Nonogram.lhs

Nonogram solver
===============

Choice of programming language
------------------------------

I chose to implement the nonogram solver in the Haskell programming language
[^haskell]. Haskell was chosen for a number of reasons:

[^haskell]: <http://www.haskell.org>

- availability of a high-performance compiler, GHC [^ghc];
- good Literate Programming support;
- semi-implicit parallelization using annotations;
- referential transparency which allows a more declarative programming model.

[^ghc]: <http://www.haskell.org/ghc/>

Implementation
--------------

The sequential and parallel nonogram solvers are implemented in the
`Nonogram.lhs` file. This is a literate Haskell file [^lhs] containing the
report as well as the carefully explained source code.

[^lhs]: <http://www.haskell.org/haskellwiki/Literate_programming>

The `Main.hs` file has an entry point for an executable to solve a puzzle
concurrently, and a simple parser for a standard nonogram file format [^fmt].

[^fmt]: <http://www.comp.lancs.ac.uk/~ss/nonogram/fmt2>

We can compile and run the program like this:

    $ make
    ghc --make -O2 -threaded Main.hs -o nonogram-solver
    [1 of 1] Compiling Nonogram         ( Nonogram.lhs, Nonogram.o )
    [2 of 2] Compiling Main             ( Main.hs, Main.o )
    Linking nonogram-solver ...
    $ ./nonogram-solver 20x20.nonogram
    Parsing 20x20.nonogram
    Solving 20x20.nonogram
    ----------XXX-------
    ---------XXXXX------
    ---------XXX-X------
    ---------XX--X------
    ------XXX-XXX-XXXX--
    ----XX--XX---XXXXXXX
    --XXXXXX-X---X------
    -XXXX---XX--XX------
    --------X---X-------
    -------XXX--X-------
    -------XXXXXX-------
    -XX---XXXXXXX-------
    XXXXXX--XXX-X-------
    X-XX--XX-X--X-------
    ---XXXX--X-X--XXX---
    --------XXXX-XX-XX--
    --------XXX--XXX-X--
    -------XXX----XXX---
    ------XXX-----------
    ------XX-X----------

Note that you might need to install the `parallel` package [^par], for
example using `cabal-install` [^cabal]. The `parallel` package is based on
the *Algorithm + Strategy = Parallelism* paper [^strategies] and gives us a
high-level interface to add parallelism to our program.

[^par]: <http://hackage.haskell.org/package/parallel>

[^cabal]: <http://hackage.haskell.org/trac/hackage/wiki/CabalInstall>

[^strategies]: Philip W. Trinder, Kevin Hammond, Hans-Wolfgang Loidl, and Simon
    L. Peyton Jones. Journal of Functional Programming, 8(1), Jan 1998.
    <http://www.macs.hw.ac.uk/~dsg/gph/papers/abstracts/strategies.html>

Parallelization conclusions
---------------------------

We can now compare the performance of the sequential program to the performance
of the parallel program. We use the excellent criterion library [^criterion],
aimed at benchmarking Haskell code. The code we used to benchmark the the
programs is located in the `Benchmark.hs` file, along with some sample puzzles.
To reproduce the benchmarks, the `Makefile` contains the `benchmark-sequential`
and `benchmark-parallel` targets, which benchmark the sequential and parallel
program.

[^criterion]: <http://www.serpentine.com/blog/2009/09/29/criterion-a-new-benchmarking-library-for-haskell/>

We ran these benchmarks on an Intel(R) Core(TM)2 Duo CPU E8400 @ 3.00GHz.
Because we use a dual-core computer, we can at most expect a 100% speedup (i.e.
a reduction of the running time by 50%).

                **5x5**      **10x10**    **15x15**    **20x20**
--------------  -----------  -----------  -----------  -----------
**sequential**  10.10453 us  202.7505 us  5.911520 ms  250.8528 ms
**parallel**    22.05260 us  392.9396 us  4.348415 ms  155.6637 ms  

Looking at the results, parallelization introduces a large overhead for the
smaller puzzles, but gives us an advantage for larger puzzles. Because of our
implementation of the branch-and-bound-based algorithm, we know that for each
branch, a `par` call is made. However, this is only useful if the branch is a
non-trivial computation: otherwise, the overhead of the `par` call outweighs the
benefits of parallelization. We can conclude that the parallelization for this
algorithm is only useful for large enough input sets: otherwise, the overhead
outweighs the speedup.

