# Splitting the Loot (#65)

There was some debate on the list about which NP-complete problem this
actually
is. I personally thought it was a variant of the partitioning problem
when I
put it together, but others made good cases for similar problems. In
the end,
two things are important: it’s difficult and tricky to get right.

Some people thought you could work with the treasures largest to
smallest to
save time. A few edge cases were pointed out for this approach though.
The
easiest example to understand being this one posted by Avi Bryant:

``````\$ ruby loot.rb 3 3 3 3 2 2 2 2 2 2 2 2 2
1: 3 2 2 2
2: 3 2 2 2
3: 3 2 2 2
``````

Here we are looking for totals of nine. If we work only with the big
values, we
will find 3 3 3 as an option, but then it is impossible to break the
remaining
even numbers into totals of nine. For this reason, you must consider
all of the
possibilities.

Manuel K. posted many of the edge cases that solutions had trouble
with, in
addition to the first solution that seems to be able to find them all.
Let’s
look at that code now, from the bottom up:

``````# ...

if \$0 == __FILE__
pirates = ARGV.shift.to_i
treasure = ARGV.map{ |gem| gem.to_i }.sort
si = SplitIt.new(pirates, treasure)
si.go
si.done
end
``````

That’s the first bit of code run when Manuel’s program is launched. We
can see
that it reads and converts arguments, builds some SplitIt object with
them, and
finally calls SplitIt#go and SplitIt#done.

Time to dig into SplitIt:

``````class SplitIt
def initialize pirates, treasure
@pirates = pirates
@treasure = treasure
@bags = []
(0...@pirates).each{ |pirate| @bags[pirate] = [[], 0] }
loot = @treasure.inject(0){ |res, gem| res + gem }
done unless loot % @pirates == 0
@share = loot/@pirates
end

# ...
``````

The only remotely tricky thing in here is the creation of @bags. Each
pirate is
given a two-element Array. The first element is another Array, which
will be
filled with the gems placed in their share. The second element is just
the
total of their share thus far.

Also note that this method may immediately force a call to SplitIt#done,
if the
treasure does not evenly divide. Let’s skip down to done now:

``````  # ...

def done
puts
if (@treasure.length == 0)
@bags.each_with_index do |bag, pirate|
puts "#{pirate+1}: #{bag[0].sort.inspect}"
end
else
puts "The #{@pirates} pirates won't be able to " +
"split their loot fairly. Take cover!"
end
exit
end
end
``````

This method just shows the final results. If all the treasure is gone,
we split
it up correctly and the shares are printed. Otherwise, we give the
error
message. Either way Kernel#exit is called, because we’re all finished.

That leaves only one method to examine:

``````  # ...

def go
done if @treasure.length == 0
gem = @treasure.pop
(0...@pirates).each do |pirate|
if @bags[pirate][1] + gem <= @share
@bags[pirate][1] += gem
@bags[pirate][0].push gem
go
@bags[pirate][0].pop
@bags[pirate][1] -= gem
# it doesn't matter which pirate is which,
# as long as their bags are empty
break if @bags[pirate][1] == 0
end
end
@treasure.push gem
end

# ...
``````

This method does all the work, with a brute-force recursive search. A
gem is
pulled off the pile here and tried in each pirate’s bag. After it is
placed in
a bag, the method recurses to try the remaining treasures.

If a recursive call returns, there was no split found (because
SplitIt#done
would have been called if there was). Since that is the case, the
treasure is
pulled back out of the bags and replaced on the pile.

The break line is just a minor optimization. If a treasure didn’t work
in one
pirate’s empty bag, it won’t work in any of the empty bags following his
and we
can skip those checks.

Since that is an exhaustive search, it will find a correct split
eventually, as
long as one exists. Let’s look at another variation of the same
technique, this
time by Simon Kroeger:

``````require 'set'

def choose_bags nr, bags, choice = Set[]
return [] if choice.size == nr
bags.each_with_index do |b, i|
c = (choice & b).empty? && choose_bags(nr, bags, choice | b)
return [i] + c if c
end && nil
end

def split_loot  nr, *treasures
each = (sum = treasures.sort!.reverse!.inject{|s, t| s + t}) / nr
return nil if (sum % nr).nonzero?

piles = Hash.new([]).merge!({0 => [[]]})
treasures.each_with_index do |t, i|
piles.dup.each do |k, v|
if k + t <= each && k + sum >= each
v.each{|a| piles[k + t] += [a + [i]]}
end
end
sum -= t
end
return nil if piles[each].empty?
return nil if !bags = choose_bags(treasures.size, piles[each])

piles[each].values_at(*bags).map{|b| b.map{|t| treasures[t]}}
end

loot = split_loot(*ARGV.map{|p| p.to_i})
puts(loot ? loot.map{|a| a.join(' ')} : 'impossible!')
``````

Here again, we see the treasures are converted from the command-line
arguments
and this time they are sent to #split_loot. The first line of
#split_loot sums
the treasures, and places the share total needed in a variable called
each. The
second line just cancels the search if the total sum is not easily
divided.

