized loop there are now 1,000,000 iterations plus some expression overhead per iteration. If it is nested twice deep there are now 1 billion iterations.

That example is exponential of 1000. Given that there is overhead associated with operation of a loop it is actually greater than exponential. It may not be quite so dramatic as a logarithmic growth curve though.

I completely concede that in reality loops vary in iteration count and so nested loops aren't likely perfectly exponential unless their iteration counts are identical. The increase of iterations from nesting loops does increase more dramatically than a simple multiplicative operation as the depth of loop nesting increases, such that the growth of total iterations is a curve on a graph. A polynomial growth operation when graphed should present a straight diagonal line without the presence of a third variable.

[tachyoff]

I wouldn’t call time complexity particularly complex math. Imagine a loop. It does something n times. Within that loop, you do something n times. For each outer looo through n,you do an inner loop through n. Trivially, this is n*n, or n^2, also known as quadratic time. If you nest another loop, it becomes n^3, or cubic time.

Anyway, there might be some confusion of terms, perhaps. Exponential time in the algorithmic complexity sense is any algorithm that takes 2^n operations to complete. If you’re talking about the number of iterations after you loop through n things n times, then that increase itself would not be exponential. But that delta in iterations is unrelated to exponential and polynomial time complexity.

[austincheney]

I am sooo not a math person.

Let's not forget there is overhead to loops. At a minimum let's assume there is a single statement in the loop body, an increment statement and a terminal condition. In a single loop of 1000 iterations there are 3000 things to evaluate. The math becomes:

(n * 3)^x

If a simple loop is nested twice (3 depths) there would be 1 billion iterations but about 27 billion evaluations. Would it be correct to say that is just slightly faster polynomial growth?

[mlevental]

>that example is exponential of 1000.

no it's not. you simply have the definitions mixed up. exponential slow down or speed up means a^x where x=1000. you are describing polynomial growth, i.e. x^a (where x=1000 and a=3)

[yorwba]

It is exponential in the number of nested loops, which is what's important for the realization that adding more nested loops is bad.

[whatyoucantsay]

> "It is exponential in the number of nested loops, which is what's important for the realization that adding more nested loops is bad."

Adding more nested loops is bad, but it's not exponential. It's polynomial.

As you nest more and more loops, the big O complexity goes from N to N^2 (quadratic) to N^3 (cubic) to N^4, etc... N^(any number) is polynomial. Exponential would be 2^N or 3^N or any number raised to the N.

See: https://en.wikipedia.org/wiki/Exponential_growth

[nickloewen]

OK I had to write things down to try and make sense of this. If anyone is like me, consider this loop that loops 5 times... maybe it will help?

  defines = 0
  tests = 0
  increments = 0

  defines++
  for (let i = 0; i < 5; i++) {
    tests++
    increments++
  }

  console.log(`defines: ${defines}, tests: ${tests}, increments: ${increments}`)

For the sake of space I'm not going too copy nested versions of that, but imagine nesting it two and then three levels deep.

  1 level will increment 5 (5^1) times    
  2 levels will increment 25 (5^2) times    
  3 levels will increment 125 (5^3) times

More interesting are the number of definitions and tests:

  levels    tests    defines
  1         15       1 
  2         30       6
  3         155      31

But I don't know is how the interpreter works and whether it has tricks to shrink those numbers; those just reflect my naive mental model of how things work.

[whatyoucantsay]

The confusion here is about math, not syntax.

N^2 is polynomial.

2^N is exponential.

https://stackoverflow.com/questions/4317414/polynomial-time-...

[kazagistar]

If N is the number of nested loops, and M is the number of times through the loop, then it is indeed a O(M^N). So indeed, complexity scales exponentially with the level of nesting. The wording was just off amd confusing due to it being a nontraditonal formulation of the problem, but what he was saying does actually make sense.

[AstralStorm]

You will probably blow up stack on counters or iterators way before you reach even close to 2^N behaviour.

[yorwba]

Something can be exponential in one context and polynomial in another. Asymptotic analysis is about the growth rate of a function in terms of some input variable. The results you get depend on which values you assume to be fixed while others vary. It is usually applied to analyze the runtime of a program in terms of the input size, which is the basis of classifying the runtime complexity, but that's not the only way you can do it.

If you have a sequence of programs with polynomial runtime, but which grows as N, N^2, N^3, N^4, ..., that's a textbook example of exponential growth. It is not generated by running a program on increasingly larger inputs, so there's nothing to be put in the exponential runtime complexity class, but it's exponential nonetheless.

[whatyoucantsay]

> If you have a sequence of programs with polynomial runtime, but which grows as N, N^2, N^3, N^4, ..., that's a textbook example of exponential growth

No, it isn't. N, N^2, N^3, N^4, ... is polynomial, not exponential. Exponential would be X^N. Look at the graph on the Wikipedia page I linked to.

[yorwba]

Each element of the sequence is a polynomial, but the sequence of polynomials grows exponentially. X^N is exponential in N, polynomial in X. N^X is exponential in X, polynomial in N.

[IncRnd]

What you are describing now is called superpolynomial time in computer science.

[dbaupp]

In this case, the number of iterations (1000) of each loop is being held fixed, and the depth of nesting is varying, i.e. the body of the inner loop executes O(1000^depth) times.

[colanderman]

Nitpick: "O(…) times" is nonsensical. O-notation applies only to behavior in the limit. Notably, O(some constant) is exactly equivalent to O(1).

[dbaupp]

What exactly are you nitpicking?

1000^depth is not a constant: the function here is "f(depth) = number of times the inner loop body executes". f(depth) = O(1000^depth).

And, "times" here is serving the same role as "comparisons" in "Mergesort takes O(n log n) comparisons": it's referring to the thing that f is actually counting.

[yorwba]

In this case depth is not constant, so 1000^depth isn't either.

And usually when you encounter O(some constant), it's meant as "of the order of magnitude of", i.e. somewhere between some constant/10 and some constant * 10. That isn't the definition used here, but seems to be the cause of most complaints about asymptotic analysis being misapplied when no asymptotic analysis was being done in the first place.

[Ar-Curunir]

You won't have random extra nested loops appear out of nowhere in your code