Data.Edison.Coll

The collection abstraction includes sets, bags and priority queues (heaps). Collections are defined in Edison as a set of eight classes.

All collections assume at least an equality relation of elements, and may also assume an ordering relation.

The hierarchy contains a root class CollX together with seven subclasses satisfying one or more of three common sub-properties:

• Uniqueness Each element in the collection is unique (no two elements in the collection are equal). These subclasses, indicated by the name Set, represent sets rather than bags (multi-sets).
• Ordering The elements have a total ordering and it is possible to process the elements in non-decreasing order. These subclasses, indicates by the Ord prefix, typically represent either priority queues (heaps) or sets/bags implemented as binary search trees.
• Observability An observable collection is one in which it is possible to view the elements in a collection. The X suffix indicates a lack of observability. This property is discussed is greater detail below.

Because collections encompass a wide range of abstractions, there is no single name that is suitable for all collection type constructors. However, most modules implementing collections will define a type constructor named either Bag, Set, or Heap.

Notes on observability

Note that the equality relation defined by the Eq class is not necessarily true equality. Very often it is merely an equivalence relation, where two equivalent values may be distinguishable by other means. For example, we might consider two binary trees to be equal if they contain the same elements, even if their shapes are different.

Because of this phenomenon, implementations of observable collections (ie, collections where it is possible to inspect the elements) are rather constrained. Such an implementation must retain the actual elements that were inserted. For example, it is not possible in general to represent an observable bag as a finite map from elements to counts, because even if we know that a given bag contains, say, three elements from some equivalence class, we do not necessarily know which three.

On the other hand, implementations of non-observable collections have much greater freedom to choose abstract representations of each equivalence class. For example, representing a bag as a finite map from elements to counts works fine if we never need to know which representatives from an equivalence class are actually present. As another example, consider the UniqueHash class defined in Data.Edison.Prelude. If we know that the hash function yields a unique integer for each equivalence class, then we can represent a collection of hashable elements simply as a collection of integers. With such a representation, we can still do many useful things like testing for membership; we just can't support functions like fold or filter that require the elements themselves, rather than the hashed values.

Superclass aliases

Monoid

empty :: CollX c a => c

The empty collection. Equivalant to mempty from the Monoid instance.

This function is always unambiguous.

union :: CollX c a => c -> c -> c

Merge two collections. For sets, it is unspecified which element is kept in the case of duplicates. Equivalant to mappend from the Monoid instance.

This function is ambiguous at set types if the sets are not disjoint. Otherwise it is unambiguous.

Non-observable collections

class (Eq a, Monoid c) => CollX c a | c -> a where

This is the root class of the collection hierarchy. However, it is perfectly adequate for many applications that use sets or bags.

Methods

singleton :: a -> c

create a singleton collection

This function is always unambiguous.

fromSeq :: Sequence seq => seq a -> c

Convert a sequence to a collection. For sets, it is unspecified which element is kept in case of duplicates.

This function is ambiguous at set types if more than one equivalent item is in the sequence. Otherwise it is unambiguous.

unionSeq :: Sequence seq => seq c -> c

Merge a sequence of collections. For sets, it is unspecified which element is kept in the case of duplicates.

This function is ambiguous at set types if the sets in the sequence are not mutually disjoint. Otherwise it is unambiguous.

insert :: a -> c -> c

Insert an element into a collection. For sets, if an equal element is already in the set, the newly inserted element is kept, and the old element is discarded.

This function is always unambiguous.

insertSeq :: Sequence seq => seq a -> c -> c

Insert a sequence of elements into a collection. For sets, the behavior with regard to multiple equal elements is unspecified.

This function is ambiguous at set types if the sequence contains more than one equivalent item or an item which is already in the set. Otherwise it is unambiguous.

delete :: a -> c -> c

Delete a single occurrence of the given element from a collection. For bags, it is unspecified which element will be deleted.

This function is ambiguous at bag types if more than one item exists in the bag equivalent to the given item. Otherwise it is unambiguous.

deleteAll :: a -> c -> c

Delete all occurrences of an element from a collection. For sets this operation is identical to delete.

