You have 4 bags, each full of the numbers 1, 3, 5, and 7 respectively. Take 10 of the numbers that when added up make 37. What numbers are they?

So far so good: that sounds easy enough. But a bit of trial and error quickly leads nowhere. Something can’t be right. So, let’s get the computer to work it out.

As I haven’t done much Erlang recently I thought I’d give it a go. And, during a casual glance at Armstrong’s Programming in Erlang I thought I’d finally understood list comprehensions, so I wrote the following program:

-module(comb).
-export([result/0]).
result() ->
[{A+B+C+D+E+F+G+H+I+J,A,B,C,D,E,F,G,H,I,J}||
A <- [1,3,5,7],
B <- [1,3,5,7],
C <- [1,3,5,7],
D <- [1,3,5,7],
E <- [1,3,5,7],
F <- [1,3,5,7],
G <- [1,3,5,7],
H <- [1,3,5,7],
I <- [1,3,5,7],
J <- [1,3,5,7],
A+B+C+D+E+F+G+H+I+J =:= 37].

I declare a module with one function, result/0. This finds me ten variables that can take any of the four specified values and add up to 37. Simples!

The list comprehension has ten generators, and one filter; it will return a tuple with the sum and the individual variables’ values.

Erlang R16B01 (erts-5.10.2) [64-bit] [smp:4:4] [async-threads:10] [hipe] [kernel-poll:false]
Eshell V5.10.2 (abort with ^G)
1> comb:result().
[]
2>

WTF???! An empty list?! So I try changing the 37 to another value, like 36.

3> comb:result().
[{36,1,1,1,1,1,3,7,7,7,7},
{36,1,1,1,1,1,5,5,7,7,7},
{36,1,1,1,1,1,5,7,5,7,7},
{36,1,1,1,1,1,5,7,7,5,7},
{36,1,1,1,1,1,5,7,7,7,5},
[etc, etc].

So it does work! Only, there doesn’t seem to be an answer to the question. And with a bit of logical reasoning it is obvious: when adding two odd numbers, you get an even number. So adding ten odd numbers also yields an even number, but 37 is odd.

What I learnt from this exercise: thinking about the problem beforehand can save you time, as there was no need to write a program at all. But then, I did get to use list comprehensions, and have learnt how powerful they are. And it neatly shows Erlang’s Prolog roots as well.

1> lists:seq(1,4).
[1,2,3,4]
2> lists:seq(634,637).    
[634,635,636,637]
3> lists:seq(1000,1003).
[1000,1001,1002,1003]

Now I would simply generate a list with a number for each row, and then send the inner loop off to do its thing, filling the slot given by the sequence number with a zero, and others with a random value.

But now it gets even better. Using a separate (tail-)recursive function for the inner loop didn’t quite seem right, so I thought a bit more about it and came to the conclusion that this is simply a mapping; mapping an integer to a list (a vector of numbers, one of which (given by the integer) is a zero). So instead of using a function for filling the row, I call lists:seq/2 again and then map the whole thing. This is the final version I arrived at, and I’m sure it can still be improved upon using list comprehensions:

random_matrix(Size, MaxVal) ->
  random:seed(),
  lists:map(
    fun(X) ->
      lists:map(
          fun(Y) ->
              case Y of 
                 X -> 0; 
                 _ -> random:uniform(MaxVal)
                 end
              end,
          lists:seq(1,Size))
      end,
    lists:seq(1,Size)).

This solution seems to be far more idiomatic, and I am beginning to think that I finally no longer think in an imperative way of loops, but more in the Erlang-way of list operations. Initially this is hard to achieve, but with any luck it will become a lot easier once one is used to it. Elegance, here I come!

Example run:

4> random_matrix(6,7).  
[[0,1,4,6,7,4],
 [3,0,5,7,5,4],
 [5,1,0,2,5,2],
 [4,2,4,0,3,1],
 [4,4,3,3,0,1],
 [5,7,3,2,2,0]]

Note: I have used random:seed/0 above, as I am happy for the function to return identical matrices on subsequent runs with the same parameters. To get truly random results, that would have to be left out. However, for my benchmarking purposes it saved me having to save the matrix to a file and read it in, as I can easily generate a new copy of the same matrix I used before.

