Data.Heap

An efficient, asymptotically optimal, implementation of a priority queues extended with support for efficient size, and Data.Foldable

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

  import Data.Heap (Heap)
  import qualified Data.Heap as Heap

The implementation of Heap is based on bootstrapped skew binomial heaps as described by:

• G. Brodal and C. Okasaki , "Optimal Purely Functional Priority Queues", Journal of Functional Programming 6:839-857 (1996), http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.48.973

All time bounds are worst-case.

Heap Type

data Heap a

A min-heap of values a.

Instances

Typeable1 Heap

Foldable Heap

Eq (Heap a)

(Ord a, Data a) => Data (Heap a)

Ord (Heap a)

(Ord a, Read a) => Read (Heap a)

Show a => Show (Heap a)

Monoid (Heap a)

Entry type

data Entry p a

Constructors

Entry

Fields

priority :: p

payload :: a

Instances

Typeable2 Entry

Functor (Entry p)

Foldable (Entry p)

Traversable (Entry p)

Eq p => Eq (Entry p a)

(Data p, Data a) => Data (Entry p a)

Ord p => Ord (Entry p a)

(Read p, Read a) => Read (Entry p a)

(Show p, Show a) => Show (Entry p a)

Basic functions

empty :: Heap a

O(1). The empty heap

 empty == fromList []
 size empty == 0

null :: Heap a -> Bool

O(1). Is the heap empty?

 Data.Heap.null empty         == True
 Data.Heap.null (singleton 1) == False

size :: Heap a -> Int

O(1). The number of elements in the heap.

 size empty == 0
 size (singleton 1) == 1
 size (fromList [4,1,2]) == 3

singleton :: Ord a => a -> Heap a

O(1). A heap with a single element

 singleton 1 == fromList [1]
 singleton 1 == insert 1 empty
 size (singleton 1) == 1

insert :: Ord a => a -> Heap a -> Heap a

O(1). Insert a new value into the heap.

 insert 2 (fromList [1,3]) == fromList [3,2,1]
 insert 5 empty            == singleton 5
 size (insert "Item" xs)    == 1 + size xs

minimum :: Heap a -> a

O(1). Assumes the argument is a non-null heap.

 minimum (fromList [3,1,2]) == 1

deleteMin :: Heap a -> Heap a

O(log n). Delete the minimum key from the heap and return the resulting heap.

 deleteMin (fromList [3,1,2]) == fromList [2,3]

meld :: Heap a -> Heap a -> Heap a

O(1). Meld the values from two heaps into one heap.

 meld (fromList [1,3,5]) (fromList [6,4,2]) = fromList [1..6]
 meld (fromList [1,1,1]) (fromList [1,2,1]) = fromList [1,1,1,1,1,2]

uncons :: Ord a => Heap a -> Maybe (a, Heap a)

O(1) access to the minimum element. O(log n) access to the remainder of the heap same operation as viewMin

 uncons (fromList [2,1,3]) == Just (1, fromList [3,2])

viewMin :: Ord a => Heap a -> Maybe (a, Heap a)

Same as uncons

Transformations

mapMonotonic :: Ord b => (a -> b) -> Heap a -> Heap b

O(n). Map a monotone increasing function over the heap. Provides a better constant factor for performance than map, but no checking is performed that the function provided is monotone increasing. Misuse of this function can cause a Heap to violate the heap property.

 map (+1) (fromList [1,2,3]) = fromList [2,3,4]
 map (*2) (fromList [1,2,3]) = fromList [2,4,6]

map :: Ord b => (a -> b) -> Heap a -> Heap b

O(n). Map a function over the heap, returning a new heap ordered appropriately for its fresh contents

 map negate (fromList [3,1,2]) == fromList [-2,-3,-1]

To/From Lists

toUnsortedList :: Heap a -> [a]

O(n). Returns the elements in the heap in some arbitrary, very likely unsorted, order.

 toUnsortedList (fromList [3,1,2]) == [1,3,2]
 fromList . toUnsortedList         == id

fromList :: Ord a => [a] -> Heap a

O(n). Build a heap from a list of values.

 size (fromList [1,5,3]) == 3
 fromList . toList = id
 toList . fromList = sort

traverse :: (Applicative t, Ord b) => (a -> t b) -> Heap a -> t (Heap b)

