dependent-map-0.3: Dependent finite maps (partial dependent products)

Data.Dependent.Map

Synopsis

# Documentation

data DMap k f Source #

Dependent maps: k is a GADT-like thing with a facility for rediscovering its type parameter, elements of which function as identifiers tagged with the type of the thing they identify. Real GADTs are one useful instantiation of k, as are Tags from Data.Unique.Tag in the 'prim-uniq' package.

Semantically, DMap k f is equivalent to a set of DSum k f where no two elements have the same tag.

More informally, DMap is to dependent products as Map is to (->). Thus it could also be thought of as a partial (in the sense of "partial function") dependent product.

Instances

data DSum (tag :: k -> Type) (f :: k -> Type) :: forall k. (k -> Type) -> (k -> Type) -> Type where #

A basic dependent sum type; the first component is a tag that specifies the type of the second; for example, think of a GADT such as:

data Tag a where
AString :: Tag String
AnInt   :: Tag Int

Then, we have the following valid expressions of type Applicative f => DSum Tag f:

AString ==> "hello!"
AnInt   ==> 42

And we can write functions that consume DSum Tag f values by matching, such as:

toString :: DSum Tag Identity -> String
toString (AString :=> Identity str) = str
toString (AnInt   :=> Identity int) = show int

By analogy to the (key => value) construction for dictionary entries in many dynamic languages, we use (key :=> value) as the constructor for dependent sums. The :=> and ==> operators have very low precedence and bind to the right, so if the Tag GADT is extended with an additional constructor Rec :: Tag (DSum Tag Identity), then Rec ==> AnInt ==> 3 + 4 is parsed as would be expected (Rec ==> (AnInt ==> (3 + 4))) and has type DSum Identity Tag. Its precedence is just above that of $, so foo bar$ AString ==> "eep" is equivalent to foo bar (AString ==> "eep").

Constructors

 (:=>) :: forall k (tag :: k -> Type) (f :: k -> Type) (a :: k). !(tag a) -> f a -> DSum tag f infixr 1
Instances

data Some (tag :: k -> Type) :: forall k. (k -> Type) -> Type where #

Existential. This is type is useful to hide GADTs' parameters.

>>> data Tag :: * -> * where TagInt :: Tag Int; TagBool :: Tag Bool
>>> instance GShow Tag where gshowsPrec _ TagInt = showString "TagInt"; gshowsPrec _ TagBool = showString "TagBool"


You can either use PatternSynonyms

>>> let x = Some TagInt
>>> x
Some TagInt

>>> case x of { Some TagInt -> "I"; Some TagBool -> "B" } :: String
"I"


or you can use functions

>>> let y = mkSome TagBool
>>> y
Some TagBool


# Combine

## Union

union :: GCompare k => DMap k f -> DMap k f -> DMap k f Source #

O(m*log(n/m + 1)), m <= n. The expression (union t1 t2) takes the left-biased union of t1 and t2. It prefers t1 when duplicate keys are encountered, i.e. (union == unionWith const).

unionWithKey :: GCompare k => (forall v. k v -> f v -> f v -> f v) -> DMap k f -> DMap k f -> DMap k f Source #

O(n+m). Union with a combining function.

unions :: GCompare k => [DMap k f] -> DMap k f Source #

The union of a list of maps: (unions == foldl union empty).

unionsWithKey :: GCompare k => (forall v. k v -> f v -> f v -> f v) -> [DMap k f] -> DMap k f Source #

The union of a list of maps, with a combining operation: (unionsWithKey f == foldl (unionWithKey f) empty).

## Difference

difference :: GCompare k => DMap k f -> DMap k g -> DMap k f Source #

O(m * log (n/m + 1)), m <= n. Difference of two maps. Return elements of the first map not existing in the second map.

differenceWithKey :: GCompare k => (forall v. k v -> f v -> g v -> Maybe (f v)) -> DMap k f -> DMap k g -> DMap k f Source #

O(n+m). Difference with a combining function. When two equal keys are encountered, the combining function is applied to the key and both values. If it returns Nothing, the element is discarded (proper set difference). If it returns (Just y), the element is updated with a new value y.

