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Commits (6)
definition module Data.SetBy
/**
* An efficient implementation of sets.
*
* This version is the same as Data.Set, except that the overloaded API is replaced
* with a higher-order function API.
*
* The naming convention is to add 'By' to a function or macro name that is overloaded
* in Data.Set but uses a higher-order function argument in Data.SetBy.
*
* For all documentation, please consult Data.Set.
*
* The `morally equivalent` function from Data.Set is added in the comment. This is not
* a strictly equivalent function because of the different types.
*
* When using the functions in Data.SetBy, make sure to use the same higher-order
* function parameter for the same data structure to ensure internal integrity.
* This higher-order function represents the < ordering on your set elements and
* should have the usual ordering properties:
*
* - if a < b and b < c then a < c
* - if a < b then not (b < a)
* - if not (a < b) and not (b < a) then a and b are considered to 'equal'
*
*/
from StdOverloaded import class ==, class < (..)
from StdClass import class Ord (..), <=, >
from StdList import foldl, map
from Data.Maybe import :: Maybe
from StdBool import not, &&
from Data.GenLexOrd import :: LexOrd
import qualified Data.Foldable
from Data.Foldable import class Foldable
:: SetBy a = TipBy
| BinBy !Int !a !(SetBy a) !(SetBy a)
/**
* True iff the two sets have the same number of elements, and these elements
* are pairwise 'equal' as described above, so the higher-order function
* parameter represents < on a, *not* == on a(!)
*
* Morally equivalent function: instance == (Set a) | == a
*/
isEqualBy :: !(a a -> Bool) !(SetBy a) !(SetBy a) -> Bool
/**
* True iff first set is `smaller` than second set, according to
* first argument (assuming the two sets are ordered with the
* same first function argument).
*
* Morally equivalent function: instance < (Set a) | < a
*/
isOrderedBy :: !(a a -> Bool) !(SetBy a) !(SetBy a) -> Bool
/**
* EQ iff the two sets have the same number of elements, occurring in the
* same order.
* LT iff the first set is the common prefix of the second set or the common
* prefix is followed in the first set with an element that is considered
* than the corresponding element in the second set.
* GT iff the second set is the common prefix of the first set or the common
* prefix is followed in the second set with an element that is considered
* greater than the corresponding element in the first set.
* The comparison of elements is done with the first function argument.
*
* Morally equivalent function: derive gLexOrd Set
*/
lexOrdBy :: !(a a -> Bool) !(SetBy a) !(SetBy a) -> LexOrd
instance Foldable SetBy
/**
* True iff this is the empty set.
* @type (SetBy a) -> Bool
* @property equivalence with size 0: A.s :: SetBy a:
* size s == 0 <==> null s
* @property equivalence with newSet: A.s :: SetBy a:
* s == newSet <==> null s
*/
null s :== case s of
TipBy -> True
(BinBy sz _ _ _) -> False
/**
* The number of elements in the set.
* @type (SetBy a) -> Int
* @property correctness: A.s :: SetBy a:
* size s =.= length (toList s)
*/
size s :== case s of
TipBy -> 0
(BinBy sz _ _ _) -> sz
/**
* Is the element in the set?
*
* Morally equivalent function: Data.Set.member x s = Data.SetBy.memberBy (<) x s
*/
memberBy :: !(a a -> Bool) !a !(SetBy a) -> Bool
/**
* Checks if an element is not in the set.
*/
notMemberBy comp x t :== not (memberBy comp x t)
/**
* Is t1 a subset of t2?
*
* Morally equivalent function: Data.Set.isSubsetOf s1 s2 = Data.SetBy.isSubsetOfBy (<) s1 s2
*/
isSubsetOfBy comp t1 t2 :== (size t1 <= size t2) && (isSubsetOfXBy comp t1 t2)
isSubsetOfXBy :: !(a a -> Bool) !(SetBy a) !(SetBy a) -> Bool
/**
* Is t1 a proper subset of t2?
*
* Morally equivalent function: Data.Set.isProperSubsetOf s1 s2 = Data.SetBy.isProperSubsetOfBy (<) s1 s2
*/
isProperSubsetOfBy comp s1 s2 :== (size s1 < size s2) && (isSubsetOfBy comp s1 s2)
/**
* The empty set.
* @complexity O(1)
* @property is null:
* null newSet
*/
newSet :: SetBy a
/**
* Create a singleton set.
