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This is a friggin' awesome problem. Lots of fun working on it. :-) But I think a solution is at hand.

My previous solution assumed that there was some sort of external criteria that would determine whether to skip generation of the powerset. Extensive private conversation with Limbic~Region plus his re-statement of the problem made it clear that that's not the case. Existence is the condition AND we need to worry about multiple sets and god knows what else. This will take time to explain.

I will run through some theory an explanation, then post the modified code, then further explain that.

The issue that other people were having was placement of the characters in the set. So if you start off with set ABC, you'll know to ignore AB. However, you need to keep track of "AB" somewhere in memory and then do a string comparison or the like. Expensive.

My original approach simply used set enumeration to flag sets as being valid or dupes or whatnot. But that wont work here. With the example set ABC, you'd flag BC (idx 6 by how my function counts). BCD comes along, but its idx 6 is CD. So you'd improperly skip over CD. Further, its BC is idx 3, which you'd duplicate. Not good.

We're going to fix this by using placeholders in the set by means of dead bits. This is far cheaper than you would initially think.

```
For set A B C, you can enumerate the powerset in binary:

111 ABC
110 AB
101 A C
100 A
011  BC
010  B
001   C
000

For BCD, it would be thus:

111 BCD
110 BC
101 B C
100 B
011  CD
010  C
001   D
000

Instead, we want to offset the values in BCD and leave an empty slot f
+or A. We'll end up with this:

0111 BCD
0110 BC
0101 B C
0100 B
0011  CD
0010  C
0001   D
0000

BC is 011 in our original set, and hey, in our newly modified set with the dead bit at position 0, BC is once again 011. So now we can safely exclude this set.

Note - this does not require the elements to be in sorted order, or even for you to know them in advance. We'll use an LZW style approach to keep assigning slots as we see them.

```
(for ABC)
Have we seen A before? No? A = slot 0
Have we seen B before? No? B = slot 1
Have we seen C before? No? C = slot 2

(for BCD)
Have we seen B before? YES - return 1
Have we seen C before? YES - return 2
Have we seen D before? No? D == slot 3

The order is irrelevent, but it does require that we build up an additional hash table mapping elements -> slots. Memory intensive, but relatively cheap - we only need to look into this index at the start of processing on a set.

But wait! You say, I glossed over a whole set of values up at the top! Let's return to them:

```
By using a dead bit for A, we actually have this:

1111 invalid
1110 invalid
1101 invalid
1100 invalid
1011 invalid
1010 invalid
1001 invalid
1000 invalid
0111 BCD
0110 BC
0101 B C
0100 B
0011  CD
0010  C
0001   D
0000

That's hugely inefficient. That's 8 checks we need to do just to get to a valid set. It spirals horribly out of control as you add more dead bits. Say, for example, you were using all letters of the alphabet. Your last set is just (z). By that point, z is in slot 25. So you'd have:

```
11111111111111111111111111  invalid
11111111111111111111111110  invalid
11111111111111111111111101  invalid
.
.
.
00000000000000000000000010  invalid
00000000000000000000000001  valid (z)

Thats 67,108,864 iterations through invalid sets before you find your one valid one. Fortunately, an xor can cause us to solve this in constant time.

Examples are in order:

```
From up above, set (z)

start at         11111111111111111111111111

this is > 0, so xor.
(start ^ deadbit) & start = 00000000000000000000000001

Bang! one and, a conditional, and an xor jumped you past 67 million so
+me odd invalid sets.

Another one, from above, set (ABCD)

start at         1111
this is > 0, so xor
(start ^ deadbit) & start = 0111    (starts at BCD

Here's a more complex example. Assume our first set was ABC, and our n
+ext was ABD. (A = 0, B = 1, C = 2, D = 3). When we hit ABD, we flag C

Let's try a complete runthrough:

1111
1111 & 0010 > 0 -> (1111 ^ 0010) & 1111 -> new index is 1101
1101  ABD    (1101 & 0010 == 0)
1100  AB    (1100 & 0010 == 0)
1011        (1011 & 0010 == 0)
1011 & 0010 > 0 -> (1011 ^ 0010) & 1011 -> new index is 1001
1001  A D    (1001 & 0010 == 0)
1000  A        (1000 & 0010 == 0)
0111
0111 & 0010 > 0 -> (0111 ^ 0010) & 0111 -> new index is 0101
0101  B D    (0101 & 0010 == 0)
0100  B        (0100 & 0010 == 0)
0011
0011 & 0010 > 0 -> (0011 ^ 0010) & 0010 -> new index is 0001
0001    D    (0001 & 0010 == 0)
0000 ()        (0000 & 0010 == 0)

The code must be modified to incorporate the deadbits, as well as given the ability to return and re-use our previously skipped indexes. Modifications are as follows. There's a fair bit of additional set up an logic required vs. the previous version, but it can spit out all the numbers as desired from node 580625 with only 39 calls to the iterator and by storing only 32 integers in memory. Most of the additional work and overhead are bitwise operations.