## Intersection

intersection :: GCompare k => DMap k f -> DMap k f -> DMap k f Source #

O(m * log (n/m + 1), m <= n. Intersection of two maps. Return data in the first map for the keys existing in both maps. (intersection m1 m2 == intersectionWith const m1 m2).

intersectionWithKey :: GCompare k => (forall v. k v -> f v -> g v -> h v) -> DMap k f -> DMap k g -> DMap k h Source #

O(m * log (n/m + 1), m <= n. Intersection with a combining function.

# Traversal

## Map

map :: (forall v. f v -> g v) -> DMap k f -> DMap k g Source #

O(n). Map a function over all values in the map.

mapWithKey :: (forall v. k v -> f v -> g v) -> DMap k f -> DMap k g Source #

O(n). Map a function over all values in the map.

traverseWithKey :: Applicative t => (forall v. k v -> f v -> t (g v)) -> DMap k f -> t (DMap k g) Source #

O(n). traverseWithKey f m == fromList $traverse ((k, v) -> (,) k$ f k v) (toList m) That is, behaves exactly like a regular traverse except that the traversing function also has access to the key associated with a value.

mapAccumLWithKey :: (forall v. a -> k v -> f v -> (a, g v)) -> a -> DMap k f -> (a, DMap k g) Source #

O(n). The function mapAccumLWithKey threads an accumulating argument throught the map in ascending order of keys.

mapAccumRWithKey :: (forall v. a -> k v -> f v -> (a, g v)) -> a -> DMap k f -> (a, DMap k g) Source #

O(n). The function mapAccumRWithKey threads an accumulating argument through the map in descending order of keys.

mapKeysWith :: GCompare k2 => (forall v. k2 v -> f v -> f v -> f v) -> (forall v. k1 v -> k2 v) -> DMap k1 f -> DMap k2 f Source #

O(n*log n). mapKeysWith c f s is the map obtained by applying f to each key of s.

The size of the result may be smaller if f maps two or more distinct keys to the same new key. In this case the associated values will be combined using c.

mapKeysMonotonic :: (forall v. k1 v -> k2 v) -> DMap k1 f -> DMap k2 f Source #

O(n). mapKeysMonotonic f s == mapKeys f s, but works only when f is strictly monotonic. That is, for any values x and y, if x < y then f x < f y. The precondition is not checked. Semi-formally, we have:

and [x < y ==> f x < f y | x <- ls, y <- ls]
==> mapKeysMonotonic f s == mapKeys f s
where ls = keys s

This means that f maps distinct original keys to distinct resulting keys. This function has better performance than mapKeys.

## Fold

foldWithKey :: (forall v. k v -> f v -> b -> b) -> b -> DMap k f -> b Source #

O(n). Fold the keys and values in the map, such that foldWithKey f z == foldr (uncurry f) z . toAscList.

This is identical to foldrWithKey, and you should use that one instead of this one. This name is kept for backward compatibility.

foldrWithKey :: (forall v. k v -> f v -> b -> b) -> b -> DMap k f -> b Source #

O(n). Post-order fold. The function will be applied from the lowest value to the highest.

foldlWithKey :: (forall v. b -> k v -> f v -> b) -> b -> DMap k f -> b Source #

O(n). Pre-order fold. The function will be applied from the highest value to the lowest.

# Conversion

keys :: DMap k f -> [Some k] Source #

O(n). Return all keys of the map in ascending order.

keys (fromList [(5,"a"), (3,"b")]) == [3,5]
keys empty == []

assocs :: DMap k f -> [DSum k f] Source #

O(n). Return all key/value pairs in the map in ascending key order.