* @complexity O(1)
*/
singleton :: !u:a -> w:(SetBy u:a), [w <= u]
/**
* Insert an element in a set. If the set already contains an element equal to
* the given value, it is replaced with the new value.
*
* Morally equivalent function: Data.Set.insert x s = Data.SetBy.insertBy (<) x s
*/
insertBy :: !(a a -> Bool) !a !.(SetBy a) -> SetBy a
/**
* Delete an element from a set.
*
* Morally equivalent function: Data.Set.delete x s = Data.SetBy (<) x s
*/
deleteBy :: !(a a -> Bool) !a !.(SetBy a) -> SetBy a
/**
* The minimal element of a set.
*
* Morally equivalent function: Data.Set.findMin
*/
findMin :: !(SetBy a) -> a
/**
* The maximal element of a set.
*
* Morally equivalent function: Data.Set.findMax
*/
findMax :: !(SetBy a) -> a
/**
* Delete the minimal element.
*
* Morally equivalent function: Data.Set.deleteMin
*/
deleteMin :: !.(SetBy a) -> SetBy a
/**
* Delete the maximal element.
*
* Morally equivalent function: Data.Set.deleteMax
*/
deleteMax :: !.(SetBy a) -> SetBy a
/**
* deleteFindMin set = (findMin set, deleteMin set)
*/
deleteFindMin :: !.(SetBy a) -> (!a, !SetBy a)
/**
* deleteFindMax set = (findMax set, deleteMax set)
*/
deleteFindMax :: !.(SetBy a) -> (!a, !SetBy a)
/**
* Retrieves the minimal key of the set, and the set stripped of that element,
* or 'Nothing' if passed an empty set.
*/
minView :: !.(SetBy a) -> .(Maybe (!a, !SetBy a))
/**
* Retrieves the maximal key of the set, and the set stripped of that element,
* or 'Nothing' if passed an empty set.
*/
maxView :: !.(SetBy a) -> .(Maybe (!a, !SetBy a))
/**
* The union of two sets, preferring the first set when equal elements are
* encountered.
*
* Morally equivalent function: Data.Set.union s1 s2 = Data.SetBy.unionBy (<) s1 s2
*/
unionBy :: !(a a -> Bool) !u:(SetBy a) !u:(SetBy a) -> SetBy a
/**
* The union of a list of sets.
*
* Morally equivalent function: Data.Set.unions ts = Data.SetBy.unionsBy (<) ts
*/
unionsBy ts :== foldl unionBy newSet ts
/**
* Difference of two sets.
*
* Morally equivalent function: Data.Set.difference s1 s2 = Data.SetBy.differenceBy (<) s1 s2
*/
differenceBy :: !(a a -> Bool) !(SetBy a) !(SetBy a) -> SetBy a
/**
* The intersection of two sets.
* Elements of the result come from the first set.
*
* Morally equivalent function: Data.Set.intersection s1 s2 = Data.SetBy.intersectionBy (<) s1 s2
*/
intersectionBy :: !(a a -> Bool) !(SetBy a) !(SetBy a) -> SetBy a
/**
* The intersection of a list of sets.
* Elements of the result come from the first set
*
* Morally equivalent function: Data.Set.intersections ts = Data.SetBy.intersectionsBy (<) ts
*/
intersectionsBy :: !(a a -> Bool) ![SetBy a] -> SetBy a
/**
* Filter all elements that satisfy the predicate.
*
* Morally equivalent function: Data.Set.filter
*/
filter :: !(a -> Bool) !(SetBy a) -> SetBy a
/**
* Partition the set into two sets, one with all elements that satisfy the
* predicate and one with all elements that don't satisfy the predicate.
*
* Morally equivalent function: Data.Set.partition
*/
partition :: !(a -> Bool) !(SetBy a) -> (!SetBy a, !SetBy a)
/**
* Split a set in elements less and elements greater than a certain pivot.
*
* Morally equivalent function: Data.Set.split x s = Data.SetBy.splitBy (<) x s
*/
splitBy :: !(a a -> Bool) !a !(SetBy a) -> (!SetBy a, !SetBy a)
/**
* Performs a 'split' but also returns whether the pivot element was found in
* the original set.
