# [perl-python] generic equivalence partition

Discussion in 'Python' started by Xah Lee, Feb 24, 2005.

1. ### Xah LeeGuest

another functional exercise with lists.

Here's the perl documentation. I'll post a perl and the translated
python version in 48 hours.

=pod

parti(aList, equalFunc)

given a list aList of n elements, we want to return a list that is a
range of numbers from 1 to n, partition by the predicate function of
equivalence equalFunc. (a predicate function is a function that
takes two arguments, and returns either True or False.)

Note: a mathematical aspect: there are certain mathematical constraints
on the a function that checks equivalence. That is to say, if a==b,
then b==a. If a==b and b==c, then a==c. And, a==a. If a equivalence
function does not satisfy these, it is inconsistent and basically give
meaningless result.

example:
parti([['x','x','x','1'],
['x','x','x','2'],
['x','x','x','2'],
['x','x','x','2'],
['x','x','x','3'],
['x','x','x','4'],
['x','x','x','5'],
['x','x','x','5']], sub {\$_[0]->[3] == \$_[1]->[3]} )

returns
[[1],['2','3','4'],['5'],['6'],['7','8']];

=cut

In the example given, the input list's elements are lists of 4
elements, and the equivalence function is one that returns True if the
last item are the same.

Note that this is a generic function. The input can be a list whose
elements are of any type. What "parti" does is to return a partitioned
range of numbers, that tells us which input element are equivalent to
which, according to the predicate given. For example, in the given
example, it tells us that the 2nd, 3rd, 4th elements are equivalent.
And they are equivalent measured by the predicate function given, which
basically tests if their last item are the same integer. (note that if
we want to view the result as indexes, then it is 1-based index. i.e.
counting starts at 1.)

PS if you didn't realize yet, nested lists/dictionaries in perl is a
complete pain in the ass.

PS note that the code "sub {\$_[0]->[3] == \$_[1]->[3]}" is what's called
the lambda form, in Perl.

Xah

http://xahlee.org/PageTwo_dir/more.html

Xah Lee, Feb 24, 2005

2. ### BryanGuest

Xah Lee wrote:
> another functional exercise with lists.
>
> Here's the perl documentation. I'll post a perl and the translated
> python version in 48 hours.
>
> =pod
>
> parti(aList, equalFunc)
>
> given a list aList of n elements, we want to return a list that is a
> range of numbers from 1 to n, partition by the predicate function of
> equivalence equalFunc. (a predicate function is a function that
> takes two arguments, and returns either True or False.)
>
> Note: a mathematical aspect: there are certain mathematical constraints
> on the a function that checks equivalence. That is to say, if a==b,
> then b==a. If a==b and b==c, then a==c. And, a==a. If a equivalence
> function does not satisfy these, it is inconsistent and basically give
> meaningless result.
>
> example:
> parti([['x','x','x','1'],
> ['x','x','x','2'],
> ['x','x','x','2'],
> ['x','x','x','2'],
> ['x','x','x','3'],
> ['x','x','x','4'],
> ['x','x','x','5'],
> ['x','x','x','5']], sub {\$_[0]->[3] == \$_[1]->[3]} )
>
> returns
> [[1],['2','3','4'],['5'],['6'],['7','8']];
>
> =cut
>
> In the example given, the input list's elements are lists of 4
> elements, and the equivalence function is one that returns True if the
> last item are the same.
>
> Note that this is a generic function. The input can be a list whose
> elements are of any type. What "parti" does is to return a partitioned
> range of numbers, that tells us which input element are equivalent to
> which, according to the predicate given. For example, in the given
> example, it tells us that the 2nd, 3rd, 4th elements are equivalent.
> And they are equivalent measured by the predicate function given, which
> basically tests if their last item are the same integer. (note that if
> we want to view the result as indexes, then it is 1-based index. i.e.
> counting starts at 1.)
>
> PS if you didn't realize yet, nested lists/dictionaries in perl is a
> complete pain in the ass.
>
> PS note that the code "sub {\$_[0]->[3] == \$_[1]->[3]}" is what's called
> the lambda form, in Perl.
>
> Xah
>
> http://xahlee.org/PageTwo_dir/more.html
>

this is the first thing that came to my mind. i'm sure there are more clever
ways to do this.