For the 20x20 puzzle, we see that the running time is reduced by 37.9461%. This
is not the 50% reduction we were naively hoping for, but it is not a bad result,
since the parallelization of the program was almost trivial (just replacing the
branching function). We also have to keep in mind that not everything can happen
in parallel: e.g. joining the results of two branches happens on one core.

![Performance comparison of the nonogram solver (log scale on y axis)](assignment-2/images/nonogram-sequential-vs-parallel.pdf)

Literate source code
--------------------

Here, we give the full source code to the programs, annotated in Literate
Programming style.

> module Nonogram
>     ( Description
>     , sequentialNonogram
>     , parallelNonogram
>     , putNonogram
>     ) where
>
> import Control.Monad (when, mplus, foldM)
> import Control.Parallel (par, pseq)
> import Data.IntMap (IntMap)
> import qualified Data.IntMap as IM

A nonogram grid is built from cells in two possible colors. For representing the
value of a cell, we use a simple datatype called `Color`.

> data Color = White | Black
>            deriving (Show, Eq)

A nonogram is then defined as a matrix of colors:

> type Nonogram = [[Color]]

The input is a list of natural numbers for every row and column. We call such a
list a `Description` and represent it as an `Int` list:

> type Description = [Int]

We use an algorithm that attempts to solve the puzzle top-down, i.e. starting at
the top row and finishing at the bottom row. It is implemented as a
branch-and-bound algorithm, searching the solution space.

As we solve more and more rows, we also learn new, partial information about the
columns -- in addition to the `Description` of the columns. We use this to
"bound" in certain cases, if the current search space does not match our partial
information.

> data Partial = MustBe Color
>              | BlackArea Int
>              deriving (Show, Eq)

Initially, this partial information is deduced from the descriptions alone.
Hence, we have a function to convert a `Description` into `Partial` information.

> fromDescription :: Description -> [Partial]
> fromDescription = map BlackArea

When we reach the bottom of the nonogram, we will have a "final state" of the
partials for every column. We then need to check if we "used up" all black
areas: this function checks if the partial information can be considered empty.
Note that we do allow `White` fields in empty partials (since they do not add to
the description counts).

> emptyPartial :: [Partial] -> Bool
> emptyPartial [] = True
> emptyPartial (MustBe Black : _) = False
> emptyPartial (BlackArea _ : _) = False
> emptyPartial (_ : xs) = emptyPartial xs

We need to keep a `[Partial]` for every column. We can't store them in a list
because we want fast random access. We use an `IntMap`, a purely functional
tree-like structure (big-endian patricia trees) which performs fast for lookups
and insertions.

> type Partials = IntMap [Partial]

The following function adds a cell to partial information. It returns a value in
the `Maybe` monad:

- if the cell is inconsistent with previous information, it returns `Nothing`;

- otherwise, it returns the updated partial information.

On update, it might consume or add to the partial information. Note that the
partial information always represents "what comes next" in the column, so we
only need to inspect/modify the first few elements of the list.

> learnCell :: Color -> [Partial] -> Maybe [Partial]
> learnCell White [] = Just []
> learnCell Black [] = Nothing
> learnCell y (MustBe x : ds) = if x == y then Just ds else Nothing
> learnCell White ds = Just ds
> learnCell Black (BlackArea n : ds) = Just $
>     replicate (n - 1) (MustBe Black) ++ (MustBe White : ds)

We also provide a convenience function to call `learnCell` for a particular
column (specified by it's 0-based index).

> learnCellAt :: Color -> Partials -> Int -> Maybe Partials
> learnCellAt cell partials index = do
>     ds <- learnCell cell $ partials IM.! index
>     return $ IM.insert index ds partials

In Haskell, we can abstract over pretty much anything. We need to write two
versions of our nonogram solving algorithm -- a simple one and a concurrent one.
Since we want as little code duplication as possible, we can do this by
abstracting over the way we deal with branches in our search tree.

More concrete, we have a branching strategy (`Branching`) which decides how two
branches which might or might not yield a result (the `Maybe a`'s) are composed.
We also provide sequential and parallel branching:

> type Branching a = Maybe a -> Maybe a -> Maybe a

> sequential :: Branching a
> sequential = mplus

> parallel :: Branching a
> parallel x y = x `par` y `pseq` mplus x y

Parallelization of Haskell programs can be easily done using `par` annotations,
which attempts to spark the calculation of a thunk on a free core (if possible).
While `par` is a very cheap function call, it is not *free*. Therefore, one
always has to make sure the computations we spark on other cores are big enough,
so we minimize the overhead caused by `par`.

