Re: matching lists

["J.D. Smith" <jdsmith(at)astro.cornell.edu>]

Craig Markwardt wrote:
> 
> "J.D. Smith" <jdsmith@astro.cornell.edu> writes:
> >
> > Mark Fardal wrote:
> > >
> > > Hi,
> > >
> > > I have been looking at properties of particles in a simulation, and
> > > sometimes need to match up the particles in two different subsets.  I
> > > typically have (quantity A, index #) for one set of particles, and
> > > (quantity B, index #) for another set, and want to compare quantities
> > > A and B for the particles that are in both sets.
> > >
> > > As of late last night I could not think of a good way to do this;
> > > WHERE inside a for-loop would be very slow.  Maybe I'm missing
> > > something easy, but in any case here's a solution inspired by the
> > > recently submitted SETINTERSECTION function.  Hope somebody finds
> > > it useful.
> > >
> >
> > The standard where_array, as posted a few years back, and modified
> > slightly for the case of the null intersection, is attached.  It
> > will work with floating point and other data types also.  It works
> > by inflating the vectors input to 2-d and testing for equality in
> > one go.  It will also handle the case of repeated entries.
> > ...
> 
> I also submit CMSET_OP, a function I recently posted on my web page.
> (Actually, I'm not sure if Mark is referring to that by
> SETINTERSECTION).
> 
> Advantages are:
>  * works on any numeric or string data type
>  * works in order (n1+n2)*log(n1+n2) time or better, rather than n1*n2
>  * uses the histogram technique for short integer lists as JD suggests
>  * also does "union" and "exclusive or"
>  * also does A and NOT B  or vice versa
> 
> Disadvantages:
>  * it removes duplicates, treating the two lists strictly as sets.
>  * returns values, not indices
> 

The flag+shift method which I came up with a few years back (search for
"Efficient comparison of arrays", circa 1997) was surrounded by a great deal of
discussion about the best type of algorithm for this method. I'll quote from one
of my postings at that time:

==============================================================================================
On another point, it is important to note that the problem of finding where b's
values exist in a (find_elements()) is really quite different from the problem
that contain() attempts to address: finding those values which are in the
intersection of the vectors a and b (which may be of similar sizes, or quite
different).  The former is a more difficult problem, in general, which
nonetheless can be solved quite rapidly as long as one vector is quite short. 
But the time taken scales as the number of elements in b, as opposed to the
comparative size of b (to the total elements in a and b) -- i.e. nearly constant
with increasing length of b.  Anyway, it is important to understand the various
scales, sizes and efficiencies in the problem you are trying to solve if you
hope to come up with an effective solution.
==============================================================================================

The point of which is that cmset_op, while a very complete and feature-packed
collection of the various *value-based* set algorithms proposed over the years,
does not solve the *index-based* set intersection operation originally
requested.  This is a harder problem, also solveable for non-sparse sets with
histogram and reverse indices, or with the full n-squared array comparison.  

A sort-based method which solves this problem, but which also lack grace when
dealing with repeated values (just selecting the last one which exists in the
intersection), is as follows:

;; Return the indices of values in a which exists anywhere in b (one only for
repeated values)
function ind_int_SORT, a, b
   flag=[replicate(0b,n_elements(a)),replicate(1b,n_elements(b))]
   s=[a,b]
   srt=sort(s)
   s=s[srt] & flag=flag[srt]
   wh=where(s eq shift(s,-1) and flag ne shift(flag, -1),cnt)
   if cnt eq 0 then return, -1
   return,srt[wh]
end

Which can be compared to the ARRAY, and HISTOGRAM methods:

function ind_int_ARRAY, a, b
   Na = n_elements(a)           
   Nb = n_elements(b)
   l = lindgen(Na,Nb)
   AA = A(l mod Na)
   BB = B(l / Na)
   
   ;;compare the two matrices we just created
   I = where(AA eq BB,cnt)
   if cnt eq 0 then return,-1
   return,I mod Na
end

function ind_int_HISTOGRAM, a, b
   minab = min(a, MAX=maxa) > min(b, MAX=maxb) 
   maxab = maxa < maxb
   ha = histogram(a, MIN=minab, MAX=maxab, reverse_indices=reva)
   hb = histogram(b, MIN=minab, MAX=maxab)
   r = where((ha ne 0) and (hb ne 0), cnt)
   if cnt eq 0 then return, -1
   return,reva[reva[r]]
end

A time comparison of the three methods, revealed the following for unique
integer vectors:

1. ARRAY is almost always slowest due to the large memory requirement.  Only for
very small input vectors (10 elements or so), will ARRAY beat SORT.  It is,
however, the only method which correctly identifies repeated entries (not used
in this test).

2. SORT is faster than HISTOGRAM for sets sparser than about 1 in 20, otherwise
histogram is faster, nearly independent of input vector size.  This will of
course depend most critically on the amount of memory you have.  

3. HISTOGRAM slows down rapidly with increasing sparseness (just more memory
required).

For fun, I also simulated the Social Security number test, with 1000x1000 number
distributed randomly between 0 and 999 99 9999:

SORT Method: 
Average Time ===>    0.0045424044
ARRAY Method: 
Average Time ===>      0.61496135
HISTOGRAM Method: 
% Array requires more memory than IDL can address.
% Execution halted at:  IND_INT3           27

As I noted, the histogram will of course fail on trying to allocate the huge
memory it needs for all those zeroes!  ARRAY only squeaked by, and increasing
the test to 10,000 SSN's yields:

SORT Method: 
Average Time ===>     0.059468949
ARRAY Method: 
% Unable to allocate memory: to make array.

and even the array method fails.  How high can sort go?  It was happy to do an
index intersection of 1 million x 1 million SSN's ( finding 41734 overlapping
indices) in 10 seconds.  

Just to be fair to ARRAY, if a and b are of very different sizes, and one of
them is quite small (say less than 5 elements or so), it can dominate over SORT
and HISTOGRAM.

So, if you want a generic solution which works in the same n log(n) time using a
fixed amount of memory for any type of integer input data, sparse or not, use
SORT.  If you know your data is non-sparse (better than 1 in 10 say), you can
see a speedup of a few with HISTOGRAM.   If you are looking for the points of
intersection in a huge array of a small set of values, you might consider
ARRAY.  If you want to do strings or floats or other types of data, you cannot
use HISTOGRAM.  And if you want information for repeated values in the same
input vector, you're stuck with ARRAY.

As always, your results will vary with individual machine and operating
systems.  Be sure to test using your data and computer if you need to optimize
for speed.

JD

-- 
 J.D. Smith                             |*|      WORK: (607) 255-5842    
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