signed-multiset-0.1: Multisets with negative membership.

Portability portable provisional stefan@vectorfabrics.com Safe-Infered

Data.SignedMultiset

Description

An efficient implementation of signed multisets.

A signed multiset is like a multiset (or bag), but additionally allows for negative membership. That is, in a signed multiset, an element can occur a negative number of times.

For a theory of signed multisets, see

• Wayne D. Blizard. Negative membership. Notre Dame Journal of Formal Logic, 31(3):346--368, 1990.

Since many function names (but not the type name) clash with Prelude names, this module is usually imported `qualified`, e.g.,

```  import Data.SignedMultiset (SignedMultiset)
import qualified Data.SignedMultiset as SignedMultiset
```

Function comments contain the function's time complexity in so-called big-O notation, with n referring to the number of multiset members involved.

Synopsis

# Type

data SignedMultiset a Source

A signed multiset with elements of type `a`.

Instances

 Typeable1 SignedMultiset Ord a => Eq (SignedMultiset a) (Ord a, Data a) => Data (SignedMultiset a) Ord a => Ord (SignedMultiset a) (Ord a, Read a) => Read (SignedMultiset a) Show a => Show (SignedMultiset a) Ord a => Monoid (SignedMultiset a) Monoid under `union`.

# Construction

O(1). The empty signed multiset, i.e., the multiset in which every element has multiplicity zero.

singleton :: a -> SignedMultiset aSource

O(1). Create a signed multiset that contains exactly one copy of the given element.

insert :: Ord a => a -> SignedMultiset a -> SignedMultiset aSource

O(log n). Insert a new copy of the given element into a signed multiset, i.e., increment the multiplicity of the element by 1.

insertMany :: Ord a => a -> Int -> SignedMultiset a -> SignedMultiset aSource

O(log n). Insert a specified number of new copies of the given element into a signed multiset, i.e., increment the multiplicity of the element by the specified number. If the specified number is negative, copies are deleted from the set.

delete :: Ord a => a -> SignedMultiset a -> SignedMultiset aSource

O(log n). Delete a copy of the given element from a signed multiset, i.e., decrement the multiplicity of the element by 1.

deleteMany :: Ord a => a -> Int -> SignedMultiset a -> SignedMultiset aSource

O(log n). Delete a specified number of copies of the given element from a signed multiset, i.e., decrement the multiplicity of the element by the specified number. If the specified number is negative, new copies of the element are inserted into the set.

deleteAll :: Ord a => a -> SignedMultiset a -> SignedMultiset aSource

O(log n). Delete all copies of the given element from a signed multiset, i.e., set the multiplicity of the element to zero.

# Queries

O(1). Return whether the signed multiset is empty, i.e., whether every element has multiplicity zero.

O(n). Return whether the signed multiset is a set, i.e., whether all elements have either multiplicity zero or else multiplicity 1.

O(1). Return the number of members of the signed multiset, i.e., the number of elements that have nonzero multiplicity.

O(n). Return the cardinality of the signed multiset, i.e., the sum of the multiplicities of all elements.

member :: Ord a => a -> SignedMultiset a -> BoolSource

O(log n). Return whether the given element is a member of the signed multiset, i.e., whether the element has nonzero multiplicity.

notMember :: Ord a => a -> SignedMultiset a -> BoolSource

O(log n). Return whether the given element is not a member of the signed multiset, i.e., whether the element has multiplicity zero.

multiplicity :: Ord a => a -> SignedMultiset a -> IntSource

O(log n). Return the multiplicity of the given element in the signed multiset.

isSubmultisetOf :: Ord a => SignedMultiset a -> SignedMultiset a -> BoolSource

O(n). Return whether the first signed multiset is a submultiset of the second, i.e., whether each element that has nonzero multiplicity `n` in the first multiset has nonzero multiplicity `m` with `n <= m` in the second.

O(n). Return whether the first signed multiset is a proper submultiset of the second, i.e., whether each element that has nonzero multiplicity `n` in the first multiset has nonzero multiplicity `m` with `n < m` in the second.

# Combining

union :: Ord a => SignedMultiset a -> SignedMultiset a -> SignedMultiset aSource

O(n). Return the union of two signed multisets. The multiplicity of an element in the returned multiset is the maximum of its nonzero multiplicites in the argument multisets.

O(n). Return the additive union of two signed multisets. The multiplicity of an element in the returned multiset is the sum of its multiplicities in the argument multisets.

O(n). Return the intersection of two signed multisets. If an element has nonzero multiplicity in both argument multisets, its multiplicity in the returned multiset is the minimum of its multiplicites in the argument multisets; otherwise, its multiplicity in the returned multiset is zero.

O(n). Return the difference of two signed multisets. The multiplicity of an element in the returned multiset is the difference between its multiplicities in the first and second argument multiplicities.

# Scalar multiplication

O(n). Return the additive union of the given number of copies of the signed multiset.

# Traversals

map :: (Ord a, Ord b) => (a -> b) -> SignedMultiset a -> SignedMultiset bSource

O(n * log n). Apply the given function to all elements of the signed multiset.

foldr :: (a -> Int -> b -> b) -> b -> SignedMultiset a -> bSource

O(n). Perform a right-associative fold on the members and elements of the signed multiset using the given operator and start value.

foldl :: (a -> b -> Int -> a) -> a -> SignedMultiset b -> aSource

O(n). Perform a left-associative fold on the members and elements of the signed multiset using the given operator and start value.

# Conversion

toList :: SignedMultiset a -> [(a, Int)]Source

O(n). Convert the signed multiset to a list that associates all members of the multiset with their multiplicity.

fromList :: Ord a => [(a, Int)] -> SignedMultiset aSource

O(n * log n). Construct a signed multiset from a list of element/multiplicity pairs.

Monoid under `additiveUnion`.