The next bit of Hash manipulation is the magic here. Each treasure is
pulled
off the pile and combined with all other totals as long as the resulting
total
will be less than or equal to the share goal and it will leave us enough
treasures to reach that goal.

How they are stored in the Hash is interesting. The keys are sums of
shares
found thus far. The values for those sums are an Array of the shares
that can
make that sum (another Array). The individual items in the shares are
indexes
into the treasure pile, instead of the treasures themselves. For
example, if I
feed Simon’s program the problem 2 3 2 1, the piles Hash ends up as:

``````{ 0 => [ []          ],
2 => [ [1]         ],
3 => [ [0], [1, 2] ] }
``````

The 0 empty set split is what the code primes the Hash with, so it has
something
to add to. The other two are totals it calculated. Higher totals
weren’t
figured because they aren’t needed. And remember 0, 1, and 2 inside
those sets
refer to indexes of treasures.

I want to mention one more gotcha in this code, before we move on. Note
that
the default Hash object is set to an Array. Usually you want to avoid
doing
that without the block form of Hash#new, because you will get the same
Array
each time. However, whenever this code calls it up, it is to add it to
another
Array, which returns a third Array (the combined results) that actually
gets
stored in the Hash.

The rest of #split_loot makes sure a solution is still possible, tries
to divide
it out with a call to #choose_bags, and finally translates all those
treasure
indexes back to actual gems. (Note that Simon’s use of these unique
indexes,
saved him the trouble of pulling non-unique elements from an Array
one-at-a-time
that was a discussion topic spawned by this problem.)

That leaves the only mystery as #choose_bags. Here’s another look at
that code:

``````require 'set'

def choose_bags nr, bags, choice = Set[]
return [] if choice.size == nr
bags.each_with_index do |b, i|
c = (choice & b).empty? && choose_bags(nr, bags, choice | b)
return [i] + c if c
end && nil
end

# ...
``````

This method obviously uses the Set library to do its work. Here each
bag is
tried one by one and checked that it doesn’t contain any already-used
elements.
That check is accomplished with a set intersection (&) test. If clear,
the
method recurses to find the other bags, after performing a set union (|)
with
the selected bag and all bags chosen thus far.

When the first line of the method informs us that we have used all of
the
treasures, we can start passing the results back up the callstack. The
result
set starts as an empty Array, but each level adds in the bag used, again
by
index of the bag. If we take one more peek at the final line of
#split_loot, we
will see that those bag indexes are resolved with a call to
Array#values_at,
just before the results are returned:

``````  # ...

piles[each].values_at(*bags).map{|b| b.map{|t| treasures[t]}}
end

# ...
``````

Finally, one last trick is worth mentioning in #choose_bags. There is a
funny
… && nil on the last line. The reason is that
Enumerable#each_with_index
always returns a true value, but this method needs a false result if a
bag could
not be added. You can get that by &&ing any true value with nil, or
Simon could
have just returned a nil at the end of the method for identical results.

As usual, all solutions were educational. My thanks to all for the code
and
excellent discussion this time around.

Tomorrow we will begin a run of at least three non-NP-complete problems
by
myself and others…

Ruby Q. schrieb:

(…)

(my go method)

The break line is just a minor optimization. If a treasure didn’t
work in one pirate’s empty bag, it won’t work in any of the empty
bags following his and we can skip those checks.

You may call it a minor optimization, and I agree it’s an easy one. But
it speeds up the whole process around factor 100, so without it my
simple brute force approach would have been intolerable slow.

As usual, all solutions were educational. My thanks to all for the
code and excellent discussion this time around.

I want to mention Luke Blanchard’s second solution. It is the fastest
solution of those I tested that worked without flaws. I never ever saw
it running longer than 0.2 seconds and I did test it a lot (because it
is around factor 10-20 faster than the next correct solution, and I
thought it must fail sometimes, but it never did). I couldn’t believe
its speed. I studied his code for ca. 30 min just to understand what
it’s doing (luckily it’s got a lot of comments, I wouldn’t have been
able to understand it otherwise). Then I could tell it is indeed
correct, but its speed is nevertheless amazing.

Another interesting solution is the one from 0x002A. It is quite slow,
but if you’ve never seen anything with lazy evaluation (“require
‘lazylist’”) you should check it. The concept was new to me and gave me
quite a headache analyzing it (the good kind of headache - I love when
the fog raises and I suddenly understand what’s going on).

Tomorrow we will begin a run of at least three non-NP-complete
problems by myself and others…

I see forward to it. I’m not home for the weekend, but I hope to have a
go at it on Monday.

