**Map** +containers

*Note:* You should use Data.Map.Strict instead of this module if:
* You will eventually need all the values stored.
* The stored values don't represent large virtual data structures to be lazily computed.
An efficient implementation of ordered maps from keys to values (dictionaries).
These modules are intended to be imported qualified, to avoid name clashes with Prelude functions, e.g.
> import qualified Data.Map as Map
The implementation of Map is based on *size balanced* binary trees (or trees of *bounded balance*) as described by:
* Stephen Adams, "*Efficient sets: a balancing act*", Journal of Functional Programming 3(4):553-562, October 1993, http://www.swiss.ai.mit.edu/~adams/BB/.
* J. Nievergelt and E.M. Reingold, "*Binary search trees of bounded balance*", SIAM journal of computing 2(1), March 1973.
Note that the implementation is *left-biased* -- the elements of a first argument are always preferred to the second, for example in union or insert.
Operation comments contain the operation time complexity in the Big-O notation (http://en.wikipedia.org/wiki/Big_O_notation).
A Map from keys k to values a.

*O(n*min(n,W))*. map f s is the set obtained by applying f to each element of s.
It's worth noting that the size of the result may be smaller if, for some (x,y), x /= y && f x == f y
*O(n)*. Map a function over all values in the map.
> map (++ "x") (fromList [(5,"a"), (3,"b")]) == fromList [(3, "bx"), (5, "ax")]

*O(n)*. Map a function over all values in the map.
> map (++ "x") (fromList [(5,"a"), (3,"b")]) == fromList [(3, "bx"), (5, "ax")]

*O(n*log n)*. map f s is the set obtained by applying f to each element of s.
It's worth noting that the size of the result may be smaller if, for some (x,y), x /= y && f x == f y
*O(n)*. The function mapAccum threads an accumulating argument through the map in ascending order of keys.
> let f a b = (a ++ b, b ++ "X")
> mapAccum f "Everything: " (fromList [(5,"a"), (3,"b")]) == ("Everything: ba", fromList [(3, "bX"), (5, "aX")])
*O(n)*. The function mapAccum threads an accumulating argument through the map in ascending order of keys.
> let f a b = (a ++ b, b ++ "X")
> mapAccum f "Everything: " (fromList [(5,"a"), (3,"b")]) == ("Everything: ba", fromList [(3, "bX"), (5, "aX")])
*O(n)*. The function mapAccumR threads an accumulating argument through the map in descending order of keys.

