We capture the pattern of meta-heuristics and local search as a process or stream of evolving solutions. These combinators provide a way to describe and manipulate these processes quickly. The basic pattern of their use is this;
loopP (strategy) seed
The strategy itself is a stream transformer. The transformer becomes a search strategy when it's output is fed back into it's input, which is the action of the loopP function. For example, the following is not a search strategy but you could write;
loopP (map (+1)) 0
Which would generate the stream [0,1,2... A real search strategy then looks like;
loopP iterativeImprover tspSeed
Many search strategies do not always produce improving sequences as the iterative improver does. For these
we provide a simple modification of
scanl which can be applied to any stream, called
Finally, these streams are usually descriptions of unlimited processes. To make them
practical we limit them using standard Haskell combinators such as
take and list index.
take 20 . bestSoFar $ loopP searchStrategy seed
Search strategies are constructed via the composition of other functions. This often resembles the composition of an arrow pipeline, and this library can be rewritten in terms of arrows, however we have found no significant advantage in doing this.
A simple TABU like search strategy, that has a memory of the recent past (10 elements) of the search process, and filters neighbourhoods accordingly can be created like this;
searchStrategy xs = map head $ zipWith (\ws->filter (flip notElem ws)) (window 10 xs) (neighbourhood xs)
A common way to improve meta-heuristics is to introduce stochastic elements, such as random decisions from a constrained set of choices, or neighbourhoods which will not generate exactly the same set of options each time a particular solution is visited. Stream transformations allow this because they can thread additional state internally, while not exposing the user of the transformation to a great deal of the process. For example in the above example, to create a random choice from the constrained set at each point you would do this;
searchStrategy rs xs = select uniformCDF $ zipWith (\ws->filter (flip notElem ws)) (window 10 xs) (neighbourhood xs)
The neighbourhood can be similarly modified. We must still provide the starting points for the extra data used by such transformers, in this case a stream of random values, or in other cases a random number generator, but one provided it is hidden, and the transformer can be composed with any other transformation.
Using the same transformation, which threads an internal state, in several places is harder. It involves merging and dividing streams in sequenced patterns. For example;
applyToBoth tr as bs = (\xs->(map head xs,map last xs)) . chunk 2 $ tr (concat $ transpose [as,bs])
- type StreamT a b = Stream a -> Stream b
- type Stream s = [s]
- type List s = [s]
- class Optimisable a where
- best :: Optimisable s => [s] -> s
- worst :: Optimisable s => [s] -> s
- bestOf :: Optimisable s => s -> s -> s
- sortO :: Optimisable a => [a] -> [a]
- bestSoFar :: Optimisable s => StreamT s s
- doMany :: Int -> StreamT s s' -> StreamT s (List s')
- chunk :: Int -> StreamT s (List s)
- variableChunk :: Stream Int -> StreamT s (List s)
- stretch :: Int -> StreamT s s
- variableStretch :: Stream Int -> StreamT s s
- window :: Int -> StreamT s (List s)
- variableDoMany :: Stream Int -> StreamT s s' -> StreamT s (List s')
- varyWindow :: (Int, Int) -> Stream (List s) -> Stream (List s)
- improvement :: Optimisable s => StreamT s (List s) -> StreamT s (List s)
- select :: (Random r, Ord r) => (Int -> List r) -> StreamT (List a) a
- select' :: (Random r, Ord r) => List r -> StreamT (List a) a
- until_ :: Stream a -> Stream Bool -> Stream (Stream a) -> Stream a
- restart :: StreamT a a -> StreamT a Bool -> StreamT a a
- restartExtract :: StreamT a a -> StreamT a Bool -> StreamT a a
- nest :: Eq n => List (n, Stream a -> Stream b) -> Stream n -> Stream a -> Stream b
- preNest :: Eq n => (Stream a -> Stream n) -> List (n, Stream a -> Stream b) -> Stream a -> Stream b
- divide :: Eq n => List n -> Stream n -> Stream v -> List (Stream v)
- join :: Eq n => List (n, Stream v) -> Stream n -> Stream v
- loopP :: StreamT s s -> s -> Stream s
- loopS :: StreamT s s -> StreamT s s
- safe :: Stream (List v) -> Stream (List v) -> Stream (List v)
Internally streams are represented as standard Haskell lists. This type synonym is provided for readability. It is assumed that streams will be infinite.
Lists are also used to represent finite collections or groups of solutions. This synonym is for where the list is being used as a finite list.
Optimisable Type Class
In previous versions I have used the standard
Ord classes. However I have then
had to assume that every problem is a minimisation. To get around this, and provide functions that
match more closely to optimisation problems, where the concept we seek is better than, rather than
greater or less than, I provide this new class. It is however very very like
It is proposed that this class is used for value of solution comparisons only, and the standard
is retained for situations where the solutions are identical, and do not just share the same value.
This returns the best solution in an input list. It can be thought of as similar to
This returns the worse solution in an input list. It can be thought of as similar to
This returns the best of two input solutions. It can be thought of as similar to
A transformer that is usually used as a final step in a process, to allow the user to only see the best possible solution at each point, and ignore the intermediate values that a strategy may produce.
Stream Transformation Combinators
This was original created to assist with making multiple selections from a population within a genetic algorithm. More generally this is a higher order function which will take a stream transformation and apply it multiple times to each element of the input stream gathering up the results.
