Data.IntSet.Internal

A set of integers.

O(\min(n, m \log \frac{2^W}{m})), m \leq n. See difference.

O(1). Is the set empty?

O(n). Cardinality of the set.

O(\min(n,W)). Is the value a member of the set?

O(\min(n,W)). Is the element not in the set?

O(\min(n,W)). Find largest element smaller than the given one.

lookupLT 3 (fromList [3, 5]) == Nothing
lookupLT 5 (fromList [3, 5]) == Just 3

O(\min(n,W)). Find smallest element greater than the given one.

lookupGT 4 (fromList [3, 5]) == Just 5
lookupGT 5 (fromList [3, 5]) == Nothing

O(\min(n,W)). Find largest element smaller or equal to the given one.

lookupLE 2 (fromList [3, 5]) == Nothing
lookupLE 4 (fromList [3, 5]) == Just 3
lookupLE 5 (fromList [3, 5]) == Just 5

O(\min(n,W)). Find smallest element greater or equal to the given one.

lookupGE 3 (fromList [3, 5]) == Just 3
lookupGE 4 (fromList [3, 5]) == Just 5
lookupGE 6 (fromList [3, 5]) == Nothing

O(\min(n, m \log \frac{2^W}{m})), m \leq n. Is this a subset? (s1 `isSubsetOf` s2) tells whether s1 is a subset of s2.

O(\min(n, m \log \frac{2^W}{m})), m \leq n. Is this a proper subset? (ie. a subset but not equal).

O(\min(n, m \log \frac{2^W}{m})), m \leq n. Check whether two sets are disjoint (i.e. their intersection is empty).

disjoint (fromList [2,4,6])   (fromList [1,3])     == True
disjoint (fromList [2,4,6,8]) (fromList [2,3,5,7]) == False
disjoint (fromList [1,2])     (fromList [1,2,3,4]) == False
disjoint (fromList [])        (fromList [])        == True

O(1). The empty set.

O(1). A set of one element.

O(n / W). Create a set from a range of integers.

fromRange (low, high) == fromList [low..high]

O(\min(n,W)). Add a value to the set. There is no left- or right bias for IntSets.

O(\min(n,W)). Delete a value in the set. Returns the original set when the value was not present.

O(\min(n,W)). (alterF f x s) can delete or insert x in s depending on whether it is already present in s.

In short:

member x <$> alterF f x s = f (member x s)

Note: alterF is a variant of the at combinator from Control.Lens.At.

O(\min(n, m \log \frac{2^W}{m})), m \leq n. The union of two sets.

The union of a list of sets.

O(\min(n, m \log \frac{2^W}{m})), m \leq n. Difference between two sets.

O(\min(n, m \log \frac{2^W}{m})), m \leq n. The intersection of two sets.

The intersection of a series of sets. Intersections are performed left-to-right.

O(\min(n, m \log \frac{2^W}{m})), m \leq n. The symmetric difference of two sets.

The result contains elements that appear in exactly one of the two sets.

symmetricDifference (fromList [0,2,4,6]) (fromList [0,3,6,9]) == fromList [2,3,4,9]

IntSets form a Semigroup under intersection.

A Monoid instance is not defined because it would be impractical to construct mempty, the IntSet containing all Ints.

O(n). Filter all elements that satisfy some predicate.

O(n). partition the set according to some predicate.

O(\min(n,W)). Take while a predicate on the elements holds. The user is responsible for ensuring that for all Ints, j < k ==> p j >= p k. See note at spanAntitone.

takeWhileAntitone p = fromDistinctAscList . Data.List.takeWhile p . toList
takeWhileAntitone p = filter p

O(\min(n,W)). Drop while a predicate on the elements holds. The user is responsible for ensuring that for all Ints, j < k ==> p j >= p k. See note at spanAntitone.

dropWhileAntitone p = fromDistinctAscList . Data.List.dropWhile p . toList
dropWhileAntitone p = filter (not . p)

O(\min(n,W)). Divide a set at the point where a predicate on the elements stops holding. The user is responsible for ensuring that for all Ints, j < k ==> p j >= p k.

spanAntitone p xs = (takeWhileAntitone p xs, dropWhileAntitone p xs)
spanAntitone p xs = partition p xs

Note: if p is not actually antitone, then spanAntitone will split the set at some unspecified point.

O(\min(n,W)). The expression (split x set) is a pair (set1,set2) where set1 comprises the elements of set less than x and set2 comprises the elements of set greater than x.

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

O(\min(n,W)). Performs a split but also returns whether the pivot element was found in the original set.

O(1). Decompose a set into pieces based on the structure of the underlying tree. This function is useful for consuming a set in parallel.

No guarantee is made as to the sizes of the pieces; an internal, but deterministic process determines this. However, it is guaranteed that the pieces returned will be in ascending order (all elements in the first submap less than all elements in the second, and so on).

Examples:

splitRoot (fromList [1..120]) == [fromList [1..63],fromList [64..120]]
splitRoot empty == []

Note that the current implementation does not return more than two subsets, but you should not depend on this behaviour because it can change in the future without notice. Also, the current version does not continue splitting all the way to individual singleton sets -- it stops at some point.

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). The

mapMonotonic f s == map f s, but works only when f is strictly increasing. Semi-formally, we have:

and [x < y ==> f x < f y | x <- ls, y <- ls]
                    ==> mapMonotonic f s == map f s
    where ls = toList s

Warning: This function should be used only if f is monotonically strictly increasing. This precondition is not checked. Use map if the precondition may not hold.

O(n). Fold the elements in the set using the given right-associative binary operator, such that foldr f z == foldr f z . toAscList.

For example,

toAscList set = foldr (:) [] set

O(n). Fold the elements in the set using the given left-associative binary operator, such that foldl f z == foldl f z . toAscList.

For example,

toDescList set = foldl (flip (:)) [] set

O(n). Map the elements in the set to a monoid and combine with (<>).

O(\min(n,W)). The minimal element of the set. Returns Nothing if the set is empty.

O(\min(n,W)). The maximal element of the set. Returns Nothing if the set is empty.

O(\min(n,W)). The minimal element of the set. Calls error if the set is empty.

O(\min(n,W)). The maximal element of the set. Calls error if the set is empty.

O(\min(n,W)). Delete the minimal element. Returns an empty set if the set is empty.

Note that this is a change of behaviour for consistency with Set – versions prior to 0.5 threw an error if the IntSet was already empty.

O(\min(n,W)). Delete the maximal element. Returns an empty set if the set is empty.

Note that this is a change of behaviour for consistency with Set – versions prior to 0.5 threw an error if the IntSet was already empty.

O(\min(n,W)). Delete and find the minimal element.

deleteFindMin set = (findMin set, deleteMin set)

O(\min(n,W)). Delete and find the maximal element.

deleteFindMax set = (findMax set, deleteMax set)

O(\min(n,W)). Retrieves the maximal key of the set, and the set stripped of that element, or Nothing if passed an empty set.

O(\min(n,W)). Retrieves the minimal key of the set, and the set stripped of that element, or Nothing if passed an empty set.

O(n \min(n,W)). Show the tree that implements the set. The tree is shown in a compressed, hanging format.

O(n \min(n,W)). The expression (showTreeWith hang wide map) shows the tree that implements the set. 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.