elements = [['x', 'x', 'x', '1'],
['x', 'x', 'x', '2'],
['x', 'x', 'x', '2'],
['x', 'x', 'x', '2'],
['x', 'x', 'x', '3'],
['x', 'x', 'x', '4'],
['x', 'x', 'x', '5'],
['x', 'x', 'x', '5']]
pos = {}

for i, element in enumerate(elements):
pos.setdefault(element[-1], []).append(i+1)

p = pos.values()
p.sort()
[[1], [2, 3, 4], [5], [6], [7, 8]]

bryan

Bryan, Feb 25, 2005

3. ### John MachinGuest

On Thu, 24 Feb 2005 17:48:47 -0800, Bryan <> wrote:

>Xah Lee wrote:
>> another functional exercise with lists.
>>
>> Here's the perl documentation. I'll post a perl and the translated
>> python version in 48 hours.
>>
>> =pod
>>
>> parti(aList, equalFunc)
>>
>> given a list aList of n elements, we want to return a list that is a
>> range of numbers from 1 to n, partition by the predicate function of
>> equivalence equalFunc. (a predicate function is a function that
>> takes two arguments, and returns either True or False.)

[snip]

>> example:
>> parti([['x','x','x','1'],
>> ['x','x','x','2'],

[snip]
>> ['x','x','x','5']], sub {\$_[0]->[3] == \$_[1]->[3]} )
>>
>> returns
>> [[1],['2','3','4'],['5'],['6'],['7','8']];
>>
>> =cut
>>
>> In the example given, the input list's elements are lists of 4
>> elements, and the equivalence function is one that returns True if the
>> last item are the same.

>

[snip]

>>

>
>this is the first thing that came to my mind. i'm sure there are more clever
>ways to do this.
>
>elements = [['x', 'x', 'x', '1'],

[snip]
> ['x', 'x', 'x', '5']]
>pos = {}
>
>for i, element in enumerate(elements):
> pos.setdefault(element[-1], []).append(i+1)
>
>p = pos.values()
>p.sort()
>[[1], [2, 3, 4], [5], [6], [7, 8]]
>
>

Bryan: Bzzzt. Xah was proposing a GENERAL function. You have HARDWIRED
his (simplistic) example.

Xah: Bzzzt. Too close to your previous exercise.

John Machin, Feb 25, 2005
4. ### David EppsteinGuest

In article <>,
"Xah Lee" <> wrote:

> parti(aList, equalFunc)
>
> given a list aList of n elements, we want to return a list that is a
> range of numbers from 1 to n, partition by the predicate function of
> equivalence equalFunc. (a predicate function is a function that
> takes two arguments, and returns either True or False.)

In Python it is much more natural to use ranges from 0 to n-1.
In the worst case, this is going to have to take quadratic time
(consider an equalFunc that always returns false) so we might as well do
something really simple rather than trying to be clever.

def parti(aList,equalFunc):
eqv = []
for i in range(len(aList)):
print i,eqv
for L in eqv:
if equalFunc(aList,aList[L[0]]):
L.append(i)
break;
else:
eqv.append()

If you really want the ranges to be 1 to n, add one to each number in
the returned list-of-lists.

--
David Eppstein
Computer Science Dept., Univ. of California, Irvine
http://www.ics.uci.edu/~eppstein/

David Eppstein, Feb 25, 2005
5. ### David EppsteinGuest

In article <>,
David Eppstein <> wrote:

> def parti(aList,equalFunc):
> eqv = []
> for i in range(len(aList)):
> print i,eqv
> for L in eqv:
> if equalFunc(aList,aList[L[0]]):
> L.append(i)
> break;
> else:
> eqv.append()

Um, take out the print, that was just there for me to debug my code.

--
David Eppstein
Computer Science Dept., Univ. of California, Irvine
http://www.ics.uci.edu/~eppstein/

David Eppstein, Feb 25, 2005
6. ### Michael SpencerGuest

David Eppstein wrote:
> In article <>,
> "Xah Lee" <> wrote:
>>given a list aList of n elements, we want to return a list that is a
>>range of numbers from 1 to n, partition by the predicate function of
>>equivalence equalFunc.