```
use strict;
use warnings; # returns _3_ closures to generate certain powersets

#arbitrary benchmark device, used to se how many times the iterator wa
+s called.
my \$calls = 0;

sub limbic_power_generator {
my \$set = shift;
# we re-define skippers as an array and allow the user to pass it in
+, so we
# can keep track of previously skipped values. The value of the hash
+ was
# that it prevents dupes. Note - it would be better to mod the old v
+ersion
# to always use a descendingly sorted array, but is left to the read
+er.
my \$skippers = shift || {};
my \$deadbits = shift || 0;

#we start with the original set and count down to the null set
my \$set_idx = 2 ** @\$set;

#these are the set indexes we should skip
my %skippers = ();

# our first closure generates subsets
my \$generator = sub {

# arbitrary benchmark device, that way you can see how may times t
+he iterator
# was called
\$calls++;
#bow out if we're out of sets
return () unless \$set_idx;

# Start decrementing the set_idx. We always do this at least once,
+ so
# we get to the next set. Our original number is 2 ** @set, so we
+start
# at 2 ** @set - 1 after the decrement
while (1) {
\$set_idx--;

# check to see if this set contains a deadbit, and if so hop ove
+r it.
# make sure that we don't accidentally jump up to a higher set
+ index.
# this can happen if you have deadbits beyond the length of yo
+ur set
\$set_idx = (\$set_idx ^ \$deadbits) & \$set_idx;
}

#bow out if this set is NOT a subset of any set we're skipping
last unless \$skippers->{\$set_idx};
#bow out of the function completely with the null set if we've h
+it 0.
return () unless \$set_idx;
}

# Now we convert our set_idx to binary. Each bit stores whether th
+e element
# is in this subset. For example, set_idx 11 would be 1011, so we
+keep
# elements 0, 2, and 3.
my @in_set = split //, unpack("b*", pack("V",\$set_idx));

# now we return a list. The first element is an arrayref which is th
+e actual
# subset we generated, the second is our set_idx.
return ([map { \$set->[\$_] } grep { \$in_set[\$_] } (0..\$#\$set)], \$se
+t_idx);
};

# our second closure allows you to add sets to skip
# it also returns the list of skipped values
my \$skipper = sub {
if (@_) {
my \$skip_key = shift;
\$skippers->{\$skip_key}++;
}
return \$skippers;
};

# return both of our closures.
return (\$generator, \$skipper)
}

# we'll use the example sets from node 580625
my \$limbic_sets = [
[qw(A B C)],
[qw(A B D)],
[qw(A B)],
[qw(B C)],
[qw(E)],
[qw(A B C E)],
[qw(A B C D E)],
];

# our index lookup hash. There are potential savings by pre-caching th
+ese values
# if all elements are known in advance.
my %idx = ();
my \$next_open_idx = 0;

# our sets to skip
my %skippers = ();

foreach my \$limbic_set (@\$limbic_sets) {

print "checks set @\$limbic_set\n";

# we need to keep track of which indexes are dead, so we copy the
# known indexes

# here we'll keep track of the bits that are dead

# we now need to iterate over our set. If we know the index of tha
+t element
# then great. That means we've seen it before, and it's currently
+live, so
# delete it from our list of dead bits.
#
# otherwise, assign it a new index.
foreach my \$elem (@\$limbic_set) {
if (defined \$idx{\$elem}) {
}
else {
\$idx{\$elem} = \$next_open_idx++;
}
}

#here we're going to store the indexes which are dead

# iterate over our dead elements list, and toss it into the deadbi
+ts string
# and add its index to the lookup
foreach my \$idx (values %dead_idx) {
}

# we need to pad out set with dead bits. So if we call with (ABC),
+ then later
# with (ABD), we need to turn that into (AB D)

foreach my \$idx (0..\$#\$limbic_set) {
# if that index is dead, then toss in a placeholder and shift
+the array
# element forward. This is using parallel indexes, there may b
+e a more
# efficient method.
redo;
}
+x];
}

my (\$limbic_iterator, \$limbic_skipper) = limbic_power_generator(\$p

#as we see an eleemnt, we're going to add it to this list, so we s
+kip it on the next pass.
my %future_skippers = ();