``Manuel``

On Feb 9, 2006, at 5:23 PM, Manuel K. wrote:

Another interesting solution is the one from 0x002A. It is quite slow,
but if you’ve never seen anything with lazy evaluation (“require
‘lazylist’”) you should check it.

I agree. I almost discussed this one, but it had a few issues, like
the speed you mentioned. I’m always limited by my time, otherwise I
would surely write-up all the solutions each week.

James Edward G. II

Manuel K. wrote:

Thanks, Manuel. Since it was 4:00 AM when I wrote the email wrapping
that guy – I finally got up to code the thing at 3 after lying awake
all night – I think my explanation was a little terse if not downright

The key idea is that the easiest way to manage searching all possible
combinations of something is with loops and recursion, meaning you just
step through all combinations right inline. Most solvers of NP-complete
problem spaces end up spending a lot of effort setting up data
structures to keep track of where they are in the process. My approach
was to just use local variables and the stack, combined with Ruby’s
magical ability to long-jump out of a deep stack by just returning from
a block (which I first used some 15 years ago in Smalltalk). No extra
bookkeeping required.

The “pick” function just walks all combinations recursively, and when it
finds one it yields a list of the indexes that make up the successful
combination. Since it is recursive, it ends up yielding to itself up
the chain. To take the 3-3s-and-9-2s example, we’ll have an execution
setup like this, where further down the page corresponds to further down
the stack and further to the right corresponds to further nesting:

values = [3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2]
pick values, 9, 0:
pick values, 6, 1:
pick values, 3, 2:
yield [2]
yield [2, 1]
yield [2, 1, 0]

The [2, 1, 0] yielded by the outermost call corresponds to the indexes
of the 3 3s. The caller of that outermost call is “find_split”, and the
way it handles this list of indexes is to dup the array of values,
delete the indices yielded from the dup, and then recurse itself on the
shorter array. In this case, it’s trying to divide 9 2s in half, and
that will fail, so it just returns. This means the original call to
pick continues, meaning all of those yields finish, and we’re nested
three levels deep again. The innermost one ends up recursing again
without success, and then the second one moves on to the 2s and recurses
a couple of times to get our next solution:

values = [3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2]
pick values, 9, 0:
pick values, 6, 1:
pick values, 4, 4:
pick values, 2, 5:
yield [5]
yield [5, 4]
yield [5, 4, 3]
yield [5, 4, 3, 0]

This time find_split’s recursion succeeds (on 2 3s and 6 2s), and so it
returns the solution, jumping out of the above stack of picks without
ever continuing them.

Isn’t that cool?

Luke B.

On Feb 9, 2006, at 7:26 PM, Luke B. wrote:

Isn’t that cool?

It sure is and that was a great explanation of just how cool it is.
I’ll be the one loosing sleep tonight, trying to wrap my head around
all that and where I can use it…

James Edward G. II

Aha. You need to be using my second solution, cleverly named
“loot2.rb”. Though actually, when I tested them with your example, they
both worked. Can you give more detail?

Luke

One group of combinations I found that Luke’s solution didn’t work
for was when the loot included a single value to be split to one
partner such as:

ruby loot.rb 3 5 4 4 1 1

It will work if you expand the line in the pick function reading:

i += 1 while i < values.size && values[i] >= sum

to:

if values[lo] == sum
i += 1
else
i += 1 while i < values.size && values[i] >= sum
end

I’m not sure how this changed the speed, but for all solvable
combinations of 3 partners with a sum of 9 Luke’s ran in 80 seconds,
and Manuel’s ran in 105 seconds.

Bill

Sorry. Wrong example. It won’t split for things including 2 values
equalling the maximum split amount like:

loot.rb 3 5 5 5
loot.rb 3 4 1 5 5
loot.rb 3 3 2 5 5
loot.rb 3 3 1 1 5 5
loot.rb 3 2 3 5 5
loot.rb 3 2 2 1 5 5
loot.rb 3 2 1 1 1 5 5
loot.rb 3 1 1 1 1 1 5 5
loot.rb 3 5 4 1 5
loot.rb 3 5 3 2 5
loot.rb 3 5 3 1 1 5
loot.rb 3 5 2 3 5
loot.rb 3 5 2 2 1 5
loot.rb 3 5 2 1 1 1 5
loot.rb 3 5 1 1 1 1 1 5
loot.rb 3 5 5 4 1
loot.rb 3 5 5 3 2
loot.rb 3 5 5 3 1 1
loot.rb 3 5 5 2 3
loot.rb 3 5 5 2 2 1
loot.rb 3 5 5 2 1 1 1
loot.rb 3 5 5 1 1 1 1 1

Bill

Wow, right you are. I guess this 3AM coding thing has its down side.