*
* Morally equivalent function: Data.Set.splitMember x s = Data.SetBy.splitMemberBy (<) x s
*/
splitMemberBy :: !(a a -> Bool) !a !(SetBy a) -> (!SetBy a, !Bool, !SetBy a)
/**
* Convert the set to an ascending list of elements.
*/
toList s :== toAscList s
/**
* Same as toList.
*/
toAscList t :== 'Data.Foldable'.foldr` (\a as -> [a:as]) [] t
/**
* Create a set from a list of elements.
*
* Morally equivalent function: Data.Set.fromList xs = Data.SetBy.fromListBy (<) xs
*/
fromListBy :: !(a a -> Bool) ![a] -> SetBy a
/**
* Map a function to all elements in a set.
*
* Morally equivalent function: Data.Set.mapSet f s = Data.SetBy.mapSetBy (<) f s
*/
mapSetBy comp_b f s :== fromListBy comp_b (map f (toList s))
/**
* Map a set without converting it to and from a list.
*
* Morally equivalent function: Data.Set.mapSetMonotonic
*/
mapSetByMonotonic :: !(a -> b) !(SetBy a) -> SetBy b
implementation module Data.SetBy
import StdClass, StdMisc, StdBool, StdFunc, StdInt
import Data.Maybe
from Data.GenLexOrd import :: LexOrd (..)
import Data.Monoid
from Data.Foldable import class Foldable (..)
import qualified StdList
from StdList import instance == [a]
/*
* This function should only be used if the argument function preserves the ordering property of
* the new set.
*/
mapSetByMonotonic :: !(a -> b) !(SetBy a) -> SetBy b
mapSetByMonotonic _ TipBy = TipBy
mapSetByMonotonic f (BinBy n x l r) = BinBy n (f x) (mapSetByMonotonic f l) (mapSetByMonotonic f r)
/*
* Sets are size balanced trees.
* A set of values @a@.
*/
:: SetBy a = TipBy
| BinBy !Int !a !(SetBy a) !(SetBy a)
isEqualBy :: !(a a -> Bool) !(SetBy a) !(SetBy a) -> Bool
isEqualBy comp s1 s2 = size s1 == size s2 && equalEltsBy comp (toAscList s1) (toAscList s2)
where
equalEltsBy :: !(a a -> Bool) ![a] ![a] -> Bool
equalEltsBy _ [] [] = True
equalEltsBy _ [] _ = False
equalEltsBy _ [_:_] [] = False
equalEltsBy comp [a:as] [b:bs]
| comp a b || comp b a = False
| otherwise = equalEltsBy comp as bs
isOrderedBy :: !(a a -> Bool) !(SetBy a) !(SetBy a) -> Bool
isOrderedBy comp s1 s2 = compare comp (toAscList s1) (toAscList s2)
where
compare :: !(a a -> Bool) ![a] ![a] -> Bool
compare _ [] [] = False
compare _ [] _ = True
compare _ [_:_] [] = False
compare comp [a:as] [b:bs]
| comp a b = True
| comp b a = False
| otherwise = compare comp as bs
lexOrdBy :: !(a a -> Bool) !(SetBy a) !(SetBy a) -> LexOrd
lexOrdBy comp s1 s2 = ordby comp (toAscList s1) (toAscList s2)
where
ordby :: !(a a -> Bool) ![a] ![a] -> LexOrd
ordby _ [] [] = EQ
ordby _ [] _ = LT
ordby _ [_:_] [] = GT
ordby comp [a:as] [b:bs]
| comp a b = LT
| comp b a = GT
| otherwise = ordby comp as bs
instance Foldable SetBy where
foldr f z (BinBy _ x l r) = foldr f (f x (foldr f z r)) l
foldr _ z _ = z
foldr` f z (BinBy _ x l r) = foldr` f (f x (foldr` f z r)) l
foldr` _ z _ = z
foldl f z (BinBy _ x l r) = foldl f (f (foldl f z l) x) r
foldl _ z _ = z
foldl` f z (BinBy _ x l r) = foldl` f (f (foldl` f z l) x) r
foldl` _ z _ = z
/*--------------------------------------------------------------------
* Query
*--------------------------------------------------------------------*/
memberBy :: !(a a -> Bool) !a !(SetBy a) -> Bool
memberBy comp x (BinBy _ y l r)
| comp x y = memberBy comp x l
| comp y x = memberBy comp x r
| otherwise = True
memberBy _ _ _ = False
/*--------------------------------------------------------------------
* Construction
*--------------------------------------------------------------------*/
newSet :: SetBy a
newSet = TipBy
singleton :: !u:a -> w:(SetBy u:a), [w <= u]
singleton x = BinBy 1 x TipBy TipBy
/*--------------------------------------------------------------------
* Insertion, Deletion
*--------------------------------------------------------------------*/