This function is always unambiguous.

deleteSeq :: Sequence seq => seq a -> c -> c

Delete a single occurrence of each of the given elements from a collection. For bags, there may be multiple occurrences of a given element in the collection, in which case it is unspecified which is deleted.

This function is ambiguous at bag types if more than one item exists in the bag equivalent to any item in the list and the number of equivalent occurrences of that item in the sequence is less than the number of occurrences in the bag. Otherwise it is unambiguous.

null :: c -> Bool

Test whether the collection is empty.

Axioms:

• null xs = (size xs == 0)

This function is always unambiguous.

size :: c -> Int

Return the number of elements in the collection.

This function is always unambiguous.

member :: a -> c -> Bool

Test whether the given element is in the collection.

Axioms:

• member x xs = (count x xs > 0)

This function is always unambiguous.

count :: a -> c -> Int

Count how many copies of the given element are in the collection. For sets, this will always return 0 or 1.

This function is always unambiguous.

strict :: c -> c

Semanticly, this function is a partial identity function. If the datastructure is infinite in size or contains exceptions or non-termination in the structure itself, then strict will result in bottom. Operationally, this function walks the datastructure forcing any closures. In many collections, the collction "shape" depends on the value of the elemnts; in such cases, the values of the elements will be forced to the extent necessary to force the structure of the collection, but no further.

This function is always unambiguous.

structuralInvariant :: c -> Bool

A method to facilitate unit testing. Returns True if the structural invariants of the implementation hold for the given collection. If this function returns False, it represents a bug; generally, either the implementation itself is flawed, or an unsafe operation has been used while violating the preconditions.

instanceName :: c -> String

The name of the module implementing c

class (CollX c a, Ord a) => OrdCollX c a | c -> a where

Collections for which the elements have an ordering relation.

Methods

deleteMin :: c -> c

Delete the minimum element from the collection. If there is more than one minimum, it is unspecified which is deleted. If the collection is empty, it will be returned unchanged.

This function is ambiguous at bag types if more than one minimum element exists in the bag. Otherwise it is unambiguous.

deleteMax :: c -> c

Delete the maximum element from the collection. If there is more than one maximum, it is unspecified which is deleted. If the collection is empty, it will be returned unchanged.

This function is ambiguous at bag types if more than one maximum element exists in the bag. Otherwise it is unambiguous.

unsafeInsertMin :: a -> c -> c

Insert an element into a collection which is guaranteed to be <= any existing elements in the collection. For sets, the precondition is strengthened to <.

This function is unambiguous, under the above preconditions.

unsafeInsertMax :: a -> c -> c

Insert an element into a collection which is guaranteed to be >= any existing elements in the collection. For sets, the precondition is strengthened to >.

This function is unambiguous, under the above preconditions.

unsafeFromOrdSeq :: Sequence seq => seq a -> c

Convert a sequence in non-decreasing order into a collection. For sets, the sequence must be in increasing order.

This function is unambiguous, under the above preconditions.

unsafeAppend :: c -> c -> c

Union two collections where every element in the first collection is <= every element in the second collection. For sets, this precondition is strengthened to <.

This function is unambiguous, under the above preconditions.

filterLT :: a -> c -> c

Extract the sub-collection of elements < the given element.

Axioms:

• filterLT x xs = filter (< x) xs

This function is always unambiguous.

filterLE :: a -> c -> c

Extract the sub-collection of elements <= the given element.

Axioms:

• filterLE x xs = filter (<= x) xs

This function is always unambiguous.

filterGT :: a -> c -> c

Extract the sub-collection of elements > the given element.

Axioms:

• filterGT x xs = filter (> x) xs

This function is always unambiguous.

filterGE :: a -> c -> c

Extract the sub-collection of elements >= the given element.

Axioms:

• filterGE x xs = filter (>= x) xs

This function is always unambiguous.

partitionLT_GE :: a -> c -> (c, c)

Split a collection into those elements < a given element and those >=.