## Lists

toList :: DMap k f -> [DSum k f] Source #

O(n). Convert to a list of key/value pairs.

fromList :: GCompare k => [DSum k f] -> DMap k f Source #

O(n*log n). Build a map from a list of key/value pairs. See also fromAscList. If the list contains more than one value for the same key, the last value for the key is retained.

fromListWithKey :: GCompare k => (forall v. k v -> f v -> f v -> f v) -> [DSum k f] -> DMap k f Source #

O(n*log n). Build a map from a list of key/value pairs with a combining function. See also fromAscListWithKey.

## Ordered lists

toAscList :: DMap k f -> [DSum k f] Source #

O(n). Convert to an ascending list.

toDescList :: DMap k f -> [DSum k f] Source #

O(n). Convert to a descending list.

fromAscList :: GEq k => [DSum k f] -> DMap k f Source #

O(n). Build a map from an ascending list in linear time. The precondition (input list is ascending) is not checked.

fromAscListWithKey :: GEq k => (forall v. k v -> f v -> f v -> f v) -> [DSum k f] -> DMap k f Source #

O(n). Build a map from an ascending list in linear time with a combining function for equal keys. The precondition (input list is ascending) is not checked.

fromDistinctAscList :: [DSum k f] -> DMap k f Source #

O(n). Build a map from an ascending list of distinct elements in linear time. The precondition is not checked.

# Filter

filter :: (a -> Bool) -> [a] -> [a] #

filter, applied to a predicate and a list, returns the list of those elements that satisfy the predicate; i.e.,

filter p xs = [ x | x <- xs, p x]

filterWithKey :: GCompare k => (forall v. k v -> f v -> Bool) -> DMap k f -> DMap k f Source #

O(n). Filter all keys/values that satisfy the predicate.

partitionWithKey :: GCompare k => (forall v. k v -> f v -> Bool) -> DMap k f -> (DMap k f, DMap k f) Source #

O(n). Partition the map according to a predicate. The first map contains all elements that satisfy the predicate, the second all elements that fail the predicate. See also split.

mapMaybe :: GCompare k => (forall v. f v -> Maybe (g v)) -> DMap k f -> DMap k g Source #

O(n). Map values and collect the Just results.

mapMaybeWithKey :: GCompare k => (forall v. k v -> f v -> Maybe (g v)) -> DMap k f -> DMap k g Source #

O(n). Map keys/values and collect the Just results.

mapEitherWithKey :: GCompare k => (forall v. k v -> f v -> Either (g v) (h v)) -> DMap k f -> (DMap k g, DMap k h) Source #

O(n). Map keys/values and separate the Left and Right results.

split :: forall k f v. GCompare k => k v -> DMap k f -> (DMap k f, DMap k f) Source #

O(log n). The expression (split k map) is a pair (map1,map2) where the keys in map1 are smaller than k and the keys in map2 larger than k. Any key equal to k is found in neither map1 nor map2.

splitLookup :: forall k f v. GCompare k => k v -> DMap k f -> (DMap k f, Maybe (f v), DMap k f) Source #

O(log n). The expression (splitLookup k map) splits a map just like split but also returns lookup k map.

# Submap

isSubmapOf :: forall k f. (GCompare k, Has' Eq k f) => DMap k f -> DMap k f -> Bool Source #

O(n+m). This function is defined as (isSubmapOf = isSubmapOfBy eqTagged)).

isSubmapOfBy :: GCompare k => (forall v. k v -> k v -> f v -> g v -> Bool) -> DMap k f -> DMap k g -> Bool Source #

O(n+m). The expression (isSubmapOfBy f t1 t2) returns True if all keys in t1 are in tree t2, and when f returns True when applied to their respective keys and values.

isProperSubmapOf :: forall k f. (GCompare k, Has' Eq k f) => DMap k f -> DMap k f -> Bool Source #

O(n+m). Is this a proper submap? (ie. a submap but not equal). Defined as (isProperSubmapOf = isProperSubmapOfBy eqTagged).

isProperSubmapOfBy :: GCompare k => (forall v. k v -> k v -> f v -> g v -> Bool) -> DMap k f -> DMap k g -> Bool Source #

O(n+m). Is this a proper submap? (ie. a submap but not equal). The expression (isProperSubmapOfBy f m1 m2) returns True when m1 and m2 are not equal, all keys in m1 are in m2, and when f returns True when applied to their respective keys and values.