insertBy :: !(a a -> Bool) !a !.(SetBy a) -> SetBy a
insertBy comp x t=:(BinBy _ y l r)
| comp x y = balanceL y (insertBy comp x l) r
| comp y x = balanceR y l (insertBy comp x r)
| otherwise = t
insertBy _ x _ = singleton x
deleteBy :: !(a a -> Bool) !a !.(SetBy a) -> SetBy a
deleteBy comp x (BinBy _ y l r)
| comp x y = balanceR y (deleteBy comp x l) r
| comp y x = balanceL y l (deleteBy comp x r)
| otherwise = glue l r
deleteBy _ _ tip = tip
/*--------------------------------------------------------------------
* Subset
*--------------------------------------------------------------------*/
isSubsetOfXBy :: !(a a -> Bool) !(SetBy a) !(SetBy a) -> Bool
isSubsetOfXBy comp (BinBy _ x l r) t
| t =: TipBy = False
#! (lt, found, gt) = splitMemberBy comp x t
= found && isSubsetOfXBy comp l lt && isSubsetOfXBy comp r gt
isSubsetOfXBy _ _ _ = True
/*--------------------------------------------------------------------
* Minimal, Maximal
*--------------------------------------------------------------------*/
findMin :: !(SetBy a) -> a
findMin (BinBy _ x TipBy _) = x
findMin (BinBy _ _ l _) = findMin l
findMin TipBy = abort "SetBy.findMin: empty set has no minimal element"
findMax :: !(SetBy a) -> a
findMax (BinBy _ x _ TipBy) = x
findMax (BinBy _ _ _ r) = findMax r
findMax TipBy = abort "SetBy.findMax: empty set has no maximal element"
deleteMin :: !.(SetBy a) -> SetBy a
deleteMin (BinBy _ _ TipBy r) = r
deleteMin (BinBy _ x l r) = balanceR x (deleteMin l) r
deleteMin TipBy = TipBy
deleteMax :: !.(SetBy a) -> SetBy a
deleteMax (BinBy _ _ l TipBy) = l
deleteMax (BinBy _ x l r) = balanceL x l (deleteMax r)
deleteMax TipBy = TipBy
/*--------------------------------------------------------------------
* Union.
*--------------------------------------------------------------------*/
unionBy :: !(a a -> Bool) !u:(SetBy a) !u:(SetBy a) -> SetBy a
unionBy _ t1 TipBy = t1
unionBy comp t1 (BinBy _ x TipBy TipBy) = insertBy comp x t1
unionBy comp (BinBy _ x TipBy TipBy) t2 = insertBy comp x t2
unionBy _ TipBy t2 = t2
unionBy comp t1=:(BinBy _ x l1 r1) t2 = link x l1l2 r1r2
where
(l2,r2) = splitS comp x t2
l1l2 = unionBy comp l1 l2
r1r2 = unionBy comp r1 r2
splitS :: !(a a -> Bool) !a !(SetBy a) -> (!SetBy a, !SetBy a)
splitS _ _ TipBy = (TipBy,TipBy)
splitS comp x (BinBy _ y l r)
| comp x y = let (lt,gt) = splitS comp x l in (lt, link y gt r)
| comp y x = let (lt,gt) = splitS comp x r in (link y l lt, gt)
| otherwise = (l,r)
/*--------------------------------------------------------------------
* Difference
*--------------------------------------------------------------------*/
differenceBy :: !(a a -> Bool) !(SetBy a) !(SetBy a) -> SetBy a
differenceBy _ TipBy _ = TipBy
differenceBy comp t1 t2 =
case t2 of
BinBy _ x l2 r2 -> case splitBy comp x t1 of
(l1, r1)
| size l1l2 + size r1r2 == size t1 -> t1
| otherwise -> merge l1l2 r1r2
where
l1l2 = differenceBy comp l1 l2
r1r2 = differenceBy comp r1 r2
_ -> t1
/*--------------------------------------------------------------------
* Intersection
*--------------------------------------------------------------------*/
intersectionsBy :: !(a a -> Bool) ![SetBy a] -> SetBy a
intersectionsBy _ [t] = t
intersectionsBy comp [t:ts] = 'StdList'.foldl (intersectionBy comp) t ts
intersectionsBy _ [] = abort "SetBy.intersectionsBy called with []\n"
intersectionBy :: !(a a -> Bool) !(SetBy a) !(SetBy a) -> SetBy a
intersectionBy _ TipBy _ = TipBy
intersectionBy _ _ TipBy = TipBy
intersectionBy comp t1 t2 = hedgeInt comp NothingS NothingS t1 t2
hedgeInt :: !(a a -> Bool) !(MaybeS a) !(MaybeS a) !(SetBy a) !(SetBy a) -> SetBy a
hedgeInt _ _ _ _ TipBy = TipBy
hedgeInt _ _ _ TipBy _ = TipBy
hedgeInt comp blo bhi (BinBy _ x l r) t2