Axioms:

• partitionLT_GE xs = partition (<) xs

This function is always unambiguous.

partitionLE_GT :: a -> c -> (c, c)

Split a collection into those elements <= a given element and those >.

Axioms:

• partitionLE_GT xs = partition (<=) xs

This function is always unambiguous.

partitionLT_GT :: a -> c -> (c, c)

Split a collection into those elements < a given element and those >. All elements equal to the given element are discarded.

Axioms:

• partitionLT_GT x xs = (filterLT x xs,filterGT x xs)

This function is always unambiguous.

class CollX c a => SetX c a | c -> a where

A collection where the set property is maintained; that is, a set contains at most one element of the equivalence class formed by the Eq instance on the elements.

Methods

intersection :: c -> c -> c

Computes the intersection of two sets. It is unspecified which element is kept when equal elements appear in each set.

This function is ambiguous, except when the sets are disjoint.

difference :: c -> c -> c

Computes the difference of two sets; that is, all elements in the first set which are not in the second set.

This function is always unambiguous.

symmetricDifference :: c -> c -> c

Computes the symmetric difference of two sets; that is, all elements which appear in exactily one of the two sets.

This function is always unambiguous.

properSubset :: c -> c -> Bool

Test whether the first set is a proper subset of the second set; that is, if every element in the first set is also a member of the second set AND there exists some element in the second set which is not present in the first.

This function is always unambiguous.

subset :: c -> c -> Bool

Test whether the first set is a subset of the second set; that is, if every element in the first set is also a member of the second set.

This function is always unambiguous.

class (OrdCollX c a, SetX c a) => OrdSetX c a | c -> a

Sets where the elements also have an ordering relation. This class contains no methods; it is only an abbreviation for the context (OrdCollX c a,SetX c a).

Observable collections

class CollX c a => Coll c a | c -> a where

Collections with observable elements. See the module documentation for comments on observability.

Methods

toSeq :: Sequence seq => c -> seq a

List the elements of the collection in an unspecified order.

This function is ambiguous iff the collection contains more than one element.

lookup :: a -> c -> a

Lookup one element equal to the given element. If no elements exist in the collection equal to the given element, an error is signaled. If multiple copies of the given element exist in the collection, it is unspecified which is returned.

This function is ambiguous at bag types, when more than one element equivalent to the given item is in the bag. Otherwise it is unambiguous.

lookupM :: Monad m => a -> c -> m a

Lookup one element equal to the given element. If no elements exist in the collection equal to the given element, fail is called. If multiple copies of the given element exist in the collection, it is unspecified which is returned.

This function is ambiguous at bag types, when more than one element equivalent to the given item is in the bag. Otherwise it is unambiguous.

lookupAll :: Sequence seq => a -> c -> seq a

Return a sequence containing all elements in the collection equal to the given element in an unspecified order.

This function is ambiguous at bag types, when more than one element equivalent to the given item is in the bag. Otherwise it is unambiguous.

lookupWithDefault

Arguments

:: a

default element

-> a

element to lookup

-> c

collection

-> a

Lookup one element equal to the (second) given element in the collection. If no elements exist in the collection equal to the given element, then the default element is returned.

This function is ambiguous at bag types, when more than one element equivalent to the given item is in the bag. Otherwise it is unambiguous.

fold :: (a -> b -> b) -> b -> c -> b

Fold over all the elements in a collection in an unspecified order.

fold f is unambiguous iff f is fold-commutative.

fold' :: (a -> b -> b) -> b -> c -> b

A strict variant of fold.

fold' f is unambiguous iff f is fold-commutative.

fold1 :: (a -> a -> a) -> c -> a

Fold over all the elements in a collection in an unspecified order. An error is signaled if the collection is empty.

fold1 f is unambiguous iff f is fold-commutative.

fold1' :: (a -> a -> a) -> c -> a

A strict variant of fold1.

fold1' f is unambiguous iff f is fold-commutative.

filter :: (a -> Bool) -> c -> c

Remove all elements not satisfying the predicate.