Pathfinder Networks (PFNETs) are networks derived from a graph representing proximity data. Basically, each node is connected to (almost) every other node by a weighted link, and that makes it hard to see what’s going on. The Pathfinder algorithm prunes the graph by removing links which are weighted higher than another path between the same nodes.

For example: A links to B with weight 5. A also links to C with weight 2, and C links to B with weight 2. Adding up the weights, A to B direct is 5, A to C to B is 4. Result: we remove the link from A to B, as the route via C is shorter. There are different ways to calculate the path lengths (using the Minkowski r-metric), but you get the general idea. The resulting PFNET has fewer links and is easier to analyse.

In Schvaneveldt (1990) an algorithm for computing PFNETs is given, but it is rather complex and computationally intensive. There is an improved algorithm called Binary Pathfinder, but that is apparently more memory intensive. Not very promising so far, but then along comes A new variant of the Pathfinder algorithm to generate large visual science maps in cubic time, by Quirin, Cordón, Santamaría, Vargas-Quesada, and Moya-Anegón. This algorithm is a lot faster (by about 450 times), but has one disadvantage: speed is traded in for flexibility. The original Pathfinder algorithm has two parameters, r (the value of the Minkowski metric to be used) and q (the maximum length of paths to be considered). The fast algorithm only has r, and always uses the maximum value for q (which is n-1). I know too little about the application of PFNETs to say whether this is important at all; for the uses I can envisage it does not seem to matter.

As added bonus, the algorithm in pseudo code in Quirin et al. is very short and straightforward. They’re using a different shortest-path algorithm to identify, erm, shorter paths. And then it’s very simple to prune the original network.

A picture tells more than 1K words, so here instead of 2000 words the before and after, graph layout courtesy of the graphviz program:

A network example (from Schvaneveldt 1990)

Here, each node is linked to every other node. Running the PFNET algorithm on it, we get the output shown in the second figure.

A PFNET generated from the previous graph:

If you compare the output with the actual result from Schvaneveldt’s book (p.6 / 7), you’ll realise that it is not identical, and the reason for that is that the example there limits the path-length, using the parameters (r = 1, q = 2) rather than (r = 1, q = n-1) as in the example shown here. As a consequence, the link from N1 to N4 (with a weight of 5) disappears, because of the shorter path (N1-N2-N3-N4, weight 4). But that path is too long if q is just 2, and so it is kept in Schvaneveldt’s example.

Implementation

It is not possible to implement the pseudo-code given in Quirin et al directly (in Erlang), as they use destructive updates of the link matrix, which we obviously cannot do in Erlang. But the first, naïve, implementation is still quite short. The input graph is represented as a matrix (n x n) where each value stands for the link weight, with zero being used to indicate non-existing links. I have written a function that creates a dot file from a matrix, which is then fed into graphviz for generating images as the ones shown above.

There are basically two steps: creating a matrix of shortest paths from the input matrix, and then generating the PFNET by comparing the two matrices; if a certain cell has the same value in both matrices, then it is a shortest path and is kept, otherwise there is a shorter path and it’s pruned. Here is the main function:

find_path(Matrix) ->
    Shortest = loop1(1,length(Matrix),Matrix),
    generate_pfnet(Matrix, Shortest, []).

Next we have the three loops (hence ‘cubic time’!) of the Floyd-Warshall shortest path algorithm to create the shortest path matrix:

loop1(K,N,Matrix) when K > N -> 
    Matrix;
loop1(K,N,Matrix) ->
    NewMatrix = loop2(K,1,Matrix,
        Matrix,[]),
    loop1(K+1,N,NewMatrix).

loop2(_,_,_,[],Acc) ->
    lists:reverse(Acc);
loop2(K,I,D,[Row|Rest],Acc) ->
    NewRow = loop3(K,I,1,D, Row, []),
    loop2(K,I+1,D,Rest,[NewRow|Acc]).

loop3(_,_,_,_, [], Acc) ->
    lists:reverse(Acc);
loop3(K,I,J,D, [X|Rest], Acc) ->
    loop3(K,I,J+1,D, Rest,
    [min(X, get(I,K,D) + get(K,J,D))|Acc]).