#! bmi = JustS x
#! l` = hedgeInt comp blo bmi l (trimBy comp blo bmi t2)
#! r` = hedgeInt comp bmi bhi r (trimBy comp bmi bhi t2)
= if (memberBy comp x t2)
(link x l` r`)
(merge l` r`)
/*--------------------------------------------------------------------
* Filter and partition
*--------------------------------------------------------------------*/
filter :: !(a -> Bool) !(SetBy a) -> SetBy a
filter p (BinBy _ x l r)
| p x = link x (filter p l) (filter p r)
| otherwise = merge (filter p l) (filter p r)
filter _ tip = tip
partition :: !(a -> Bool) !(SetBy a) -> (!SetBy a, !SetBy a)
partition p (BinBy _ x l r)
#! (l1,l2) = partition p l
#! (r1,r2) = partition p r
| p x = (link x l1 r1,merge l2 r2)
| otherwise = (merge l1 r1,link x l2 r2)
partition _ t = (t, t)
/*--------------------------------------------------------------------
* Lists
*--------------------------------------------------------------------*/
fromListBy :: !(a a -> Bool) ![a] -> SetBy a
fromListBy comp xs = 'StdList'.foldl (ins comp) newSet xs
where
ins :: !(a a -> Bool) !(SetBy a) !a -> SetBy a
ins comp t x = insertBy comp x t
/*--------------------------------------------------------------------
Utility functions that return sub-ranges of the original
tree. Some functions take a comparison function as argument to
allow comparisons against infinite values. A function [cmplo x]
should be read as [compare lo x].
[trimBy comp cmplo cmphi t] A tree that is either empty or where [cmplo x == LT]
and [cmphi x == GT] for the value [x] of the root.
[splitBy comp k t] Returns two trees [l] and [r] where all values
in [l] are <[k] and all keys in [r] are >[k].
[splitMemberBy comp k t] Just like [splitBy] but also returns whether [k]
was found in the tree.
--------------------------------------------------------------------*/
:: MaybeS a = NothingS | JustS !a
/*--------------------------------------------------------------------
[trimBy comp lo hi t] trims away all subtrees that surely contain no
values between the range [lo] to [hi]. The returned tree is either
empty or the key of the root is between @lo@ and @hi@.
--------------------------------------------------------------------*/
trimBy :: !(a a -> Bool) !(MaybeS a) !(MaybeS a) !(SetBy a) -> SetBy a
trimBy _ NothingS NothingS t = t
trimBy comp (JustS lx) NothingS t = greater comp lx t
where
greater comp lo (BinBy _ x _ r) | not (comp lo x) = greater comp lo r
greater _ _ t` = t`
trimBy comp NothingS (JustS hx) t = lesser comp hx t
where
lesser comp hi (BinBy _ x l _) | not (comp x hi) = lesser comp hi l
lesser _ _ t` = t`
trimBy comp (JustS lx) (JustS hx) t = middle comp lx hx t
where
middle comp lo hi (BinBy _ x _ r) | not (comp lo x) = middle comp lo hi r
middle comp lo hi (BinBy _ x l _) | not (comp x hi) = middle comp lo hi l
middle _ _ _ t` = t`
/*--------------------------------------------------------------------
* Split
*--------------------------------------------------------------------*/
splitBy :: !(a a -> Bool) !a !(SetBy a) -> (!SetBy a, !SetBy a)
splitBy comp x (BinBy _ y l r)
| comp x y
#! (lt, gt) = splitBy comp x l
= (lt, link y gt r)
| comp y x
#! (lt,gt) = splitBy comp x r
= (link y l lt,gt)
| otherwise = (l, r)
splitBy _ _ t = (t, t)
splitMemberBy :: !(a a -> Bool) !a !(SetBy a) -> (!SetBy a, !Bool, !SetBy a)
splitMemberBy comp x (BinBy _ y l r)
| comp x y
#! (lt, found, gt) = splitMemberBy comp x l
= (lt, found, link y gt r)
| comp y x
#! (lt, found, gt) = splitMemberBy comp x r
= (link y l lt, found, gt)
| otherwise = (l, True, r)
splitMemberBy _ _ t = (t, False, t)
/*--------------------------------------------------------------------
Utility functions that maintain the balance properties of the tree.