This function is always unambiguous.

partition :: (a -> Bool) -> c -> (c, c)

Returns two collections, the first containing all the elements satisfying the predicate, and the second containing all the elements not satisfying the predicate.

This function is always unambiguous.

strictWith :: (a -> b) -> c -> c

Similar to strict, this function walks the datastructure forcing closures. However, strictWith will additionally apply the given function to the collection elements, force the result using seq, and then ignore it. This function can be used to perform various levels of forcing on the sequence elements. In particular:

 strictWith id xs

will force the spine of the datastructure and reduce each element to WHNF.

This function is always unambiguous.

class (Coll c a, OrdCollX c a) => OrdColl c a | c -> a where

Collections with observable elements where the elements additionally have an ordering relation. See the module documentation for comments on observability.

Methods

minView :: Monad m => c -> m (a, c)

Return the minimum element in the collection, together with the collection without that element. If there are multiple copies of the minimum element, it is unspecified which is chosen. Note that minView, minElem, and deleteMin may make different choices. Calls fail if the collection is empty.

This function is ambiguous at bag types, if more than one minimum element exists in the bag. Otherwise, it is unambiguous.

minElem :: c -> a

Return the minimum element in the collection. If there are multiple copies of the minimum element, it is unspecified which is chosen. Note that minView, minElem, and deleteMin may make different choices. Signals an error if the collection is empty.

This function is ambiguous at bag types, if more than one minimum element exists in the bag. Otherwise, it is unambiguous.

maxView :: Monad m => c -> m (a, c)

Return the maximum element in the collection, together with the collection without that element. If there are multiple copies of the maximum element, it is unspecified which is chosen. Note that maxView, maxElem and deleteMax may make different choices. Calls fail if the collection is empty.

This function is ambiguous at bag types, if more than one maximum element exists in the bag. Otherwise, it is unambiguous.

maxElem :: c -> a

Return the maximum element in the collection. If there are multiple copies of the maximum element, it is unspecified which is chosen. Note that maxView, maxElem and deleteMax may make different choices. Signals an error if the collection is empty.

This function is ambiguous at bag types, if more than one maximum element exists in the bag. Otherwise, it is unambiguous.

foldr :: (a -> b -> b) -> b -> c -> b

Fold across the elements in non-decreasing order with right associativity. (For sets, this will always be increasing order)

This function is unambiguous if the combining function is fold-commutative, at all set types, and at bag types where no two equivalent elements exist in the bag. Otherwise it is ambiguous.

foldr' :: (a -> b -> b) -> b -> c -> b

A strict variant of foldr.

This function is unambiguous if the combining function is fold-commutative, at all set types, and at bag types where no two equivalent elements exist in the bag. Otherwise it is ambiguous.

foldl :: (b -> a -> b) -> b -> c -> b

Fold across the elements in non-decreasing order with left associativity. (For sets, this will always be increasing order)

This function is unambiguous if the combining function is fold-commutative, at all set types, and at bag types where no two equivalent elements exist in the bag. Otherwise it is ambiguous.

foldl' :: (b -> a -> b) -> b -> c -> b

A strict variant of foldl.

This function is unambiguous if the combining function is fold-commutative, at all set types, and at bag types where no two equivalent elements exist in the bag. Otherwise it is ambiguous.

foldr1 :: (a -> a -> a) -> c -> a

Fold across the elements in non-decreasing order with right associativity, or signal an error if the collection is empty. (For sets, this will always be increasing order)

This function is unambiguous if the combining function is fold-commutative, at all set types, and at bag types where no two equivalent elements exist in the bag. Otherwise it is ambiguous.

foldr1' :: (a -> a -> a) -> c -> a

A strict variant of foldr1.

This function is unambiguous if the combining function is fold-commutative, at all set types, and at bag types where no two equivalent elements exist in the bag. Otherwise it is ambiguous.

foldl1 :: (a -> a -> a) -> c -> a

Fold across the elements in non-decreasing order with left associativity, or signal an error if the collection is empty. (For sets, this will always be increasing order)

This function is unambiguous if the combining function is fold-commutative, at all set types, and at bag types where no two equivalent elements exist in the bag. Otherwise it is ambiguous.

foldl1' :: (a -> a -> a) -> c -> a

A strict variant of foldl1.