It can be used to make multiple selections from a population in a GA, or to create a neighbourhood function from a perturbation operation. For example;
nF = doMany 5 perturbFunction
Where 5 is just an arbitrary example constant and perturbFunction is a placeholder.
If the perturb function is defined with a parameter controlling the perturbation, for example a random number, such as which pair of cities in a TSP to exchange then you can do this for a deterministic operation;
nF = doMany 5 (zipWith perturbationFunction (cycle [0..4]))
For a more stochastic effect;
nF = doMany 5 (zipWith perturbationFunction (randoms g))
Where g is assumed to be of the
Breaks down a stream into a series of chunks, frequently finds use in preparing sets of random numbers
for various functions, but also in the
makePop function that is important for genetic algorithms.
A more flexible version of chunk which can vary the size of each chunk, in accordance with a list providing the sizes required of each grouping. For example;
chunk i = variableChunk (repeat i) variableChunk (cycle [1,2]) [0..] = [,[1,2],,[4,5]...
An operation which changes the structure of an underlying stream, but not the data type, by
replicating the elements of the stream in place. Can be thought of as changing the speed of the stream.
This finds use in
doMany, genetic algorithms and ant colony optimisation.
stretch 2 "abcd" = "aabbccdd"
A more flexible version of stretch which can vary how far each value in the underlying stream is stretched, in accordance with a list providing the sizes required of each grouping. For example;
stretch i = variableStretch (repeat i) variableStretch [3,3,7,2] "abcd" = "aaabbbcccccccdd"
Creates a rolling window over a stream of values up to the size parameter. The windows are then produced as a stream of lists. This can also be done using a queue data structure, however this was found to be slightly faster.
variableDoMany perfoms the same action as
doMany, but allows the programmer to vary how many times
the transformation is applied to each underlying solution, through the use of a stream of sizes.
A function for transforming a stream of
windows, taking random numbers of elements from the front
of each window. This can be used in the implementation of variations of the TABU search algorithm.
Usage as follows (with example values) ;
varyWindow (3,6) . window 10
This function transforms a neighbourhood function to give a transformation that yields improving
neighbourhoods, that is, neighbourhoods that only contain solutions that improve upon the seed solution.
In the case that there are no improving solutions (local minima), the output is a singleton list
containing the seed solution. This functionality is provided byt the
safe helper function.
The generalised selection routine. This takes a stream of lists and selects one element from each list,
to construct the new stream. The selection routine takes a
DistributionMaker, and selects based upon this.
This only really makes sense when there is some internal structure in the stream of lists. For example;
select (poissonCDF 1) . map sortO
Each list in the input stream is sorted, so that the best solutions appear first. This is then selected from using a Poisson distribution, with the mean at 1. This means that the early (better) solutions are much more likely to be selected.
This function acts like
select, however the distribution is fixed, not being constructed anew for each
list in the input stream. This is much more efficient, but does assume that the size of each list in the
input is the same (a fixed size neighbourhood is perfect for this).
This function is very similar to the until function of FRP. It takes a stream and
returns the elements of that stream until
True appears on the trigger stream. At this point
one of the potential futures is chosen and becomes the remainder of the stream.
The potential futures are zipped with the triggers, so the choice is fixed, the current potential
future is the choice. More generally this concept could be elaborated in the future.
- The initial stream of values to place before the trigger (2) The stream of triggers (3) The stream of potential future streams
Could be used to provide a temperature strategy that restarts once in Simulated Annealing like so;
let t = iterate (+1) 0 in until_ t triggerStream (repeat t)
This alternative will restart every time
True appears on the trigger stream, not just the first time.
let t = until_ (iterate (+1) 0) triggerStream (repeat t) in t
Restart is a little like loop. It will construct a stream of values by applying a stream transformation to one value, and then the successor and so on. It differs in that it also takes a triggering mechanism that can choose to stop the current sequence and continue from a different value (the next value on the initial stream). For example, the following will start counting initially from 0, then -5, then -10, and will count until it reaches 11 each time.
restart (map (+1)) (map (>10)) [0,-5,-10]
restart, except that it will only return the result of an iteration of transformation,
not the intermediate values. For example;
restartExtract (map (+1)) (map ((==0) . flip mod 4)) [1,-5,-10,-13]
Is the combination of
join. It takes a set of indices and stream transformations,
divides the input stream, using the indices and a stream of indices, transforms each substream
by the related stream transformation, and then puts them all back together again as a new stream.
For example, to apply a transformation (f) to only every third value in a stream, you could do this;
nest [(True,id),(False,f)] (cycle [True,True,False])
This function acts like
nest, but the name stream makes choices (or can do) based upon the initial value
of solutions on the input stream.
Divide splits a stream of values into a list of substreams. The division is controlled by a list of indices and then a stream of these indices.
Join is the inverse of
divide, it combined substreams into a stream fo values.
It takes a list of substreams, and the indices that indicate them, and then a stream of the indices.
For each value in the stream of indices, the next value in the appropriate substream is chosen and
A more specific version of
loopS and implemented in terms of it. Rather than allowing a
number of initial values, this allows only 1.
The standard function for tying the knot on a stream described process. This links the outputs of the stream process to the inputs, with an initial set of values, and provides a single stream of values to the user.
A helper function that chooses between elements of the two input streams at each point in the stream, and returns one which is non-empty. The check looks at values in the second stream first, if this list is not empty, it is returned.