All constructors assume that all values in [l] < [x] and all values
in [r] > [x], and that [l] and [r] are valid trees.
In order of sophistication:
[BinBy sz x l r] The type constructor.
[bin x l r] Maintains the correct size, assumes that both [l]
and [r] are balanced with respect to each other.
[balance x l r] Restores the balance and size.
Assumes that the original tree was balanced and
that [l] or [r] has changed by at most one element.
[join x l r] Restores balance and size.
Furthermore, we can construct a new tree from two trees. Both operations
assume that all values in [l] < all values in [r] and that [l] and [r]
are valid:
[glue l r] Glues [l] and [r] together. Assumes that [l] and
[r] are already balanced with respect to each other.
[merge l r] Merges two trees and restores balance.
Note: in contrast to Adam's paper, we use (<=) comparisons instead
of (<) comparisons in [join], [merge] and [balance].
Quickcheck (on [difference]) showed that this was necessary in order
to maintain the invariants. It is quite unsatisfactory that I haven't
been able to find out why this is actually the case! Fortunately, it
doesn't hurt to be a bit more conservative.
--------------------------------------------------------------------*/
/*--------------------------------------------------------------------
* Join
*--------------------------------------------------------------------*/
link :: !a !(SetBy a) !(SetBy a) -> SetBy a
link x l=:(BinBy sizeL y ly ry) r=:(BinBy sizeR z lz rz)
| delta*sizeL < sizeR = balanceL z (link x l lz) rz
| delta*sizeR < sizeL = balanceR y ly (link x ry r)
| otherwise = bin x l r
link x TipBy r = insertMin x r
link x l _ = insertMax x l
// insertMin and insertMax don't perform potentially expensive comparisons.
insertMax :: !a !(SetBy a) -> SetBy a
insertMax x (BinBy _ y l r) = balanceR y l (insertMax x r)
insertMax x _ = singleton x
insertMin :: !a !(SetBy a) -> SetBy a
insertMin x (BinBy _ y l r) = balanceL y (insertMin x l) r
insertMin x _ = singleton x
/*--------------------------------------------------------------------
* [merge l r]: merges two trees.
*--------------------------------------------------------------------*/
merge :: !(SetBy a) !(SetBy a) -> SetBy a
merge l=:(BinBy sizeL x lx rx) r=:(BinBy sizeR y ly ry)
| delta*sizeL < sizeR = balanceL y (merge l ly) ry
| delta*sizeR < sizeL = balanceR x lx (merge rx r)
| otherwise = glue l r
merge TipBy r = r
merge l _ = l
/*--------------------------------------------------------------------
* [glue l r]: glues two trees together.
* Assumes that [l] and [r] are already balanced with respect to each other.