O(n log n). Traverse the elements of the heap in sorted order and produce a new heap using Applicative side-effects.

mapM :: (Monad m, Ord b) => (a -> m b) -> Heap a -> m (Heap b)

O(n log n). Traverse the elements of the heap in sorted order and produce a new heap using Monadic side-effects.

concatMap :: Ord b => (a -> Heap b) -> Heap a -> Heap b

O(n). Construct heaps from each element in another heap, and meld them together.

concatMap (a -> fromList [a,a+1]) (fromList [1,4]) == fromList [1,2,4,5]

Filtering

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

O(n). Filter the heap, retaining only values that satisfy the predicate.

 filter (>'a') (fromList "ab") == singleton 'b'
 filter (>'x') (fromList "ab") == empty
 filter (<'a') (fromList "ab") == empty

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

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

 partition (>'a') (fromList "ab") (singleton 'b', singleton 'a')

split :: a -> Heap a -> (Heap a, Heap a, Heap a)

O(n). Partition the heap into heaps of the elements that are less than, equal to, and greater than a given value.

 split 'h' (fromList "hello") == (singleton 'e', singleton 'h', fromList "lol")

break :: (a -> Bool) -> Heap a -> (Heap a, Heap a)

O(n log n). break applied to a predicate p and a heap xs returns a tuple where the first element is a heap consisting of the longest prefix the least elements of xs that do not satisfy p and the second element is the remainder of the elements in the heap.

 break (\x -> x `mod` 4 == 0) (fromList [3,5,7,12,13,16]) == (fromList [3,5,7], fromList [12,13,16])

break p is equivalent to span (not . p).

span :: (a -> Bool) -> Heap a -> (Heap a, Heap a)

O(n log n). span applied to a predicate p and a heap xs returns a tuple where the first element is a heap consisting of the longest prefix the least elements of xs that satisfy p and the second element is the remainder of the elements in the heap.

 span (\x -> x `mod` 4 == 0) (fromList [4,8,12,14,16]) == (fromList [4,8,12],fromList [14,16])

span p xs is equivalent to (takeWhile p xs, 'dropWhile p xs)

take :: Int -> Heap a -> Heap a

O(n log n). Return a heap consisting of the least n elements of a given heap.

 take 3 (fromList [10,2,4,1,9,8,2]) == fromList [1,2,2]

drop :: Int -> Heap a -> Heap a

O(n log n). Return a heap consisting of all members of given heap except for the n least elements.

splitAt :: Int -> Heap a -> (Heap a, Heap a)

O(n log n). Split a heap into two heaps, the first containing the n least elements, the latter consisting of all members of the heap except for those elements.

takeWhile :: (a -> Bool) -> Heap a -> Heap a

O(n log n). takeWhile applied to a predicate p and a heap xs returns a heap consisting of the longest prefix the least elements of xs that satisfy p.

 takeWhile (\x -> x `mod` 4 == 0) (fromList [4,8,12,14,16]) == fromList [4,8,12]

dropWhile :: (a -> Bool) -> Heap a -> Heap a

O(n log n). dropWhile p xs returns the suffix of the heap remaining after takeWhile p xs.

 dropWhile (\x -> x `mod` 4 == 0) (fromList [4,8,12,14,16]) == fromList [14,16]

Grouping

group :: Heap a -> Heap (Heap a)

O(n log n). Group a heap into a heap of heaps, by melding together duplicates.

 group (fromList "hello") == fromList [fromList "h", fromList "e", fromList "ll", fromList "o"]

groupBy :: (a -> a -> Bool) -> Heap a -> Heap (Heap a)

O(n log n). Group using a user supplied function.

nub :: Heap a -> Heap a

O(n log n). Remove duplicate entries from the heap.

 nub (fromList [1,1,2,6,6]) == fromList [1,2,6]

Intersection

intersect :: Heap a -> Heap a -> Heap a

O(n log n + m log m). Intersect the values in two heaps, returning the value in the left heap that compares as equal

intersectWith :: Ord b => (a -> a -> b) -> Heap a -> Heap a -> Heap b

O(n log n + m log m). Intersect the values in two heaps using a function to generate the elements in the right heap.

Duplication

replicate :: Ord a => a -> Int -> Heap a

O(log n). Create a heap consisting of multiple copies of the same value.

 replicate 'a' 10 == fromList "aaaaaaaaaa"