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I have the following Python code.

def collatz(n):
    if n <= 1:
        return True
    elif (n%2==0):
        return collatz(n/2)
    else:
        return collatz(3*n+1)

What is the running-time of this algorithm?

Try:

If $T(n)$ denotes the running time of the function collatz(n). Then I think I have \begin{cases} T(n)=1 \text{ for } n\le 1\\ T(n)=T(n/2) \text{ for } n\text{ even}\\ T(n)=T(3n+1) \text{ for } n\text{ odd}\\ \end{cases}

I think $T(n)$ will be $\lg n$ if $n$ is even but how to calculate the recurrence in general?

Raphael
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9bi7
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5 Answers5

31

This is Collatz conjecture - still open problem.
Conjecture is about proof that this sequence stops for any input, since this is unresolved, we do not know how to solve this runtime recurrence relation, moreover it may not halt at all - so until proven, the running time is unknown and may be $\infty$.

Evil
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15

You translated the code correctly. There are many methods for solving recurrences.

However, it is currently unknown if collatz even halts for all n; the claim that it does is known as Collatz conjecture. Therefore, no known method will work on this recurrence.

I think $T(n)$ will be $\lg n$ if $n$ is even

How so? I guess you are thinking of $n=2^k$, for which your claim is correct. This goes to show that this recurrence is not one we can solve up to $\Theta$ by investigating exponentially few points (see also here).

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Raphael
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13

The time complexity function is

\begin{cases} T(n)= O(1) \text{ for } n\le 1\\ T(n)=T(n/2) + O(1) \text{ for } n\text{ even}\\ T(n)=T(3n+1) + O(1)\text{ for } n\text{ odd}\\ \end{cases}

which can be rewritten as the following if you are interested in asymptotic time complexity.

\begin{cases} T(n)= 1 \text{ for } n\le 1\\ T(n)=T(n/2) + 1 \text{ for } n\text{ even}\\ T(n)=T(3n+1) + 1\text{ for } n\text{ odd}\\ \end{cases}

It is not even known that the resulting TM $\langle M,1^n\rangle \in Halt$ (See Halting Problem) for every $n$. So naturally we can't measure the time taken to halt if we do not even know if it halts for every $n$. Also refer https://math.stackexchange.com/questions/2694/what-is-the-importance-of-the-collatz-conjecture.

Collatz conjecture is a very famous conjecture which Collatz proposed in 1937. Many eminent mathematicians have spent (read wasted) countless hours in trying to solve this conjecture but to little avail. Even Paul Erdős said about the Collatz conjecture, "Mathematics is not yet ready for such problems."

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Sarvottamananda
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2

I would add to the other answers the fact that the third branch of the function i.e the case of $n$ is $odd$ suggests that in that case the function will be computed on a larger instance ($n$ is lower than $3n+1$). So there is more work to be done on that computation branch.

Other recurrence relations that we encounter (for example in the case of merge sort $T(n) = 2T(n/2) +n$) adhere to the division of a problem into subproblems for instances with smaller size. Those reccurences degenerate naturally into base case(s), namely $T(0)$ or $T(1)$ which are known. So, standard mathematical tools for analyzing the reccurence can be used, like the master theorem.

In the case of collatz function we are not able to infer directly from the reccurence that the case of $3n+1$ will eventually deteriorate using standard mathematical tools of reccurences. This is a topic to be resolved by looking at the problem itself, not the reccurence that describes it.

Sorrop
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0

You have T (n) = T (n/2) + 1 if n is even. But then n/2 is quite likely not even, so you are stuck there.

What happens is that the nice little rules that you learned are confronted with a real problem, and they don't work. They hit a brick wall, face first, and it hurts. Do yourself a favour, and follow the recursion for T (7) manually, and then you tells if you still believe this is related lg n.

If you think this isn't related to the original question because 7 is not even: Whenever n is odd, T (n) = T (3n + 1), and 3n + 1 is even, so if T (n) were log n if n is even, it would be log (3n + 1) + 1 whenever n > 1 is odd.

gnasher729
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