This function is unambiguous if the combining function is fold-commutative, at all set types, and at bag types where no two equivalent elements exist in the bag. Otherwise it is ambiguous.

toOrdSeq :: Sequence seq => c -> seq a

List the elements in non-decreasing order. (For sets, this will always be increasing order)

At set types, this function is unambiguous. At bag types, it is unambiguous if no two equivalent elements exist in the bag; otherwise it is ambiguous.

unsafeMapMonotonic :: (a -> a) -> c -> c

Map a monotonic function across all elements of a collection. The function is required to satisfy the following precondition:

 forall x y. x < y ==> f x < f y

This function is unambiguous, under the precondition.

class (Coll c a, SetX c a) => Set c a | c -> a where

Collections with observable elements where the set property is maintained; that is, a set contains at most one element of the equivalence class formed by the Eq instance on the elements.

WARNING: Each of the following \"With\" functions is unsafe. The passed in combining functions are used to choose which element is kept in the case of duplicates. They are required to satisfy the precondition that, given two equal elements, they return a third element equal to the other two. Usually, the combining function just returns its first or second argument, but it can combine elements in non-trivial ways.

The combining function should usually be associative. Where the function involves a sequence of elements, the elements will be combined from left-to-right, but with an unspecified associativity.

For example, if x == y == z, then fromSeqWith (+) [x,y,z] equals either single (x + (y + z)) or single ((x + y) + z)

Methods

fromSeqWith :: Sequence seq => (a -> a -> a) -> seq a -> c

Same as fromSeq but with a combining function to resolve duplicates.

This function is unambiguous under the "with" precondition if the combining function is associative. Otherwise it is ambiguous.

insertWith :: (a -> a -> a) -> a -> c -> c

Same as insert but with a combining function to resolve duplicates.

This function is unambiguous under the "with" precondition.

insertSeqWith :: Sequence seq => (a -> a -> a) -> seq a -> c -> c

Same as insertSeq but with a combining function to resolve duplicates.

This function is unambiguous under the "with" precondition if the combining function is associative. Otherwise it is ambiguous.

unionl :: c -> c -> c

Left biased union.

Axioms:

• unionl = unionWith (\x y -> x)

This function is always unambiguous.

unionr :: c -> c -> c

Right biased union.

Axioms:

• unionr = unionWith (\x y -> y)

This function is always unambiguous.

unionWith :: (a -> a -> a) -> c -> c -> c

Same as union, but with a combining function to resolve duplicates.

This function is unambiguous under the "with" precondition.

unionSeqWith :: Sequence seq => (a -> a -> a) -> seq c -> c

Same as unionSeq, but with a combining function to resolve duplicates.

This function is unambiguous under the "with" precondition if the combining function is associative. Otherwise it is ambiguous.

intersectionWith :: (a -> a -> a) -> c -> c -> c

Same as intersection, but with a combining function to resolve duplicates.

This function is unambiguous under the "with" precondition.

class (OrdColl c a, Set c a) => OrdSet c a | c -> a

Collections with observable elements where the set property is maintained and where additionally, there is an ordering relation on the elements. This class introduces no new methods, and is simply an abbreviation for the context:

(OrdColl c a,Set c a)

Specializations of all the sequence operations to lists

fromList :: CollX c a => [a] -> c

insertList :: CollX c a => [a] -> c -> c

unionList :: CollX c a => [c] -> c

deleteList :: CollX c a => [a] -> c -> c

unsafeFromOrdList :: OrdCollX c a => [a] -> c

toList :: Coll c a => c -> [a]

lookupList :: Coll c a => a -> c -> [a]

toOrdList :: OrdColl c a => c -> [a]

fromListWith :: Set c a => (a -> a -> a) -> [a] -> c

insertListWith :: Set c a => (a -> a -> a) -> [a] -> c -> c

unionListWith :: Set c a => (a -> a -> a) -> [c] -> c