The final line implements the Minkowski metric with r = 1; this could be expanded to include other values as well, eg r = 2 for Euclidean, or r = ∞ (which seem to be the most common values in use; the latter means using the maximum weight of any path component along the full path).

And here are two utility methods, one to find the smaller of two values, and one to retrieve an element from a matrix (which is a list of rows). There is something of a hack to deal with the fact that zero does not mean a very small weight, but refers to a non-existing link:

min(X, Y) when X < Y -> X;
min(_X, Y) -> Y.

get(Row, Col, Matrix) ->
    case lists:nth(Col,
      lists:nth(Row, Matrix)) of
        0 -> 999999999;
        X -> X
    end.

And finally, generating the PFNET by comparing the two matrices (again, awful hack included):

generate_pfnet([],[],Result) ->
    lists:reverse(Result);
generate_pfnet([R1|Rest1], [R2|Rest2], Acc) ->
    Row = generate_pfrow(R1,R2,[]),
    generate_pfnet(Rest1, Rest2, [Row|Acc]).

generate_pfrow([],[],Result) ->
    lists:reverse(Result);
generate_pfrow([C|Rest1], [C|Rest2], Acc) ->
    case C of
        999999999 -> C1 = 0;
        _ -> C1 = C
    end,
    generate_pfrow(Rest1, Rest2, [C1|Acc]);
generate_pfrow([C1|Rest1], [C2|Rest2], Acc) ->
    generate_pfrow(Rest1, Rest2, [0|Acc]).

Discussion

So this is the basic code. It works, but there is scope for improvement.

• it only currently generates PFNETs with (r = 1, q = n-1)
• there is no parallelism, hence it’s not really making use of Erlang’s strengths.
• the three loops don’t look very elegant, and could probably be replaced by list comprehensions

Because of the matrix being updated, it doesn’t look that easy to parallelise the processing, but it would work at the level of updating the individual rows. If that can be done in parallel, it would probably provide some speed-up (provided the matrix is not of a trivial size). So the first step would be to change the matrix processing using lists:map/2, and replacing this then by pmap.

Once the code is up to scratch with full parallelism, and tested on larger matrices I will probably put it up on Google Code in case other people are interested in using it. If you have any suggestions, tell me in the comments!

References

Schvaneveldt, R. W. (Ed.) (1990)

Pathfinder Associative Networks: Studies in Knowledge Organization. Norwood, NJ: Ablex. The book is out of print. A copy can be downloaded from the Wikipedia page

Quirin, A; Cordón, O; Santamaría, J; Vargas-Quesada, B; Moya-Anegón, F (2008)

“A new variant of the Pathfinder algorithm to generate large visual science maps in cubic time”, Information Processing and Management, 44, p.1611-1623.

Now, the first thing you learn about natural language is that it is full of ambiguities, and so there will always be several alternatives available, several possible paths through the network which defines the grammar. The traditional solution is to dump all the alternatives on a stack, and look at them when the current path has been finished with. You can either go depth-first, where you complete the current path before you get the next one off the stack, or breadth-first, where you advance all paths by one step at a time, kind of pseudo-parallel.

Having to deal with a stack is tedious, as you need to keep track of the current configuration: which network are you at, what node, what position in the sentence, etc. But then, it occurred to me, there’s an easier way to do it (at least it’s easier in Erlang!): every time you come to a point where you have multiple alternatives, you spawn a new process and pursue all of them in parallel.

The only overhead you need is a loop which keeps track of all the processes currently running. This loop receives the results of successful paths, and gets notified of unsuccessful ones (where the process terminates without having found a valid structure). No need for a stack, and hopefully very efficient processing on multi-core machines as a free side-effect.

I’m still amazed how easy it was to implement. I wouldn’t have fancied doing that in Java or even C. For my test sentences I had about 8 to 10 processes running in parallel most of the time, but it depends on the size of the grammar and the length of the sentence really. What I liked about this was that it seemed the natural way to do in Erlang, where working with processes is just so easy.

And also, another nail in the coffin for the claim that you can’t use Erlang for handling texts easily!