Math.SetCover.Exact

This module provides a solver for exact set cover problems. http://en.wikipedia.org/wiki/Exact_cover

Documentation

data Assign label set

Assign allows to associate a set with a label. If a particular set is chosen for a set cover, then its label is included in the output of partitions.

I have decided to separate sets and labels this way, since it is the easiest way to assign a meaning to a set. If you really want to know the sets in a partition, then you can fill the label field with the set.

Constructors

Assign

Fields

label :: label

labeledSet :: set

Instances

Functor (Assign label)

assign :: label -> set -> Assign label set

Construction of a labeled set.

bitVectorFromSetAssigns :: Ord a => [Assign label (Set a)] -> [Assign label (Set Integer)]

You may use this to post-process a set of Assigns in order to speedup the solver considerably. You must process the whole set of Assigns at once, i.e. do not process only parts of the assignment list. The output of bitVectorFromSetAssigns should go into the solver as is.

intSetFromSetAssigns :: Ord a => [Assign label (Set a)] -> [Assign label IntSet]

Like bitVectorFromSetAssigns but generates IntSet instead of Integer bitvectors. Since containers-0.5.5 as shipped with GHC-7.8.4, IntSet should usually be more efficient than Integer.

partitions :: Set set => [Assign label set] -> [[label]]

partitions [assign '0' set0, assign '1' set1, assign '2' set2] computes unions [set0, set1, set2] and tries to partition the union set using the sets set0, set1, set2. partitions returns all such partitions. If a set is chosen for a partition, then its label is included in the output. E.g. set0 = Set.fromList [0,1], set1 = Set.fromList [2], set2 = Set.fromList [0,1,2], then partitions returns [01, 2].

The order of partitions and the order of labels depends on the implementation and you must not rely on them.

You may use listToMaybe in order to select only the first solution.

search :: Set set => State label set -> [[label]]

Start the search for partitions on a certain search state. This can be an initState or the result of performing some search steps. In the examples we use this for parallelization: We perform some steps manually and then run search on the results in parallel.

step :: Set set => State label set -> [State label set]

This is the key of the search algorithm. The search algorithm tries to build partitions by adding sets to a partition list successively. A step starts on a partial partition and looks for new sets that could be added. The goal is to avoid to check a set again down in a search branch and to quickly determine search directions that lead to a dead end. To this end a search step selects a certain set element and tries all sets that contain that element and that do not overlap with the partial partition. Practically, step selects an element with the minimal number of non-overlapping sets it is contained in. If this number is zero, then the search can be aborted in this branch.

Most oftenly the power of the algorithm originates from the formulation of a problem as a set-cover problem and from the equal treatment of all elements. E.g. in the Soma cube example the algorithm chooses whether to do a case analysis on all bricks that cover a certain position, or to do a case analysis on all positions that are possible for a certain brick.

The algorithm might not be extraordinarily fast, but in all cases it consumes only little memory since it only has to maintain the current state of search.

Precondition: freeElements of the input state must not be empty.

data State label set

The state of the search. usedSubsets contains the partial partition built up so far. availableSubsets is the list of sets we can still try to put into a partition. The lists usedSubsets and availableSubsets are disjoint, but their union is not necessarily equal to the list of initially given sets. There are sets not contained in the partial partition that overlap with the partial partition. Those sets are not available for extending the partition.

freeElements contains the elements that are not covered by the partial partition in usedSubsets. unions usedSubset and freeElements are disjoint and their union is the set of all elements.

Constructors

State

Fields

availableSubsets :: [Assign label set]

freeElements :: set

usedSubsets :: [label]

Instances

Functor (State label)

initState :: Set set => [Assign label set] -> State label set

updateState :: Set set => Assign label set -> State label set -> State label set

class Set set where

This class provides all operations needed for the set cover algorithm. It allows to use the same algorithm both for containers' Set and for sets represented by bit vectors.

Methods

null :: set -> Bool

disjoint :: set -> set -> Bool

unions :: [set] -> set

difference :: set -> set -> set

minimize :: set -> [Assign label set] -> [Assign label set]

minimize free assigns finds a set element x from free that is contained in the least number of sets in assigns. Then it returns the assigns where x is contained in the associated set. This formulation allows us not to name x and thus we do not need a second type variable for set elements and no type family from set to its element type.

Unchecked preconditions: free must be a superset of all sets in the assign list. free must be non-empty. The assigns list may be empty. The output of assigns must be a subsequence of the input assigns, that is, it must be a subset of the input and it must be in the same order. This requirement was originally needed by minimize for Map, but currently it is not utilized anywhere.

Instances

Set IntSet

Ord a => Set (Set a)

C a => Set (Set a)

(Ord k, Set set) => Set (Map k set)

This instance supports Maps of Sets. This way you can structure your sets hierarchically. You may also use it to combine several low-level bitsets. A Map must not contain empty subsets.

data Tree label set

Constructors

Leaf

Branch set [(label, Tree label set)]

Instances

(Eq label, Eq set) => Eq (Tree label set)

decisionTree :: Choose set => [Assign label set] -> Tree label set

completeTree :: Choose set => State label set -> Tree label set

class Set set => Choose set where

Methods

chooseMinimize :: set -> [Assign label set] -> (set, [Assign label set])

Instances

Choose IntSet

Ord a => Choose (Set a)

C a => Choose (Set a)

(Ord k, Choose set) => Choose (Map k set)