> In the worst case, this is going to have to take quadratic time
> (consider an equalFunc that always returns false) so we might as well do
> something really simple rather than trying to be clever.
>
> def parti(aList,equalFunc):
> eqv = []
> for i in range(len(aList)):
> print i,eqv
> for L in eqv:
> if equalFunc(aList,aList[L[0]]):
> L.append(i)
> break;
> else:
> eqv.append()
>
>

Unless we can inspect the predicate function and derive a hash function such
that hash(a) == hash(b) => predicate(a,b) is True. Then the partition can take
linear time
i.e.,
>>> def equal(a,b):

... return a[-1] == b[-1]
...
>>> def hashFunc(obj):

... return hash(obj[-1])
...
>>> def parti(aList, hashFunc):

... eqv = {}
... for i,obj in enumerate(aList):
... eqv.setdefault(hashFunc(obj),[]).append(i)
... return eqv.values()
...

In the case where the predicate is a "black box", would a logistic regression
over a sample of inputs enable a hash function to be derived experimentally?

Michael

Michael Spencer, Feb 25, 2005
7. ### Paul MooreGuest

David Eppstein <> writes:

> In article <>,
> "Xah Lee" <> wrote:
>
>> parti(aList, equalFunc)
>>
>> given a list aList of n elements, we want to return a list that is a
>> range of numbers from 1 to n, partition by the predicate function of
>> equivalence equalFunc. (a predicate function is a function that
>> takes two arguments, and returns either True or False.)

>
> In Python it is much more natural to use ranges from 0 to n-1.
> In the worst case, this is going to have to take quadratic time
> (consider an equalFunc that always returns false) so we might as well do
> something really simple rather than trying to be clever.

As you say, with the spec as it stands, you can't do better than
quadratic time (although it's O(n*m) where m is the number of
partitions, rather than O(n^2)).

You can do a lot better if you can use a "key" function, rather than
an "equivalence" function, much as list.sort has a "key" argument, and
itertools.groupby (which is pretty close in function to this
partitioning problem) uses a key argument.

In fact, I'd have difficulty thinking of an example where I'd want a
partition function as specified, in Python. In Perl, it makes a lot of
sense, as Perl's array indexing operations lend themselves to slinging
round lists of indices like this. But in Python, I'd be far more
likely to use list.sort followed by itertools.groupby - sort is stable
(so doesn't alter the relative order within equivalence classes), and
groupby then picks out the equivalence classes:

>>> elements = [['x', 'x', 'x', '1'],

.... ['x', 'x', 'x', '2'],
.... ['x', 'x', 'x', '2'],
.... ['x', 'x', 'x', '2'],
.... ['x', 'x', 'x', '3'],
.... ['x', 'x', 'x', '4'],
.... ['x', 'x', 'x', '5'],
.... ['x', 'x', 'x', '5']]

>>> # No need to sort here, as the elements are already sorted!

>>> from pprint import pprint
>>> pprint([(k, list(v)) for k, v in groupby(elements, itemgetter(3))])

[('1', [['x', 'x', 'x', '1']]),
('2', [['x', 'x', 'x', '2'], ['x', 'x', 'x', '2'], ['x', 'x', 'x', '2']]),
('3', [['x', 'x', 'x', '3']]),
('4', [['x', 'x', 'x', '4']]),
('5', [['x', 'x', 'x', '5'], ['x', 'x', 'x', '5']])]

If you avoid the sort, the whole thing is highly memory efficient, as
well, because by using iterators, we don't ever take a copy of the
original list.

Having cleverly redefined the question so that it fits the answer I
wanted to give, I'll shut up now

Paul.
--
To attain knowledge, add things every day; to attain wisdom, remove
things every day. -- Lao-Tse

Paul Moore, Feb 25, 2005
8. ### Xah LeeGuest

Re: generic equivalence partition

# the following solution is submitted by
# Sean Gugler and David Eppstein independently
# 20050224.

@def parti(aList, equalFunc):
@ result = []
@ for i in range(len(aList)):
@ for s in result:
@ if equalFunc( aList, aList[s[0]] ):
@ s.append(i)
@ break
@ else:
@ result.append( )
@ return [[x+1 for x in L] for L in result] # add 1 to all numbers
@
@---------------

as for my original perl code, i realized it is written to work on a
sorted input. Here it is and the translated Python code.

