{-# OPTIONS_GHC -XNPlusKPatterns #-}
-- (c) MP-I (1998/9-2006/7) and CP (2005/6-2019/20)
module BTree where
import Cp
import Data.List
import Data.Monoid
-- (1) Datatype definition -----------------------------------------------------
data BTree a = Empty | Node(a, (BTree a, BTree a)) deriving Show
inBTree :: Either () (b,(BTree b,BTree b)) -> BTree b
inBTree = either (const Empty) Node
outBTree :: BTree a -> Either () (a,(BTree a,BTree a))
outBTree Empty = Left ()
outBTree (Node (a,(t1,t2))) = Right(a,(t1,t2))
baseBTree g f = id -|- g >< (f >< f)
-- (2) Ana + cata + hylo -------------------------------------------------------
recBTree f = baseBTree id f
cataBTree g = g . (recBTree (cataBTree g)) . outBTree
anaBTree g = inBTree . (recBTree (anaBTree g) ) . g
hyloBTree f g = cataBTree f . anaBTree g
-- (3) Map ---------------------------------------------------------------------
instance Functor BTree
where fmap f = cataBTree ( inBTree . baseBTree f id )
-- equivalent to:
-- where fmap f = anaBTree ( baseBTree f id . outBTree )
-- (4) Examples ----------------------------------------------------------------
-- (4.1) Inversion (mirror) ----------------------------------------------------
mirrorBTree = cataBTree (inBTree . (id -|- id >< swap))
-- (4.2) Count and depth -------------------------------------------------------
countBTree = cataBTree (either (const 0) (succ . (uncurry (+)) . p2))
depthBTree = cataBTree (either zero (succ.umax.p2))
-- (4.3) Serialization ---------------------------------------------------------
inordt = cataBTree inord -- in-order traversal
preordt = cataBTree preord -- pre-order traversal
postordt = cataBTree posord -- post-order traversal
-- where
preord = either nil f where f(x,(l,r))=x:l++r
inord = either nil f where f(x,(l,r))=l++[x]++r
posord = either nil f where f(x,(l,r))=l++r++[x]
-- (4.4) Quicksort -------------------------------------------------------------
qSort :: Ord a => [a] -> [a]
qSort = hyloBTree inord qsep -- the same as (cataBTree inord) . (anaBTree qsep)
-- where
qsep [] = i1 ()
qsep (h:t) = i2 (h,(s,l)) where (s,l) = part ( Bool) -> [a] -> ([a], [a])
part p [] = ([],[])
part p (h:t) | p h = let (s,l) = part p t in (h:s,l)
| otherwise = let (s,l) = part p t in (s,h:l)
{-- pointwise versions:
qSort [] = []
qSort (h:t) = let (t1,t2) = part (= h ]
--}
-- (4.5) Traces ----------------------------------------------------------------
traces :: Eq a => BTree a -> [[a]]
traces = cataBTree (either (const [[]]) tunion)
where tunion(a,(l,r)) = union (map (a:) l) (map (a:) r)
-- (4.6) Towers of Hanoi -------------------------------------------------------
-- pointwise:
-- hanoi(d,0) = []
-- hanoi(d,n+1) = (hanoi (not d,n)) ++ [(n,d)] ++ (hanoi (not d, n))
hanoi = hyloBTree present strategy
--- where
present = inord -- same as in qSort
strategy(d,0) = Left ()
strategy(d,n+1) = Right ((n,d),((not d,n),(not d,n)))
{--
The Towers of Hanoi problem comes from a puzzle marketed in 1883
by the French mathematician Édouard Lucas, under the pseudonym
Claus. The puzzle is based on a legend according to which
there is a temple, apparently in Bramah rather than in Hanoi as
one might expect, where there are three giant poles fixed in the
ground. On the first of these poles, at the time of the world's
creation, God placed sixty four golden disks, each of different
size, in decreasing order of size. The Bramin monks were given
the task of moving the disks, one per day, from one pole to another
subject to the rule that no disk may ever be above a smaller disk.
The monks' task would be complete when they had succeeded in moving
all the disks from the first of the poles to the second and, on
the day that they completed their task the world would come to
an end!
There is a wellknown inductive solution to the problem given
by the pseudocode below. In this solution we make use of the fact
that the given problem is symmetrical with respect to all three
poles. Thus it is undesirable to name the individual poles. Instead
we visualize the poles as being arranged in a circle; the problem
is to move the tower of disks from one pole to the next pole in
a specified direction around the circle. The code defines H n d
to be a sequence of pairs (k,d') where n is the number of disks,
k is a disk number and d and d' are directions. Disks are numbered
from 0 onwards, disk 0 being the smallest. (Assigning number 0
to the smallest rather than the largest disk has the advantage
that the number of the disk that is moved on any day is independent
of the total number of disks to be moved.) Directions are boolean
values, true representing a clockwise movement and false an anticlockwise
movement. The pair (k,d') means move the disk numbered k from
its current position in the direction d'. The semicolon operator
concatenates sequences together, [] denotes an empty sequence
and [x] is a sequence with exactly one element x. Taking the pairs
in order from left to right, the complete sequence H n d prescribes
how to move the n smallest disks onebyone from one pole to the
next pole in the direction d following the rule of never placing
a larger disk on top of a smaller disk.
H 0 d = [],
H (n+1) d = H n ¬d ; [ (n, d) ] ; H n ¬d.
(excerpt from R. Backhouse, M. Fokkinga / Information Processing
Letters 77 (2001) 71--76)
--}
-- (5) Depth and balancing (using mutual recursion) --------------------------
balBTree = p1.baldepth
baldepthBTree = p2.baldepth
baldepth = cataBTree g where
g = either (const(True,1)) (h.(id> (a -> c) -> Either () (a,(c, c)) -> c
tnat f = either (const mempty) (theta . (f >< theta))
where theta = uncurry mappend
-- monoid reduction
monBTree f = cataBTree (tnat f)
-- alternative to (4.2) serialization ----------------------------------------
preordt' = monBTree singl
-- alternative to (4.1) counting ---------------------------------------------
countBTree' = monBTree (const (Sum 1))
-- (7) Zipper ----------------------------------------------------------------
data Deriv a = Dr Bool a (BTree a)
type Zipper a = [ Deriv a ]
plug :: Zipper a -> BTree a -> BTree a
plug [] t = t
plug ((Dr False a l):z) t = Node (a,(plug z t,l))
plug ((Dr True a r):z) t = Node (a,(r,plug z t))
---------------------------- end of library ----------------------------------