Math Forum Discussions

JT

Posts: 436
Registered: 4/7/12

Re: Another count sort that certainly must exist, it do not have any
restrictions upon size of (S number of possibilities)
Posted: Jan 4, 2013 8:29 PM

Plain Text

On 4 Jan, 18:07, David Bernier <david...@videotron.ca> wrote:
> On 01/04/2013 10:46 AM, JT wrote:
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> > On 4 Jan, 15:46, JT<jonas.thornv...@gmail.com> wrote:
> >> I remember doing this in a tentamen during my education in information
> >> theory beleiving what i did was binary sort but my teacher informed me
> >> it wasn't so what is it.
> >> By creating a Pascal pointer binary tree with each leaf holding a
> >> integer, you move the binary numbers to the tree from least digit to
> >> highest using left legs for 0's and right for 1's. (Basicly creating
> >> leaves for new numbers, and at last digit you add 1 to the leaf slot.
> >> So after you moved all values into the tree and created all the nodes,
> >> you simply read out all the none zero values holded into the slot of
> >> the leaves within the binary tree.
>
> >> What is this sort called?
> >> Of course you cannot have more leaves then memory, but this does not
> >> need to hold memory for slots never used like the array slots, it is
> >> therefore my beleif that this sort could be useful also for database
> >> purposes sorting basicly anything. What do you think?
>
> > I can see there would be problems reading out the sizes of a binary
> > tree from smallest to biggest, if you have legs with different
> > lengths? Is there any algorithmic solution to this problem.
> > I have kind of a foul play solution, you create a binary tree for
> > every digit bigger then 2^20 the smaller ones you run with the array
> > approach. So for 21,22,23... bits and so on each numbers run on their
> > own computers, with 2048 computers you could sort enormous amount of
> > data of different size. So basicly the "heaps?" all have legs with
> > same sizes and is easy to read out in order.
> > Is this a working idea or just plain silly, maybe it is just easier to
> > use one computer and read out the values from the heap and sort them
> > with quicksort after you filled up the tree? (Is it called tree or
> > heap, what is the difference betwee a heap and a tree?).
>
> > So what you think about the mix using this kind of sort for counting
> > in values, and then quicksort to sort the none null tree nodes by
> > sizes.
>
> Oops.. below is about factoring. The best algorithms
> have been getting better since Maurice Kraitchik's [1920s]
> improvement on Fermat's method of expressing a number
> as a difference of squares, n = a^2 - b^2, so
> n = (a-b) (a+b).
>
> There's a very good article called "A Tale of Two Sieves"
> by Carl Pomerance: Notices of the AMS, vol. 43, no. 12,
> December 1996:
> <http://www.ams.org/notices/199612/index.html>
>
> The 9th Fermat number F_9 = 2^(512)+1 had been factored
> around 1990 by the Lenstras et al using the Number Field
> sieve (which had supplanted the quadratic sieve).
>
> The Quadratic sieve is easier to understand than the
> Number Field Sieve, which I don't understand.
>
> F_10 and F_11 were fully factored then, using the elliptic
> curve method (which can find smallish prime factors).
>
> F_12 was listed as not completely factored, with
> F_12 being a product of 5 distinct odd primes and
> the 1187-digit composite:
>
> C_1187 =
> 22964766349327374158394934836882729742175302138572\
> 22257593176439130841895160961323826592803808643123\
> 15776330453915314460450194556572637889591520959595\
> 00781101126096495656976145338084323609391242570049\
> 59146146100932078255130896682422242552873156911153\
> 49491277441664272360127694182069497019299146312879\
> 53679124328078403443589001544785043209243005176672\
> 36512498567556601129618233580642646148465607080211\
> 50483896593552361820682419503442019994498256473415\
> 56766313684295383743697537161298411893329950259437\
> 02457251084955979786901113201153080673107947314499\
> 89885761657097352227077484815352368256239445951125\
> 33741234160090993221997405711848497115626313770615\
> 84634017936609811822404415794282448107580150138831\
> 67949250345497227202182371779894151535731419443909\
> 33701532957472310726727304029461192020120667119324\
> 40906462375814643855500503626564314311613740004222\
> 88239457400101057642788560965414596506825478363862\
> 10032027169896230115182649724551245475912070548418\
> 45921140740300676916471986974995922243980616471547\
> 01759458614628952014532145179607626863555620392963\
> 07129357252744645128034273466002900209575716007479\
> 66912966168394403107609922082657201649660373439896\
> 3042158832323677881589363722322001921.
>
> At 3942 bits for C_1187 above, what's the
> probability density function of expected time
> till C_1187 is fully factored?
>
> Or, centiles: e.g. 50% chance fully factored
> within <= 10 years. 95% chance fully factored within
> <= 95 years, etc. ...
>
> dave

Is this idea the same sort as the other counting sort, it seem more
adjustable to sort anysized number when just fitting them into the
binary tree.