*--------------------------------------------------------------------*/
glue :: !.(SetBy a) !.(SetBy a) -> SetBy a
glue TipBy r = r
glue l TipBy = l
glue l r
| size l > size r
#! (m, l`) = deleteFindMax l
= balanceR m l` r
| otherwise
#! (m, r`) = deleteFindMin r
= balanceL m l r`
deleteFindMin :: !.(SetBy a) -> (!a, !SetBy a)
deleteFindMin (BinBy _ x TipBy r) = (x, r)
deleteFindMin (BinBy _ x l r)
#! (xm, l`) = deleteFindMin l
= (xm, balanceR x l` r)
deleteFindMin TipBy = (abort "SetBy.deleteFindMin: can not return the minimal element of an empty set", TipBy)
deleteFindMax :: !.(SetBy a) -> (!a, !SetBy a)
deleteFindMax (BinBy _ x l TipBy ) = (x, l)
deleteFindMax (BinBy _ x l r)
#! (xm, r`) = deleteFindMax r
= (xm, balanceL x l r`)
deleteFindMax TipBy = (abort "SetBy.deleteFindMax: can not return the maximal element of an empty set", TipBy)
minView :: !.(SetBy a) -> .(Maybe (!a, !SetBy a))
minView TipBy = Nothing
minView x = Just (deleteFindMin x)
maxView :: !.(SetBy a) -> .(Maybe (!a, !SetBy a))
maxView TipBy = Nothing
maxView x = Just (deleteFindMax x)
/*--------------------------------------------------------------------
[balance x l r] balances two trees with value x.
The sizes of the trees should balance after decreasing the
size of one of them. (a rotation).
[delta] is the maximal relative difference between the sizes of
two trees, it corresponds with the [w] in Adams' paper,
or equivalently, [1/delta] corresponds with the $\alpha$
in Nievergelt's paper. Adams shows that [delta] should
be larger than 3.745 in order to garantee that the
rotations can always restore balance.
[ratio] is the ratio between an outer and inner sibling of the
heavier subtree in an unbalanced setting. It determines
whether a double or single rotation should be performed
to restore balance. It is correspondes with the inverse
of $\alpha$ in Adam's article.
Note that:
- [delta] should be larger than 4.646 with a [ratio] of 2.
- [delta] should be larger than 3.745 with a [ratio] of 1.534.
- A lower [delta] leads to a more 'perfectly' balanced tree.
- A higher [delta] performs less rebalancing.
- Balancing is automatic for random data and a balancing
scheme is only necessary to avoid pathological worst cases.
Almost any choice will do in practice
- Allthough it seems that a rather large [delta] may perform better
than smaller one, measurements have shown that the smallest [delta]
of 4 is actually the fastest on a wide range of operations. It
especially improves performance on worst-case scenarios like
a sequence of ordered insertions.
Note: in contrast to Adams' paper, we use a ratio of (at least) 2
to decide whether a single or double rotation is needed. Allthough
he actually proves that this ratio is needed to maintain the
invariants, his implementation uses a (invalid) ratio of 1.
He is aware of the problem though since he has put a comment in his
original source code that he doesn't care about generating a
slightly inbalanced tree since it doesn't seem to matter in practice.
However (since we use quickcheck :-) we will stick to strictly balanced
trees.
--------------------------------------------------------------------*/
delta :== 4
ratio :== 2
// Functions balanceL and balanceR are specialised versions of balance.
// balanceL only checks whether the left subtree is too big,
// balanceR only checks whether the right subtree is too big.
// balanceL is called when left subtree might have been inserted to or when
// right subtree might have been deleted from.
balanceL :: !a !(SetBy a) !(SetBy a) -> SetBy a
balanceL x l r = case r of
BinBy rs _ _ _ -> case l of
BinBy ls lx ll lr
| ls > delta*rs
# (BinBy lls _ _ _ ) = ll
# (BinBy lrs lrx lrl lrr) = lr
| lrs < ratio*lls -> BinBy (1+ls+rs) lx ll (BinBy (1+rs+lrs) x lr r)
| otherwise -> BinBy (1+ls+rs) lrx (BinBy (1+lls+size lrl) lx ll lrl) (BinBy (1+rs+size lrr) x lrr r)
| otherwise -> BinBy (1+ls+rs) x l r
_ -> BinBy (1+rs) x TipBy r
_ -> case l of
BinBy ls lx ll=:(BinBy lls _ _ _) lr=:(BinBy lrs lrx lrl lrr)
| lrs < ratio*lls -> BinBy (1+ls) lx ll (BinBy (1+lrs) x lr TipBy)
| otherwise -> BinBy (1+ls) lrx (BinBy (1+lls+size lrl) lx ll lrl) (BinBy (1+size lrr) x lrr TipBy)
BinBy _ lx TipBy (BinBy _ lrx _ _) -> BinBy 3 lrx (BinBy 1 lx TipBy TipBy) (BinBy 1 x TipBy TipBy)
BinBy _ lx ll=:(BinBy _ _ _ _) TipBy -> BinBy 3 lx ll (BinBy 1 x TipBy TipBy)
BinBy _ _ _ _ -> BinBy 2 x l TipBy
_ -> BinBy 1 x TipBy TipBy
// balanceR is called when right subtree might have been inserted to or when
// left subtree might have been deleted from.
