Date: Jan 4, 2013 12:07 PM Author: David Bernier Subject: Re: Another count sort that certainly must exist, it do not have<br> any restrictions upon size of (S number of possibilities) On 01/04/2013 10:46 AM, JT wrote:

> 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