#and start cruising over our powersets.
while ( my (\$set, \$idx) = \$limbic_iterator->() ) {
#fancy crap to get it to print out properly.
my \$display = {map {\$_ => 1} grep {defined} @\$set};
my \$format = "%2s" x scalar(@\$padded_limbic_set)  . " (%d)\n";
printf(\$format, (map {defined \$_ && \$display->{\$_} ? \$_ : ' '} @

#we don't skip anything in this pass, but we'll do it the next t
+ime around.
\$future_skippers{\$idx}++;
}

@skippers{keys %future_skippers} = values %future_skippers;

}

print "TOTAL CALLS \$calls\n";

Update: Fixed a minor logic glitch that could cause dupes with specially crafted sets. This code should now be complete.

Update 2: Changed to pack as a little endian long (V) instead of just a long (L) to make cross platform compatible.

Update 3: Okay, I'm man enough to admit that all of my slick binary logic actually introduced a horrrible inefficiency (looping to see if a set should be skipped). Fixed with a hash lookup instead. Duh.

Update4: I think I've squeezed as much raw power out of this as I'm going to get. The newest version here requires the set elements be known in advance and it doen't print out the null set. Otherwise? It does work. Comparing against Limbic~Region's code below, mine is indeed still slower. Dang.

Current state of the art. Expects the data file to be passed on the command line:

```
#!/usr/bin/perl

use strict;
use warnings;

sub set_generator {
my (\$set, \$set_idx, \$skippers, \$deadbits) = @_;

# Start decrementing the set_idx. We always do this at least once,
+ so
# we get to the next set. Our original number is 2 ** @set, so we
+start
# at 2 ** @set - 1 after the decrement
while (\$set_idx--) {

# check to see if this set contains a deadbit, and if so hop ove
+r it.
# make sure that we don't accidentally jump up to a higher set i
+ndex.
# this can happen if you have deadbits beyond the length of your
+ set
\$set_idx = (\$set_idx ^ \$deadbits) & \$set_idx;

#bow out if this set is NOT a subset of any set we're skipping
last unless \$skippers->{\$set_idx};
#bow out of the function completely with the null set if we've h
+it 0.
}

#bow out if we're out of sets
return () unless \$set_idx;

# Now we convert our set_idx to binary. Each bit stores whether th
+e element
# is in this subset. For example, set_idx 11 would be 1011, so we
+keep
# elements 0, 2, and 3.
my @in_set = split //, unpack("b*", pack("L",\$set_idx));

# now we return a list. The first element is an arrayref which is th
+e actual
# subset we generated, the second is our set_idx.
return ([map { \$set->[\$_] } grep { \$in_set[\$_] } (0..\$#\$set)], \$se
+t_idx);
};

# our index lookup hash. Assume letters 'a'-'z'. The original version
+of
# dynamically assigning indexes had a few bugs. Whoops. So now you're
+required
# to know all elements in advance. Damn. I'll fix this later, I guess.
my @letters = ('a'..'z');
my %idx = map {\$letters[\$_] => \$_} (0..25);

# our sets to skip
my %skippers = ();

while (my \$limbic_string = <>) {
chomp \$limbic_string;
my \$limbic_set = [split //, \$limbic_string];

# we need to keep track of which indexes are dead, so we copy the
# known indexes

# here we'll keep track of the bits that are dead

#remove the live bits from the dead set.

# iterate over our dead elements list, and toss it into the deadbi
+ts string
# and splice in an undef to our list.
foreach my \$idx (sort {\$a <=> \$b} values %dead_idx) {
splice @\$limbic_set, \$idx, 0, undef unless \$idx >= @\$limbic_se
+t;
}

#as we see an element, we're going to add it to this list, so we s
+kip it on the next pass.
my @future_skippers = ();

my \$set_idx = 2 ** @\$limbic_set;

#and start cruising over our powersets.
while (my (\$set, \$newidx) = set_generator(\$limbic_set, \$set_idx, \
print @\$set,"\n";

#note our new set index
\$set_idx = \$newidx;
#we don't skip anything in this pass, but we'll do it the next t
+ime around.
push @future_skippers, \$set_idx;
}

@skippers{@future_skippers} = (1) x @future_skippers;
}

In reply to Re^3: Powerset short-circuit optimization by jimt
in thread Powerset short-circuit optimization by Limbic~Region

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