# perl
sub parti(\$\$) {
my @li = @{\$_[0]};
my \$sameQ = \$_[1];

my @tray=(1);
my @result;

for (my \$i=1; \$i <= ((scalar @li)-1); \$i++) {
if (&\$sameQ(\$li[\$i-1], \$li[\$i])) {
push @tray, \$i+1}
else {
push @result, [@tray]; @tray=(\$i+1);
}
}
push @result, [@tray];
return \@result;
}

@#python
@def parti(li,sameQ):
@ tray=[1];
@ result=[];
@
@ for i in range(1, len(li) ):
@ if sameQ(li[i-1],li):
@ tray.append(i+1)
@ else:
@ result.append(tray)
@ tray=[i+1]
@ result.append(tray)
@ return result
@

http://xahlee.org/perl-python/gen_parti_by_equiv.html

Xah

http://xahlee.org/PageTwo_dir/more.html

Xah Lee, Feb 26, 2005
9. ### Xah LeeGuest

Re: generic equivalence partition

folks:

when using google to post a reply, it sometimes truncates the subject
line. i.e. [perl-python] is lost. This software error is obvious, they
could not have not noticed it.

another thing more egregious is that google _intentionally_ edit with
people's posts. (e.g. they change email address lines without author's
permission, and they also change program codes so it no longer run).
software responsibility and online forum issues.

Ostensible incorrect behavior like these by google is egregious enough
to generate a law suit and if such company do not take software
correctness seriously, we must punish them.

Xah

http://xahlee.org/PageTwo_dir/more.html

Xah Lee, Feb 26, 2005
10. ### Xah LeeGuest

Re: generic equivalence partition

People,

.... sorry for the latching on on this broadside issue, but it is
impotant ...

here's are some germane points from another online discussion:

the bug-reporting issue has came up so many times by so many people i
thought i'd make a comment of my view.

when a software is ostensibly incorrect, and if it is likely in
connection to egregious irresponsibility as most software companies are
thru their irresponsible licensing, the thing one should not do is to
fawn up to their ass as in filing a bug report, and that is also the
least effective in correcting the software.

the common attitude of bug-reporting is one reason that contributed to
the tremendous egregious irresponsible fuckups in computer software
industry that each of us have to endure daily all the time. (e.g.
software A clashed, software B can't do this, C can't do that, D i
don't know how to use, E download location currently broken, F i need
to join discussion group to find a work-around, G is all pretty and
dysfunctional... )

when a software is ostensibly incorrect and when the company is
irresponsible with their licensing, the most effective and moral
attitude is to do legal harm to the legal entity. This one an do by
filing a law suit or spreading the fact. Filing a law suit is
appropriate in severe and serious cases, and provided you have such
devotion to the cause. For most cases, we should just spread the fact.
When a company see facts flying about their incompetence or
irresponsibility, they will immediately mend the problem source, or
cease to exist.

Another harm sprang from the fucking bug-reporting attitude rampant
among IT morons is the multiplication of pop-ups that bug users for
bug-reporting, complete with their privacy intrusion legalese.

Xah

http://xahlee.org/PageTwo_dir/more.html

Xah Lee wrote:
> folks:
>
> when using google to post a reply, it sometimes truncates the subject
> line. i.e. [perl-python] is lost. This software error is obvious,

they
> could not have not noticed it.
>
> another thing more egregious is that google _intentionally_ edit with
> people's posts. (e.g. they change email address lines without

author's
> permission, and they also change program codes so it no longer run).
> software responsibility and online forum issues.
>
> Ostensible incorrect behavior like these by google is egregious

enough
> to generate a law suit and if such company do not take software
> correctness seriously, we must punish them.
>
>
> Xah
>
> http://xahlee.org/PageTwo_dir/more.html

Xah Lee, Feb 27, 2005
11. ### Erik Max FrancisGuest

Re: generic equivalence partition

Xah Lee wrote:

> ... sorry for the latching on on this broadside issue, but it is
> impotant ...

You made a typo in that last word there. Obviously you meant to write
an _e_ instead of an _a_.

--
Erik Max Francis && && http://www.alcyone.com/max/
San Jose, CA, USA && 37 20 N 121 53 W && AIM erikmaxfrancis
All bad poetry springs from genuine feeling.
-- Oscar Wilde

Erik Max Francis, Feb 27, 2005