balanceR :: !a !(SetBy a) !(SetBy a) -> SetBy a
balanceR x l r = case l of
BinBy ls _ _ _ -> case r of
BinBy rs rx rl rr
| rs > delta*ls
# (BinBy rls rlx rll rlr) = rl
# (BinBy rrs _ _ _ ) = rr
| rls < ratio*rrs -> BinBy (1+ls+rs) rx (BinBy (1+ls+rls) x l rl) rr
| otherwise -> BinBy (1+ls+rs) rlx (BinBy (1+ls+size rll) x l rll) (BinBy (1+rrs+size rlr) rx rlr rr)
| otherwise -> BinBy (1+ls+rs) x l r
_ -> BinBy (1+ls) x l TipBy
_ -> case r of
BinBy rs rx rl=:(BinBy rls rlx rll rlr) rr=:(BinBy rrs _ _ _)
| rls < ratio*rrs -> BinBy (1+rs) rx (BinBy (1+rls) x TipBy rl) rr
| otherwise -> BinBy (1+rs) rlx (BinBy (1+size rll) x TipBy rll) (BinBy (1+rrs+size rlr) rx rlr rr)
BinBy _ rx TipBy rr=:(BinBy _ _ _ _) -> BinBy 3 rx (BinBy 1 x TipBy TipBy) rr
BinBy _ rx (BinBy _ rlx _ _) TipBy -> BinBy 3 rlx (BinBy 1 x TipBy TipBy) (BinBy 1 rx TipBy TipBy)
BinBy _ _ _ _ -> BinBy 2 x TipBy r
_ -> BinBy 1 x TipBy TipBy
// rotate
rotateL :: !a !(SetBy a) !(SetBy a) -> SetBy a
rotateL x l r=:(BinBy _ _ ly ry)
| size ly < ratio*size ry = singleL x l r
| otherwise = doubleL x l r
rotateL _ _ TipBy = abort "rotateL TipBy"
rotateR :: !a !(SetBy a) !(SetBy a) -> SetBy a
rotateR x l=:(BinBy _ _ ly ry) r
| size ry < ratio*size ly = singleR x l r
| otherwise = doubleR x l r
rotateR _ TipBy _ = abort "rotateL TipBy"
// basic rotations
singleL :: !a !(SetBy a) !(SetBy a) -> SetBy a
singleL x1 t1 (BinBy _ x2 t2 t3) = bin x2 (bin x1 t1 t2) t3
singleL _ _ TipBy = abort "singleL"
singleR :: !a !(SetBy a) !(SetBy a) -> SetBy a
singleR x1 (BinBy _ x2 t1 t2) t3 = bin x2 t1 (bin x1 t2 t3)
singleR _ TipBy _ = abort "singleR"
doubleL :: !a !(SetBy a) !(SetBy a) -> SetBy a
doubleL x1 t1 (BinBy _ x2 (BinBy _ x3 t2 t3) t4) = bin x3 (bin x1 t1 t2) (bin x2 t3 t4)
doubleL _ _ _ = abort "doubleL"
doubleR :: !a !(SetBy a) !(SetBy a) -> SetBy a
doubleR x1 (BinBy _ x2 t1 (BinBy _ x3 t2 t3)) t4 = bin x3 (bin x2 t1 t2) (bin x1 t3 t4)
doubleR _ _ _ = abort "doubleR"
/*--------------------------------------------------------------------
* The bin constructor maintains the size of the tree
*--------------------------------------------------------------------*/
//bin :: !a !(SetBy a) !(SetBy a) -> SetBy a
bin x l r :== BinBy (size l + size r + 1) x l r
...@@ -138,6 +138,7 @@ import qualified Data.OrdList ...@@ -138,6 +138,7 @@ import qualified Data.OrdList
import qualified Data.Queue import qualified Data.Queue
import qualified Data.Real import qualified Data.Real
import qualified Data.Set import qualified Data.Set
import qualified Data.SetBy
import qualified Data.Set.GenJSON import qualified Data.Set.GenJSON
import qualified Data.Set.Gast import qualified Data.Set.Gast
import qualified Data.Stack import qualified Data.Stack
......