| Safe Haskell | Safe |
|---|---|
| Language | Haskell98 |
Math.SetCover.Exact
Description
This module provides a solver for exact set cover problems. http://en.wikipedia.org/wiki/Exact_cover
- data Assign label set = Assign {
- label :: label
- labeledSet :: set
- assign :: label -> set -> Assign label set
- partitions :: Set set => [Assign label set] -> [[label]]
- search :: Set set => State label set -> [[label]]
- step :: Set set => State label set -> [State label set]
- data State label set = State {
- availableSubsets :: [Assign label set]
- freeElements :: set
- usedSubsets :: [Assign label set]
- 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
Documentation
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
| |
partitions :: Set set => [Assign label set] -> [[label]] Source
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.
step :: Set set => State label set -> [State label set] Source
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.
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
| |
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
disjoint :: set -> set -> Bool Source
difference :: set -> set -> set Source
minimize :: set -> [Assign label set] -> [Assign label set] Source
Unchecked preconditions:
set must be a superset of all sets in the assign list.
set must be non-empty.
The list of assignments must be non-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