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

*O(n)*. The function mapAccumWithKey threads an accumulating argument through the map in ascending order of keys.
> let f a k b = (a ++ " " ++ (show k) ++ "-" ++ b, b ++ "X")
> mapAccumWithKey f "Everything:" (fromList [(5,"a"), (3,"b")]) == ("Everything: 3-b 5-a", fromList [(3, "bX"), (5, "aX")])
*O(n)*. The function mapAccumWithKey threads an accumulating argument through the map in ascending order of keys.
> let f a k b = (a ++ " " ++ (show k) ++ "-" ++ b, b ++ "X")
> mapAccumWithKey f "Everything:" (fromList [(5,"a"), (3,"b")]) == ("Everything: 3-b 5-a", fromList [(3, "bX"), (5, "aX")])
*O(n)*. Map values and separate the Left and Right results.
> let f a = if a < "c" then Left a else Right a
> mapEither f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
> == (fromList [(3,"b"), (5,"a")], fromList [(1,"x"), (7,"z")])
>
> mapEither (\ a -> Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
> == (empty, fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
*O(n)*. Map values and separate the Left and Right results.
> let f a = if a < "c" then Left a else Right a
> mapEither f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
> == (fromList [(3,"b"), (5,"a")], fromList [(1,"x"), (7,"z")])
>
> mapEither (\ a -> Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
> == (empty, fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
*O(n)*. Map keys/values and separate the Left and Right results.
> let f k a = if k < 5 then Left (k * 2) else Right (a ++ a)
> mapEitherWithKey f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
> == (fromList [(1,2), (3,6)], fromList [(5,"aa"), (7,"zz")])
>
> mapEitherWithKey (\_ a -> Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
> == (empty, fromList [(1,"x"), (3,"b"), (5,"a"), (7,"z")])
*O(n)*. Map keys/values and separate the Left and Right results.
> let f k a = if k < 5 then Left (k * 2) else Right (a ++ a)
> mapEitherWithKey f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
> == (fromList [(1,2), (3,6)], fromList [(5,"aa"), (7,"zz")])
>
> mapEitherWithKey (\_ a -> Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
> == (empty, fromList [(1,"x"), (3,"b"), (5,"a"), (7,"z")])
*O(n*min(n,W))*. mapKeys 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 value at the greatest of the original keys is retained.
> mapKeys (+ 1) (fromList [(5,"a"), (3,"b")]) == fromList [(4, "b"), (6, "a")]
> mapKeys (\ _ -> 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "c"
> mapKeys (\ _ -> 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "c"
*O(n*log n)*. mapKeys 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 value at the greatest of the original keys is retained.
> mapKeys (+ 1) (fromList [(5,"a"), (3,"b")]) == fromList [(4, "b"), (6, "a")]
> mapKeys (\ _ -> 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "c"
> mapKeys (\ _ -> 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "c"
*O(n*min(n,W))*. 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
>
This means that f maps distinct original keys to distinct resulting keys. This function has slightly better performance than mapKeys.
> mapKeysMonotonic (\ k -> k * 2) (fromList [(5,"a"), (3,"b")]) == fromList [(6, "b"), (10, "a")]
*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
>
This means that f maps distinct original keys to distinct resulting keys. This function has better performance than mapKeys.
> mapKeysMonotonic (\ k -> k * 2) (fromList [(5,"a"), (3,"b")]) == fromList [(6, "b"), (10, "a")]
> valid (mapKeysMonotonic (\ k -> k * 2) (fromList [(5,"a"), (3,"b")])) == True
> valid (mapKeysMonotonic (\ _ -> 1) (fromList [(5,"a"), (3,"b")])) == False
*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.
> mapKeysWith (++) (\ _ -> 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "cdab"
> mapKeysWith (++) (\ _ -> 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "cdab"
*O(n*min(n,W))*. 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.
> mapKeysWith (++) (\ _ -> 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "cdab"
> mapKeysWith (++) (\ _ -> 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "cdab"
*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.
> mapKeysWith (++) (\ _ -> 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "cdab"
> mapKeysWith (++) (\ _ -> 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "cdab"
*O(n)*. Map values and collect the Just results.
> let f x = if x == "a" then Just "new a" else Nothing
> mapMaybe f (fromList [(5,"a"), (3,"b")]) == singleton 5 "new a"
*O(n)*. Map values and collect the Just results.
> let f x = if x == "a" then Just "new a" else Nothing
> mapMaybe f (fromList [(5,"a"), (3,"b")]) == singleton 5 "new a"
*O(n)*. Map keys/values and collect the Just results.
> let f k _ = if k < 5 then Just ("key (:) " ++ (show k)) else Nothing
> mapMaybeWithKey f (fromList [(5,"a"), (3,"b")]) == singleton 3 "key (:) 3"
*O(n)*. Map keys/values and collect the Just results.
> let f k _ = if k < 5 then Just ("key (:) " ++ (show k)) else Nothing
> mapMaybeWithKey f (fromList [(5,"a"), (3,"b")]) == singleton 3 "key (:) 3"
*O(n)*. The
mapMonotonic f s == map f s, but works only when f is monotonic. *The precondition is not checked.* Semi-formally, we have:
> and [x < y ==> f x < f y | x <- ls, y <- ls]
> ==> mapMonotonic f s == map f s
>
A generalization of fmap, mapWithIndex takes a mapping function that also depends on the element's index, and applies it to every element in the sequence.
*O(n)*. Map a function over all values in the map.
> let f key x = (show key) ++ ":" ++ x
> mapWithKey f (fromList [(5,"a"), (3,"b")]) == fromList [(3, "3:b"), (5, "5:a")]

*O(n)*. Map a function over all values in the map.
> let f key x = (show key) ++ ":" ++ x
> mapWithKey f (fromList [(5,"a"), (3,"b")]) == fromList [(3, "3:b"), (5, "5:a")]

An efficient implementation of maps from integer keys to values (dictionaries).
This module re-exports the value lazy Data.IntMap.Lazy API, plus several deprecated value strict functions. Please note that these functions have different strictness properties than those in Data.IntMap.Strict: they only evaluate the result of the combining function. For example, the default value to insertWith' is only evaluated if the combining function is called and uses it.
These modules are intended to be imported qualified, to avoid name clashes with Prelude functions, e.g.
> import Data.IntMap (IntMap)
> import qualified Data.IntMap as IntMap
The implementation is based on *big-endian patricia trees*. This data structure performs especially well on binary operations like union and intersection. However, my benchmarks show that it is also (much) faster on insertions and deletions when compared to a generic size-balanced map implementation (see Data.Map).
* Chris Okasaki and Andy Gill, "*Fast Mergeable Integer Maps*", Workshop on ML, September 1998, pages 77-86, http://citeseer.ist.psu.edu/okasaki98fast.html
* D.R. Morrison, "/PATRICIA -- Practical Algorithm To Retrieve Information Coded In Alphanumeric/", Journal of the ACM, 15(4), October 1968, pages 514-534.
Operation comments contain the operation time complexity in the Big-O notation http://en.wikipedia.org/wiki/Big_O_notation. Many operations have a worst-case complexity of *O(min(n,W))*. This means that the operation can become linear in the number of elements with a maximum of *W* -- the number of bits in an Int (32 or 64).
A map of integers to values a.

*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 values. For example, the following expressions are all True:
> isProperSubmapOfBy (==) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
> isProperSubmapOfBy (<=) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
But the following are all False:
> isProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)])
> isProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)])
> isProperSubmapOfBy (<) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
*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 values. For example, the following expressions are all True:
> isProperSubmapOfBy (==) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
> isProperSubmapOfBy (<=) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
But the following are all False:
> isProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)])
> isProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)])
> isProperSubmapOfBy (<) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
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