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Topic: binary search with arbitrary or random split
Replies: 2   Last Post: Jul 26, 2012 11:35 PM

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Ben Bacarisse

Posts: 1,149
Registered: 7/4/07
Re: binary search with arbitrary or random split
Posted: Jul 26, 2012 10:02 PM
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pam <pamelafluente@libero.it> writes:

> On Jul 27, 1:29 am, Ben Bacarisse <ben.use...@bsb.me.uk> wrote:
<snip>
>> I think these examples make it clear why you need a new notation.  You
>> can't re-define O(f) with a probabilistic meaning without there being
>> way too much scope for confusion.

>
> Alas, it looks like it has already invented since long time. So i
> arrived late :-(


Except that the paper you cite does not seem to use the same definition
that you were proposing. Of course, if you like this definition, go
ahead and you their notation -- no need to proliferate symbols -- but
you'll need a way to write it for Usenet posts as your quote below shows
so clearly.

> Making a quick check, it looks like that (while someone here was
> accusing me of "gibberish" :-)


It's possible that what you wrote was gibberish (I'm sorry but I don't
recall). What's clear is that the paper you cite does not use the same
reasoning you do. It uses what looks like a significantly different
definition. In particular it does not rely on "almost sure"
convergence. It's possible, of course, that their definition is that
same as yours. It would make an interesting exercise to see if that is
the case.

> using the big O in the probabilistic sense is the norm even in
> official CS literature (SIAM) when
> dealing with probabilistic formulations.


That's not a clear way to report it. The paper you cite uses big O as
everyone else does. They define another relation (with it's own
notation) that relies on probabilities. I'd need a few more references
to know if either this new notation or the specific definition they use
is anything like "the norm".

What is still not in doubt is that big O is used, even in the context
probabilistic algorithms, to mean what it has always meant. The
textbook I cited earlier does so, and so does the paper you've
referenced here. (I'm sorry to keep banging on about this but you've
worded the above remark in such a way that I felt it needed to be said
again.)

> See for instance:
>
> http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.30.2557
> http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=DCD58376005D28855D7D062968CC925F?doi=10.1.1.30.2557&rep=rep1&type=pdf
>
> You can read at pag 4, under 1.3 just the same things i have been
> saying here ;-)


Well, not "just the same"; "similar" is the best I can do right now.
They don't use almost sure convergence and it's not clear how their
definition relates to your algorithm. Have you been able to show that
your uniform splitting search is OPr(log(n))? (I'm using my suggested
OPr because the notation they use does not obviously translate to
ASCII.)

> "Just like big-O function serves to represent the complexity bounds of
> deterministic algorithms, we employ O to represent complexity bounds
> of randomized algorithms. We say a randomized algorithm has resource
> (like time, space, etc.) bound O(g(n)) if there is a constant c such
> that the amount of resource used by the algorithm (on any input of
> size n) is no more than c g(n) with probability 1 1=n for any > 1
> ... "


That went wrong. For everyone else (is anyone else reading this?) the
paper uses O with a tilde over it for this probabilistic bound, and the
definition is that

We say a randomized algorithm has resource [...] bound O(g(n)) if
there is a constant c such that the amount of resource used by the
algorithm (on any input of size n) is no more than c a g(n) with
probability >= 1 - 1/(n^a) for any a > 1.

> (i found similar stuff in several other papers) Hmm, it's time i
> begin to read some CS article too ... :-)


Do they all use this definition? Any agreement between them on the
notation to be used?

--
Ben.



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