# Indexed

lookupIndex :: forall k f v. GCompare k => k v -> DMap k f -> Maybe Int Source #

O(log n). Lookup the index of a key. The index is a number from 0 up to, but not including, the size of the map.

findIndex :: GCompare k => k v -> DMap k f -> Int Source #

O(log n). Return the index of a key. The index is a number from 0 up to, but not including, the size of the map. Calls error when the key is not a member of the map.

elemAt :: Int -> DMap k f -> DSum k f Source #

O(log n). Retrieve an element by index. Calls error when an invalid index is used.

updateAt :: (forall v. k v -> f v -> Maybe (f v)) -> Int -> DMap k f -> DMap k f Source #

O(log n). Update the element at index. Does nothing when an invalid index is used.

deleteAt :: Int -> DMap k f -> DMap k f Source #

O(log n). Delete the element at index. Defined as (deleteAt i map = updateAt (k x -> Nothing) i map).

# Min/Max

findMin :: DMap k f -> DSum k f Source #

O(log n). The minimal key of the map. Calls error is the map is empty.

findMax :: DMap k f -> DSum k f Source #

O(log n). The maximal key of the map. Calls error is the map is empty.

lookupMin :: DMap k f -> Maybe (DSum k f) Source #

lookupMax :: DMap k f -> Maybe (DSum k f) Source #

deleteMin :: DMap k f -> DMap k f Source #

O(log n). Delete the minimal key. Returns an empty map if the map is empty.

deleteMax :: DMap k f -> DMap k f Source #

O(log n). Delete the maximal key. Returns an empty map if the map is empty.

deleteFindMin :: DMap k f -> (DSum k f, DMap k f) Source #

O(log n). Delete and find the minimal element.

deleteFindMin (fromList [(5,"a"), (3,"b"), (10,"c")]) == ((3,"b"), fromList[(5,"a"), (10,"c")])
deleteFindMin                                            Error: can not return the minimal element of an empty map

deleteFindMax :: DMap k f -> (DSum k f, DMap k f) Source #

O(log n). Delete and find the maximal element.

deleteFindMax (fromList [(5,"a"), (3,"b"), (10,"c")]) == ((10,"c"), fromList [(3,"b"), (5,"a")])
deleteFindMax empty                                      Error: can not return the maximal element of an empty map

updateMinWithKey :: (forall v. k v -> f v -> Maybe (f v)) -> DMap k f -> DMap k f Source #

O(log n). Update the value at the minimal key.

updateMaxWithKey :: (forall v. k v -> f v -> Maybe (f v)) -> DMap k f -> DMap k f Source #

O(log n). Update the value at the maximal key.

minViewWithKey :: forall k f. DMap k f -> Maybe (DSum k f, DMap k f) Source #

O(log n). Retrieves the minimal (key :=> value) entry of the map, and the map stripped of that element, or Nothing if passed an empty map.

maxViewWithKey :: forall k f. DMap k f -> Maybe (DSum k f, DMap k f) Source #

O(log n). Retrieves the maximal (key :=> value) entry of the map, and the map stripped of that element, or Nothing if passed an empty map.

# Debugging

showTree :: (GShow k, Has' Show k f) => DMap k f -> String Source #

O(n). Show the tree that implements the map. The tree is shown in a compressed, hanging format. See showTreeWith.

showTreeWith :: (forall v. k v -> f v -> String) -> Bool -> Bool -> DMap k f -> String Source #

O(n). The expression (showTreeWith showelem hang wide map) shows the tree that implements the map. Elements are shown using the showElem function. If hang is True, a hanging tree is shown otherwise a rotated tree is shown. If wide is True, an extra wide version is shown.

valid :: GCompare k => DMap k f -> Bool Source #

O(n). Test if the internal map structure is valid.