This implies it could be a better strategy to only use parallel branching if we
are in the upper part of the search tree (i.e. depth is less than a given `n`).
However, after experimenting with this, this `n` was highly dependent of the
used input set, and the speedup was only marginally better than the simpler
solution using just `parallel` -- so I decided not to use this more advanced
strategy.

The strategy is used in the `solve` function, which holds most of the main
solver logic.

The `solve` function is actually a wrapper around a `solve'` function which does
the actual work. This is an optimization called the static argument
transformation [^sat].

[^sat]: <http://blog.johantibell.com/2010/09/static-argument-transformation.html>

> solve :: Branching Nonogram -> Int -> Int -> [Description] -> Partials
>       -> Maybe Nonogram
> solve branch width = solve'
>   where
>     solve' column descriptions partials

There are a number of cases that need to be considered. First, suppose there are
no more descriptions for the rows. In this case, we know that the knowledge we
have about the column partials must be empty: we can no longer place any black
cells. If this is the case, we have a correct solution (the empty one),
otherwise, our solver fails by returning `Nothing`.

>         | null descriptions =
>             if all emptyPartial (map snd $ IM.toList partials)
>                 then return [[]]
>                 else Nothing

If we have at least one row description, we check to see if that row description
is empty. Suppose this is the case. This means no more black areas should be
placed in this row, so we just fill it up with white cells. We then recursively
call `solve'` to solve the other rows.

>         | null rd = do
>             ps <- foldM (learnCellAt White) partials [column .. width - 1]
>             rows <- solve' 0 rds ps
>             return $ replicate (width - column) White : rows

Since we know now that the row description is not empty, we have at least one
black area on this row. We consider two possibilities:

- the black areas starts at the beginning of the row (consider the beginning of
  the row as indicated by the `column` argument);
- we have at least one white cell, followed by a row with the same black areas.

This is where our search tree branches between the two cases, `branch` and
`skip`, defined later.

>         | otherwise = branch place skip

Some definitions of previously used values:

>       where
>         (rd : rds) = descriptions
>         (l : ds) = rd

The case where the black area is at the beginning at the row has a pretty long
definition but the logic behind the code is actually quite straightforward:

- we fail if the black area cannot be placed due to the fact there are
  insuficient cells left;
- we update the the column partial knowledge with these new black cells;
- we place a white cell after the black area, two because black areas cannot be
  contiguous (if the end of the black area touches the border of the grid, we
  can skip this);
- we recursively call `solve'` to solve the rest of the row, then add the cells
  we just calculated to the returned solution.

>         place = do
>             when (column + l > width) Nothing
>             ps <- foldM (learnCellAt Black) partials
>                       [column .. column + l - 1]
>
>             let atEnd = column + l == width
>
>             ps' <- if atEnd then return ps
>                             else learnCellAt White ps (column + l)
>
>             (row : rows) <- solve' (column + l + 1) (ds : rds) ps'
>             let row' = if atEnd then row else White : row
>             return $ (replicate l Black ++ row') : rows

If we choose not to place the black area at the beginning of the row, we just
need to add a white cell and recursively call `solve'` again. This case can fail
as well -- when we've reached the far right side of the grid, we can no longer
place white cells.

>         skip = do
>             when (column >= width) Nothing
>             ps <- learnCellAt White partials column
>             (row : rows) <- solve' (column + 1) ((l : ds) : rds) ps
>             return $ ((White : row)) : rows

The `nonogram` function helps us in converting the column descriptions into the
partial information we need, and thus provides a nicer interface to the
programmer than `solve` does.

> nonogram :: Branching Nonogram -> [Description] -> [Description]
>          -> Maybe Nonogram
> nonogram branch rows columns = solve branch (length columns) 0 rows state
>   where
>     state = IM.fromList (zip [0 ..] $ map fromDescription columns)

We provide a sequential and a parallel program:

> sequentialNonogram :: [Description] -> [Description] -> Maybe Nonogram
> sequentialNonogram = nonogram sequential

> parallelNonogram :: [Description] -> [Description] -> Maybe Nonogram
> parallelNonogram = nonogram parallel

At last, the `putNonogram` function allows us to print a solution we found
(of the type `Maybe Nonogram`) to standard output.

> putNonogram :: Maybe Nonogram -> IO ()
> putNonogram Nothing  = putStrLn "No solution found"
> putNonogram (Just s) = mapM_ (putStrLn . concatMap showCell) s
>   where
>     showCell Black = "X"
>     showCell White = "-"