Wednesday, November 13, 2013

Chocolate, Logic Puzzles, and Dancing Links

Early last year, I visited a cafe that awards chocolate for solving logic puzzles. Naturally, I couldn’t resist free chocolate, and afterwards, just as naturally, I couldn’t resist thinking about programs that solve logic puzzles.

I’ve had to write such programs before for homework assignments or to test out frameworks. But oddly, I had never put much effort into it. I loved logic puzzles as a kid, even going so far as to buy a magazine or two that were filled with grids and clues. Why hadn’t I already written a decent tool to help?

Better late than never. After a spate of intense coding, I had a program that read clues in a terse text format and used brute force to find the solution. I spent most of my time devising the mini-language for the clues rather than the algorithm, as I figured the bottleneck would be the human entering the clues.

My solver worked well enough on a few examples, including the puzzle I solved to get a chocolate. But then I tried my program on the Zebra Puzzle. I was too impatient to let it finish. After a little thought, it was clear why.

On the grid

Logic grid puzzles can be abstracted as follows. We are given a table with M rows and N columns, and each cell contains a unique symbol. Our goal is to rearrange the symbols within each row except for those in the first row so that given constraints are satisfied. To be clear, symbols must stay in the row they started in, but apart from the first row, they can change places with other symbols in their row.

Some examples of constraints:

• symbols A and B must be in the same column.

• symbols A, B, and C must be in distinct columns.

• symbol A’s column must be exactly one to the left of symbol B’s column.

We fix the first row because of contraints such as the last example: clues often refer to the order of the elements in the first row in the input table. Let us call them order contraints. This inspires the following convenient relabeling. Without loss of generality, let the symbols of the first row be 1, …, N from left to right.

My brute force solver generates every possible table and prints the ones that satisfy all given constraints. That means it has to examine up to N!(M-1) cases: there are N! permutations of the symbols in each row, and we have M-1 rows to rearrange. For the Zebra Puzzle, this is 1205.

Got it covered

I needed a smarter approach. Since I had already coded a sudoku solver, I chose the same approach, namely, represent a puzzle as an instance of the exact cover problem, then use the dancing links algorithm.

Firstly, we populate a set X with all the symbols in the table. Next, instead of generating every table, generate every possible column. Each such column corresponds to the subset of X consisting of the symbols in the column. Generating every possible column means brute force is still present, but to a lesser degree.

An exact cover of X corresponds to a collection of columns such that no two columns contain the same symbol, and furthermore, each symbol appears in one of the columns. Thus these columns can be joined together to form a candidate solution to the logic puzzle: we merely order them so that the first row reads 1, …, N.

It remains to disallow covers that violate the constraints. For some constraints, we achieve this by omitting certain columns. For example, suppose A and B must be in the same column. When we generate a column that only contains one of A or B, we omit it from the exact cover instance. Similarly, if a constraint requires that A and B lie in distinct columns, we omit subsets that contain both symbols from the exact cover instance.

Out of order

The above suffices for many puzzles, but falls short for those with order constraints such as "A and B lie in adjacent columns". For this particular constraint, we add N elements to X. Let us call them x[1], …, x[N]. Given a generated column whose first row contains the number n (recall we have relabeled so that the first row of the input table is 1, …, N), if the column contains:

• both A and B: eliminate the column from consideration.

• A and not B: we add x[n] to the column’s corresponding subset.

• B and not A: we add x[k] to the column’s corresponding subset for all k in 1..N where |n-k| is not 1.

Lastly, we mark x[1] ,…, x[N] as optional, meaning that they need not be covered by our solution. (We could instead augment our collection of subsets with singleton subsets {x[n]} for each n, but with dancing links there’s a faster way.)

Thus any exact cover must have A and B in adjacent columns to avoid conflicts in the x[1], …, x[N] elements. We can handle other order constraints in a similar fashion.

The size of X is the number of symbols, MN, plus N for each order constraint. The number of subsets is the number of possible columns, which is NM, because we have M rows, and each can be one of N different symbols. Each subset has size M, one for each row, plus up to N more elements for each order constraint that involves it.

That’ll do

Though NM may be exponential in input size, typical logic grid puzzles are so small that my program solves them in a few milliseconds on my laptop. The bottleneck is now indeed the human entering the clues. [I briefly thought about writing a program to automate this by searching for keywords in each sentence.]

I was annoyed the above algorithm is adequate. Firstly, trying entire columns in a single step bears no resemblance to actions performed by human solvers. Secondly, it was too easy: I wish I were forced to think harder about the algorithm! Perhaps I’ll devise a larger puzzle that demands a better approach.

Around this time I lost focus because I felt my old code could use a few edits. I got sidetracked rewriting my dancing-links sudoku solver and comparing it against other ways of solving sudoku, and soon after that I moved on.

Luckily for my code, I recently felt compelled to clean it up enough for public release. It seemed a pity to let it languish in a forgotten corner until rendered useless from bit rot and lack of documentation.

The DLX library is now available at a git repository near you: