{-# OPTIONS_GHC -fglasgow-exts #-} ----------------------------------------------------------------------------- -- | -- Module : Data.Tree.AVL.Internals.HSet -- Copyright : (c) Adrian Hey 2004,2005 -- License : BSD3 -- -- Maintainer : http://homepages.nildram.co.uk/~ahey/em.png -- Stability : stable -- Portability : portable -- -- Set primitives on AVL trees with (height information supplied where needed). -- All the functions in this module use essentially the same symetric \"Divide and Conquer\" algorithm. ----------------------------------------------------------------------------- module Data.Tree.AVL.Internals.HSet (-- * Union primitives. unionH,unionMaybeH, -- * Intersection primitives. intersectionH,intersectionMaybeH, -- * Difference primitives. differenceH,differenceMaybeH,symDifferenceH, ) where import Data.Tree.AVL.Types(AVL(..)) import Data.Tree.AVL.Internals.HJoin(spliceH,joinH) import Data.COrdering #ifdef __GLASGOW_HASKELL__ import GHC.Base #include "ghcdefs.h" #else #include "h98defs.h" #endif -- | Uses the supplied combining comparison to evaluate the union of two sets represented as -- sorted AVL trees of known height. Whenever the combining comparison is applied, the first -- comparison argument is an element of the first tree and the second comparison argument is -- an element of the second tree. -- -- Complexity: Not sure, but I'd appreciate it if someone could figure it out. -- (Faster than Hedge union from Data.Set at any rate). unionH :: (e -> e -> COrdering e) -> AVL e -> UINT -> AVL e -> UINT -> UBT2(AVL e,UINT) unionH c = u where -- u :: AVL e -> UINT -> AVL e -> UINT -> UBT2(AVL e,UINT) u E _ t1 h1 = UBT2(t1,h1) u t0 h0 E _ = UBT2(t0,h0) u (N l0 e0 r0) h0 (N l1 e1 r1) h1 = u_ l0 DECINT2(h0) e0 r0 DECINT1(h0) l1 DECINT2(h1) e1 r1 DECINT1(h1) u (N l0 e0 r0) h0 (Z l1 e1 r1) h1 = u_ l0 DECINT2(h0) e0 r0 DECINT1(h0) l1 DECINT1(h1) e1 r1 DECINT1(h1) u (N l0 e0 r0) h0 (P l1 e1 r1) h1 = u_ l0 DECINT2(h0) e0 r0 DECINT1(h0) l1 DECINT1(h1) e1 r1 DECINT2(h1) u (Z l0 e0 r0) h0 (N l1 e1 r1) h1 = u_ l0 DECINT1(h0) e0 r0 DECINT1(h0) l1 DECINT2(h1) e1 r1 DECINT1(h1) u (Z l0 e0 r0) h0 (Z l1 e1 r1) h1 = u_ l0 DECINT1(h0) e0 r0 DECINT1(h0) l1 DECINT1(h1) e1 r1 DECINT1(h1) u (Z l0 e0 r0) h0 (P l1 e1 r1) h1 = u_ l0 DECINT1(h0) e0 r0 DECINT1(h0) l1 DECINT1(h1) e1 r1 DECINT2(h1) u (P l0 e0 r0) h0 (N l1 e1 r1) h1 = u_ l0 DECINT1(h0) e0 r0 DECINT2(h0) l1 DECINT2(h1) e1 r1 DECINT1(h1) u (P l0 e0 r0) h0 (Z l1 e1 r1) h1 = u_ l0 DECINT1(h0) e0 r0 DECINT2(h0) l1 DECINT1(h1) e1 r1 DECINT1(h1) u (P l0 e0 r0) h0 (P l1 e1 r1) h1 = u_ l0 DECINT1(h0) e0 r0 DECINT2(h0) l1 DECINT1(h1) e1 r1 DECINT2(h1) u_ l0 hl0 e0 r0 hr0 l1 hl1 e1 r1 hr1 = case c e0 e1 of -- e0 < e1, so (l0 < e0 < e1) & (e0 < e1 < r1) Lt -> case forkR r0 hr0 e1 of UBT5(rl0,hrl0,e1_,rr0,hrr0) -> case forkL e0 l1 hl1 of -- (e0 < rl0 < e1) & (e0 < e1 < rr0) UBT5(ll1,hll1,e0_,lr1,hlr1) -> -- (ll1 < e0 < e1) & (e0 < lr1 < e1) -- (l0 + ll1) < e0 < (rl0 + lr1) < e1 < (rr0 + r1) case u l0 hl0 ll1 hll1 of UBT2(l,hl) -> case u rl0 hrl0 lr1 hlr1 of UBT2(m,hm) -> case u rr0 hrr0 r1 hr1 of UBT2(r,hr) -> case spliceH m hm e1_ r hr of UBT2(t,ht) -> spliceH l hl e0_ t ht -- e0 = e1 Eq e -> case u l0 hl0 l1 hl1 of UBT2(l,hl) -> case u r0 hr0 r1 hr1 of UBT2(r,hr) -> spliceH l hl e r hr -- e1 < e0, so (l1 < e1 < e0) & (e1 < e0 < r0) Gt -> case forkL e0 r1 hr1 of UBT5(rl1,hrl1,e0_,rr1,hrr1) -> case forkR l0 hl0 e1 of -- (e1 < rl1 < e0) & (e1 < e0 < rr1) UBT5(ll0,hll0,e1_,lr0,hlr0) -> -- (ll0 < e1 < e0) & (e1 < lr0 < e0) -- (ll0 + l1) < e1 < (lr0 + rl1) < e0 < (r0 + rr1) case u ll0 hll0 l1 hl1 of UBT2(l,hl) -> case u lr0 hlr0 rl1 hrl1 of UBT2(m,hm) -> case u r0 hr0 rr1 hrr1 of UBT2(r,hr) -> case spliceH l hl e1_ m hm of UBT2(t,ht) -> spliceH t ht e0_ r hr -- We need 2 different versions of fork (L & R) to ensure that comparison arguments are used in -- the right order (c e0 e1) -- forkL :: e -> AVL e -> UINT -> UBT5(AVL e,UINT,e,AVL e,UINT) forkL e0 t1 ht1 = forkL_ t1 ht1 where forkL_ E _ = UBT5(E, L(0), e0, E, L(0)) forkL_ (N l e r) h = forkL__ l DECINT2(h) e r DECINT1(h) forkL_ (Z l e r) h = forkL__ l DECINT1(h) e r DECINT1(h) forkL_ (P l e r) h = forkL__ l DECINT1(h) e r DECINT2(h) forkL__ l hl e r hr = case c e0 e of Lt -> case forkL_ l hl of UBT5(l0,hl0,e0_,l1,hl1) -> case spliceH l1 hl1 e r hr of UBT2(l1_,hl1_) -> UBT5(l0,hl0,e0_,l1_,hl1_) Eq e0_ -> UBT5(l,hl,e0_,r,hr) Gt -> case forkL_ r hr of UBT5(l0,hl0,e0_,l1,hl1) -> case spliceH l hl e l0 hl0 of UBT2(l0_,hl0_) -> UBT5(l0_,hl0_,e0_,l1,hl1) -- forkR :: AVL e -> UINT -> e -> UBT5(AVL e,UINT,e,AVL e,UINT) forkR t0 ht0 e1 = forkR_ t0 ht0 where forkR_ E _ = UBT5(E, L(0), e1, E, L(0)) forkR_ (N l e r) h = forkR__ l DECINT2(h) e r DECINT1(h) forkR_ (Z l e r) h = forkR__ l DECINT1(h) e r DECINT1(h) forkR_ (P l e r) h = forkR__ l DECINT1(h) e r DECINT2(h) forkR__ l hl e r hr = case c e e1 of Lt -> case forkR_ r hr of UBT5(l0,hl0,e1_,l1,hl1) -> case spliceH l hl e l0 hl0 of UBT2(l0_,hl0_) -> UBT5(l0_,hl0_,e1_,l1,hl1) Eq e1_ -> UBT5(l,hl,e1_,r,hr) Gt -> case forkR_ l hl of UBT5(l0,hl0,e1_,l1,hl1) -> case spliceH l1 hl1 e r hr of UBT2(l1_,hl1_) -> UBT5(l0,hl0,e1_,l1_,hl1_) ----------------------------------------------------------------------- -------------------------- unionH Ends Here --------------------------- ----------------------------------------------------------------------- -- | Similar to _unionH_, but the resulting tree does not include elements in cases where -- the supplied combining comparison returns @(Eq Nothing)@. -- -- Complexity: Not sure, but I_d appreciate it if someone could figure it out. unionMaybeH :: (e -> e -> COrdering (Maybe e)) -> AVL e -> UINT -> AVL e -> UINT -> UBT2(AVL e,UINT) unionMaybeH c = u where -- u :: AVL e -> UINT -> AVL e -> UINT -> UBT2(AVL e,UINT) u E _ t1 h1 = UBT2(t1,h1) u t0 h0 E _ = UBT2(t0,h0) u (N l0 e0 r0) h0 (N l1 e1 r1) h1 = u_ l0 DECINT2(h0) e0 r0 DECINT1(h0) l1 DECINT2(h1) e1 r1 DECINT1(h1) u (N l0 e0 r0) h0 (Z l1 e1 r1) h1 = u_ l0 DECINT2(h0) e0 r0 DECINT1(h0) l1 DECINT1(h1) e1 r1 DECINT1(h1) u (N l0 e0 r0) h0 (P l1 e1 r1) h1 = u_ l0 DECINT2(h0) e0 r0 DECINT1(h0) l1 DECINT1(h1) e1 r1 DECINT2(h1) u (Z l0 e0 r0) h0 (N l1 e1 r1) h1 = u_ l0 DECINT1(h0) e0 r0 DECINT1(h0) l1 DECINT2(h1) e1 r1 DECINT1(h1) u (Z l0 e0 r0) h0 (Z l1 e1 r1) h1 = u_ l0 DECINT1(h0) e0 r0 DECINT1(h0) l1 DECINT1(h1) e1 r1 DECINT1(h1) u (Z l0 e0 r0) h0 (P l1 e1 r1) h1 = u_ l0 DECINT1(h0) e0 r0 DECINT1(h0) l1 DECINT1(h1) e1 r1 DECINT2(h1) u (P l0 e0 r0) h0 (N l1 e1 r1) h1 = u_ l0 DECINT1(h0) e0 r0 DECINT2(h0) l1 DECINT2(h1) e1 r1 DECINT1(h1) u (P l0 e0 r0) h0 (Z l1 e1 r1) h1 = u_ l0 DECINT1(h0) e0 r0 DECINT2(h0) l1 DECINT1(h1) e1 r1 DECINT1(h1) u (P l0 e0 r0) h0 (P l1 e1 r1) h1 = u_ l0 DECINT1(h0) e0 r0 DECINT2(h0) l1 DECINT1(h1) e1 r1 DECINT2(h1) u_ l0 hl0 e0 r0 hr0 l1 hl1 e1 r1 hr1 = case c e0 e1 of -- e0 < e1, so (l0 < e0 < e1) & (e0 < e1 < r1) Lt -> case forkR r0 hr0 e1 of UBT5(rl0,hrl0,mbe1_,rr0,hrr0) -> case forkL e0 l1 hl1 of -- (e0 < rl0 < e1) & (e0 < e1 < rr0) UBT5(ll1,hll1,mbe0_,lr1,hlr1) -> -- (ll1 < e0 < e1) & (e0 < lr1 < e1) -- (l0 + ll1) < e0 < (rl0 + lr1) < e1 < (rr0 + r1) case u l0 hl0 ll1 hll1 of UBT2(l,hl) -> case u rl0 hrl0 lr1 hlr1 of UBT2(m,hm) -> case u rr0 hrr0 r1 hr1 of UBT2(r,hr) -> case (case mbe1_ of Just e1_ -> spliceH m hm e1_ r hr Nothing -> joinH m hm r hr ) of UBT2(t,ht) -> case mbe0_ of Just e0_ -> spliceH l hl e0_ t ht Nothing -> joinH l hl t ht -- e0 = e1 Eq mbe -> case u l0 hl0 l1 hl1 of UBT2(l,hl) -> case u r0 hr0 r1 hr1 of UBT2(r,hr) -> case mbe of Just e -> spliceH l hl e r hr Nothing -> joinH l hl r hr -- e1 < e0, so (l1 < e1 < e0) & (e1 < e0 < r0) Gt -> case forkL e0 r1 hr1 of UBT5(rl1,hrl1,mbe0_,rr1,hrr1) -> case forkR l0 hl0 e1 of -- (e1 < rl1 < e0) & (e1 < e0 < rr1) UBT5(ll0,hll0,mbe1_,lr0,hlr0) -> -- (ll0 < e1 < e0) & (e1 < lr0 < e0) -- (ll0 + l1) < e1 < (lr0 + rl1) < e0 < (r0 + rr1) case u ll0 hll0 l1 hl1 of UBT2(l,hl) -> case u lr0 hlr0 rl1 hrl1 of UBT2(m,hm) -> case u r0 hr0 rr1 hrr1 of UBT2(r,hr) -> case (case mbe1_ of Just e1_ -> spliceH l hl e1_ m hm Nothing -> joinH l hl m hm ) of UBT2(t,ht) -> case mbe0_ of Just e0_ -> spliceH t ht e0_ r hr Nothing -> joinH t ht r hr -- We need 2 different versions of fork (L & R) to ensure that comparison arguments are used in -- the right order (c e0 e1) -- forkL :: e -> AVL e -> UINT -> UBT5(AVL e,UINT,Maybe e,AVL e,UINT) forkL e0 t1 ht1 = forkL_ t1 ht1 where forkL_ E _ = UBT5(E, L(0), Just e0, E, L(0)) forkL_ (N l e r) h = forkL__ l DECINT2(h) e r DECINT1(h) forkL_ (Z l e r) h = forkL__ l DECINT1(h) e r DECINT1(h) forkL_ (P l e r) h = forkL__ l DECINT1(h) e r DECINT2(h) forkL__ l hl e r hr = case c e0 e of Lt -> case forkL_ l hl of UBT5(l0,hl0,mbe0_,l1,hl1) -> case spliceH l1 hl1 e r hr of UBT2(l1_,hl1_) -> UBT5(l0,hl0,mbe0_,l1_,hl1_) Eq mbe0_ -> UBT5(l,hl,mbe0_,r,hr) Gt -> case forkL_ r hr of UBT5(l0,hl0,mbe0_,l1,hl1) -> case spliceH l hl e l0 hl0 of UBT2(l0_,hl0_) -> UBT5(l0_,hl0_,mbe0_,l1,hl1) -- forkR :: AVL e -> UINT -> e -> UBT5(AVL e,UINT,Maybe e,AVL e,UINT) forkR t0 ht0 e1 = forkR_ t0 ht0 where forkR_ E _ = UBT5(E, L(0), Just e1, E, L(0)) forkR_ (N l e r) h = forkR__ l DECINT2(h) e r DECINT1(h) forkR_ (Z l e r) h = forkR__ l DECINT1(h) e r DECINT1(h) forkR_ (P l e r) h = forkR__ l DECINT1(h) e r DECINT2(h) forkR__ l hl e r hr = case c e e1 of Lt -> case forkR_ r hr of UBT5(l0,hl0,mbe1_,l1,hl1) -> case spliceH l hl e l0 hl0 of UBT2(l0_,hl0_) -> UBT5(l0_,hl0_,mbe1_,l1,hl1) Eq mbe1_ -> UBT5(l,hl,mbe1_,r,hr) Gt -> case forkR_ l hl of UBT5(l0,hl0,mbe1_,l1,hl1) -> case spliceH l1 hl1 e r hr of UBT2(l1_,hl1_) -> UBT5(l0,hl0,mbe1_,l1_,hl1_) ----------------------------------------------------------------------- ----------------------- unionMaybeH Ends Here ------------------------- ----------------------------------------------------------------------- -- | Uses the supplied combining comparison to evaluate the intersection of two sets represented as -- sorted AVL trees. This function requires no height information at all for -- the two tree inputs. The absolute height of the resulting tree is returned also. -- -- Complexity: Not sure, but I_d appreciate it if someone could figure it out. intersectionH :: (a -> b -> COrdering c) -> AVL a -> AVL b -> UBT2(AVL c,UINT) intersectionH comp = i where -- i :: AVL a -> AVL b -> UBT2(AVL c,UINT) i E _ = UBT2(E,L(0)) i _ E = UBT2(E,L(0)) i (N l0 e0 r0) (N l1 e1 r1) = i_ l0 e0 r0 l1 e1 r1 i (N l0 e0 r0) (Z l1 e1 r1) = i_ l0 e0 r0 l1 e1 r1 i (N l0 e0 r0) (P l1 e1 r1) = i_ l0 e0 r0 l1 e1 r1 i (Z l0 e0 r0) (N l1 e1 r1) = i_ l0 e0 r0 l1 e1 r1 i (Z l0 e0 r0) (Z l1 e1 r1) = i_ l0 e0 r0 l1 e1 r1 i (Z l0 e0 r0) (P l1 e1 r1) = i_ l0 e0 r0 l1 e1 r1 i (P l0 e0 r0) (N l1 e1 r1) = i_ l0 e0 r0 l1 e1 r1 i (P l0 e0 r0) (Z l1 e1 r1) = i_ l0 e0 r0 l1 e1 r1 i (P l0 e0 r0) (P l1 e1 r1) = i_ l0 e0 r0 l1 e1 r1 i_ l0 e0 r0 l1 e1 r1 = case comp e0 e1 of -- e0 < e1, so (l0 < e0 < e1) & (e0 < e1 < r1) Lt -> case forkR r0 e1 of UBT5(rl0,_,mbc1,rr0,_) -> case forkL e0 l1 of -- (e0 < rl0 < e1) & (e0 < e1 < rr0) UBT5(ll1,_,mbc0,lr1,_) -> -- (ll1 < e0 < e1) & (e0 < lr1 < e1) -- (l0 + ll1) < e0 < (rl0 + lr1) < e1 < (rr0 + r1) case i rr0 r1 of UBT2(r,hr) -> case i rl0 lr1 of UBT2(m,hm) -> case i l0 ll1 of UBT2(l,hl) -> case (case mbc1 of Just c1 -> spliceH m hm c1 r hr Nothing -> joinH m hm r hr ) of UBT2(t,ht) -> case mbc0 of Just c0 -> spliceH l hl c0 t ht Nothing -> joinH l hl t ht -- e0 = e1 Eq c -> case i l0 l1 of UBT2(l,hl) -> case i r0 r1 of UBT2(r,hr) -> spliceH l hl c r hr -- e1 < e0, so (l1 < e1 < e0) & (e1 < e0 < r0) Gt -> case forkL e0 r1 of UBT5(rl1,_,mbc0,rr1,_) -> case forkR l0 e1 of -- (e1 < rl1 < e0) & (e1 < e0 < rr1) UBT5(ll0,_,mbc1,lr0,_) -> -- (ll0 < e1 < e0) & (e1 < lr0 < e0) -- (ll0 + l1) < e1 < (lr0 + rl1) < e0 < (r0 + rr1) case i r0 rr1 of UBT2(r,hr) -> case i lr0 rl1 of UBT2(m,hm) -> case i ll0 l1 of UBT2(l,hl) -> case (case mbc0 of Just c0 -> spliceH m hm c0 r hr Nothing -> joinH m hm r hr ) of UBT2(t,ht) -> case mbc1 of Just c1 -> spliceH l hl c1 t ht Nothing -> joinH l hl t ht -- We need 2 different versions of fork (L & R) to ensure that comparison arguments are used in -- the right order (c e0 e1) -- forkL :: a -> AVL b -> UBT5(AVL b,UINT,Maybe c,AVL b,UINT) forkL e0 t1 = forkL_ t1 L(0) where forkL_ E h = UBT5(E,h,Nothing,E,h) -- Relative heights!! forkL_ (N l e r) h = forkL__ l DECINT2(h) e r DECINT1(h) forkL_ (Z l e r) h = forkL__ l DECINT1(h) e r DECINT1(h) forkL_ (P l e r) h = forkL__ l DECINT1(h) e r DECINT2(h) forkL__ l hl e r hr = case comp e0 e of Lt -> case forkL_ l hl of UBT5(l0,hl0,mbc0,l1,hl1) -> case spliceH l1 hl1 e r hr of UBT2(l1_,hl1_) -> UBT5(l0,hl0,mbc0,l1_,hl1_) Eq c0 -> UBT5(l,hl,Just c0,r,hr) Gt -> case forkL_ r hr of UBT5(l0,hl0,mbc0,l1,hl1) -> case spliceH l hl e l0 hl0 of UBT2(l0_,hl0_) -> UBT5(l0_,hl0_,mbc0,l1,hl1) -- forkR :: AVL a -> b -> UBT5(AVL a,UINT,Maybe c,AVL a,UINT) forkR t0 e1 = forkR_ t0 L(0) where forkR_ E h = UBT5(E,h,Nothing,E,h) -- Relative heights!! forkR_ (N l e r) h = forkR__ l DECINT2(h) e r DECINT1(h) forkR_ (Z l e r) h = forkR__ l DECINT1(h) e r DECINT1(h) forkR_ (P l e r) h = forkR__ l DECINT1(h) e r DECINT2(h) forkR__ l hl e r hr = case comp e e1 of Lt -> case forkR_ r hr of UBT5(l0,hl0,mbc1,l1,hl1) -> case spliceH l hl e l0 hl0 of UBT2(l0_,hl0_) -> UBT5(l0_,hl0_,mbc1,l1,hl1) Eq c1 -> UBT5(l,hl,Just c1,r,hr) Gt -> case forkR_ l hl of UBT5(l0,hl0,mbc1,l1,hl1) -> case spliceH l1 hl1 e r hr of UBT2(l1_,hl1_) -> UBT5(l0,hl0,mbc1,l1_,hl1_) ----------------------------------------------------------------------- ---------------------- intersectionH Ends Here ------------------------ ----------------------------------------------------------------------- -- | Similar to _intersectionH_, but the resulting tree does not include elements in cases where -- the supplied combining comparison returns @(Eq Nothing)@. -- -- Complexity: Not sure, but I_d appreciate it if someone could figure it out. intersectionMaybeH :: (a -> b -> COrdering (Maybe c)) -> AVL a -> AVL b -> UBT2(AVL c,UINT) intersectionMaybeH comp = i where -- i :: AVL a -> AVL b -> UBT2(AVL c,UINT) i E _ = UBT2(E,L(0)) i _ E = UBT2(E,L(0)) i (N l0 e0 r0) (N l1 e1 r1) = i_ l0 e0 r0 l1 e1 r1 i (N l0 e0 r0) (Z l1 e1 r1) = i_ l0 e0 r0 l1 e1 r1 i (N l0 e0 r0) (P l1 e1 r1) = i_ l0 e0 r0 l1 e1 r1 i (Z l0 e0 r0) (N l1 e1 r1) = i_ l0 e0 r0 l1 e1 r1 i (Z l0 e0 r0) (Z l1 e1 r1) = i_ l0 e0 r0 l1 e1 r1 i (Z l0 e0 r0) (P l1 e1 r1) = i_ l0 e0 r0 l1 e1 r1 i (P l0 e0 r0) (N l1 e1 r1) = i_ l0 e0 r0 l1 e1 r1 i (P l0 e0 r0) (Z l1 e1 r1) = i_ l0 e0 r0 l1 e1 r1 i (P l0 e0 r0) (P l1 e1 r1) = i_ l0 e0 r0 l1 e1 r1 i_ l0 e0 r0 l1 e1 r1 = case comp e0 e1 of -- e0 < e1, so (l0 < e0 < e1) & (e0 < e1 < r1) Lt -> case forkR r0 e1 of UBT5(rl0,_,mbc1,rr0,_) -> case forkL e0 l1 of -- (e0 < rl0 < e1) & (e0 < e1 < rr0) UBT5(ll1,_,mbc0,lr1,_) -> -- (ll1 < e0 < e1) & (e0 < lr1 < e1) -- (l0 + ll1) < e0 < (rl0 + lr1) < e1 < (rr0 + r1) case i rr0 r1 of UBT2(r,hr) -> case i rl0 lr1 of UBT2(m,hm) -> case i l0 ll1 of UBT2(l,hl) -> case (case mbc1 of Just c1 -> spliceH m hm c1 r hr Nothing -> joinH m hm r hr ) of UBT2(t,ht) -> case mbc0 of Just c0 -> spliceH l hl c0 t ht Nothing -> joinH l hl t ht -- e0 = e1 Eq mbc -> case i l0 l1 of UBT2(l,hl) -> case i r0 r1 of UBT2(r,hr) -> case mbc of Just c -> spliceH l hl c r hr Nothing -> joinH l hl r hr -- e1 < e0, so (l1 < e1 < e0) & (e1 < e0 < r0) Gt -> case forkL e0 r1 of UBT5(rl1,_,mbc0,rr1,_) -> case forkR l0 e1 of -- (e1 < rl1 < e0) & (e1 < e0 < rr1) UBT5(ll0,_,mbc1,lr0,_) -> -- (ll0 < e1 < e0) & (e1 < lr0 < e0) -- (ll0 + l1) < e1 < (lr0 + rl1) < e0 < (r0 + rr1) case i r0 rr1 of UBT2(r,hr) -> case i lr0 rl1 of UBT2(m,hm) -> case i ll0 l1 of UBT2(l,hl) -> case (case mbc0 of Just c0 -> spliceH m hm c0 r hr Nothing -> joinH m hm r hr ) of UBT2(t,ht) -> case mbc1 of Just c1 -> spliceH l hl c1 t ht Nothing -> joinH l hl t ht -- We need 2 different versions of fork (L & R) to ensure that comparison arguments are used in -- the right order (c e0 e1) -- forkL :: a -> AVL b -> UBT5(AVL b,UINT,Maybe c,AVL b,UINT) forkL e0 t1 = forkL_ t1 L(0) where forkL_ E h = UBT5(E,h,Nothing,E,h) -- Relative heights!! forkL_ (N l e r) h = forkL__ l DECINT2(h) e r DECINT1(h) forkL_ (Z l e r) h = forkL__ l DECINT1(h) e r DECINT1(h) forkL_ (P l e r) h = forkL__ l DECINT1(h) e r DECINT2(h) forkL__ l hl e r hr = case comp e0 e of Lt -> case forkL_ l hl of UBT5(l0,hl0,mbc0,l1,hl1) -> case spliceH l1 hl1 e r hr of UBT2(l1_,hl1_) -> UBT5(l0,hl0,mbc0,l1_,hl1_) Eq mbc0_ -> UBT5(l,hl,mbc0_,r,hr) Gt -> case forkL_ r hr of UBT5(l0,hl0,mbc0,l1,hl1) -> case spliceH l hl e l0 hl0 of UBT2(l0_,hl0_) -> UBT5(l0_,hl0_,mbc0,l1,hl1) -- forkR :: AVL a -> b -> UBT5(AVL a,UINT,Maybe c,AVL a,UINT) forkR t0 e1 = forkR_ t0 L(0) where forkR_ E h = UBT5(E,h,Nothing,E,h) -- Relative heights!! forkR_ (N l e r) h = forkR__ l DECINT2(h) e r DECINT1(h) forkR_ (Z l e r) h = forkR__ l DECINT1(h) e r DECINT1(h) forkR_ (P l e r) h = forkR__ l DECINT1(h) e r DECINT2(h) forkR__ l hl e r hr = case comp e e1 of Lt -> case forkR_ r hr of UBT5(l0,hl0,mbc1,l1,hl1) -> case spliceH l hl e l0 hl0 of UBT2(l0_,hl0_) -> UBT5(l0_,hl0_,mbc1,l1,hl1) Eq mbc1_ -> UBT5(l,hl,mbc1_,r,hr) Gt -> case forkR_ l hl of UBT5(l0,hl0,mbc1,l1,hl1) -> case spliceH l1 hl1 e r hr of UBT2(l1_,hl1_) -> UBT5(l0,hl0,mbc1,l1_,hl1_) ----------------------------------------------------------------------- -------------------- intersectionMaybeH Ends Here --------------------- ----------------------------------------------------------------------- -- | Uses the supplied comparison to evaluate the difference between two sets represented as -- sorted AVL trees. -- -- N.B. This function works with relative heights for the first tree and needs no height -- information for the second tree, so it_s OK to initialise the height of the first to zero, -- rather than calculating the absolute height. However, if you do this the height of the resulting -- tree will be incorrect also (it will have the same fixed offset as the first tree). -- -- Complexity: Not sure, but I_d appreciate it if someone could figure it out. differenceH :: (a -> b -> Ordering) -> AVL a -> UINT -> AVL b -> UBT2(AVL a,UINT) differenceH comp = d where -- d :: AVL a -> UINT -> AVL b -> UBT2(AVL a,UINT) d E h0 _ = UBT2(E ,h0) -- Relative heights!! d t0 h0 E = UBT2(t0,h0) d (N l0 e0 r0) h0 (N l1 e1 r1) = d_ l0 DECINT2(h0) e0 r0 DECINT1(h0) l1 e1 r1 d (N l0 e0 r0) h0 (Z l1 e1 r1) = d_ l0 DECINT2(h0) e0 r0 DECINT1(h0) l1 e1 r1 d (N l0 e0 r0) h0 (P l1 e1 r1) = d_ l0 DECINT2(h0) e0 r0 DECINT1(h0) l1 e1 r1 d (Z l0 e0 r0) h0 (N l1 e1 r1) = d_ l0 DECINT1(h0) e0 r0 DECINT1(h0) l1 e1 r1 d (Z l0 e0 r0) h0 (Z l1 e1 r1) = d_ l0 DECINT1(h0) e0 r0 DECINT1(h0) l1 e1 r1 d (Z l0 e0 r0) h0 (P l1 e1 r1) = d_ l0 DECINT1(h0) e0 r0 DECINT1(h0) l1 e1 r1 d (P l0 e0 r0) h0 (N l1 e1 r1) = d_ l0 DECINT1(h0) e0 r0 DECINT2(h0) l1 e1 r1 d (P l0 e0 r0) h0 (Z l1 e1 r1) = d_ l0 DECINT1(h0) e0 r0 DECINT2(h0) l1 e1 r1 d (P l0 e0 r0) h0 (P l1 e1 r1) = d_ l0 DECINT1(h0) e0 r0 DECINT2(h0) l1 e1 r1 d_ l0 hl0 e0 r0 hr0 l1 e1 r1 = case comp e0 e1 of -- e0 < e1, so (l0 < e0 < e1) & (e0 < e1 < r1) LT -> case forkR r0 hr0 e1 of UBT4(rl0,hrl0, rr0,hrr0) -> case forkL e0 l1 of -- (e0 < rl0 < e1) & (e0 < e1 < rr0) UBT5(ll1,_ ,be0,lr1,_ ) -> -- (ll1 < e0 < e1) & (e0 < lr1 < e1) -- (l0 + ll1) < e0 < (rl0 + lr1) < e1 < (rr0 + r1) case d rr0 hrr0 r1 of -- right UBT2(r,hr) -> case d rl0 hrl0 lr1 of -- middle UBT2(m,hm) -> case d l0 hl0 ll1 of -- left UBT2(l,hl) -> case joinH m hm r hr of -- join middle right UBT2(y,hy) -> if be0 then spliceH l hl e0 y hy else joinH l hl y hy -- e0 = e1 EQ -> case d r0 hr0 r1 of -- right UBT2(r,hr) -> case d l0 hl0 l1 of -- left UBT2(l,hl) -> joinH l hl r hr -- e1 < e0, so (l1 < e1 < e0) & (e1 < e0 < r0) GT -> case forkL e0 r1 of UBT5(rl1,_ ,be0,rr1,_ ) -> case forkR l0 hl0 e1 of -- (e1 < rl1 < e0) & (e1 < e0 < rr1) UBT4(ll0,hll0, lr0,hlr0) -> -- (ll0 < e1 < e0) & (e1 < lr0 < e0) -- (ll0 + l1) < e1 < (lr0 + rl1) < e0 < (r0 + rr1) case d r0 hr0 rr1 of -- right UBT2(r,hr) -> case d lr0 hlr0 rl1 of -- middle UBT2(m,hm) -> case d ll0 hll0 l1 of -- left UBT2(l,hl) -> case joinH l hl m hm of -- join left middle UBT2(x,hx) -> if be0 then spliceH x hx e0 r hr else joinH x hx r hr -- We need 2 different versions of fork (L & R) to ensure that comparison arguments are used in -- the right order (c e0 e1), and for other algorithmic reasons in this case. -- N.B. forkL returns True if t1 does not contain e0 (I.E. If e0 is an element of the result). -- forkL :: a -> AVL b -> UBT5(AVL b, UINT, Bool, AVL b, UINT) forkL e0 t1 = forkL_ t1 L(0) where forkL_ E h = UBT5(E,h,True,E,h) -- Relative heights!! forkL_ (N l e r) h = forkL__ l DECINT2(h) e r DECINT1(h) forkL_ (Z l e r) h = forkL__ l DECINT1(h) e r DECINT1(h) forkL_ (P l e r) h = forkL__ l DECINT1(h) e r DECINT2(h) forkL__ l hl e r hr = case comp e0 e of LT -> case forkL_ l hl of UBT5(x0,hx0,be0,x1,hx1) -> case spliceH x1 hx1 e r hr of UBT2(x1_,hx1_) -> UBT5(x0,hx0,be0,x1_,hx1_) EQ -> UBT5(l,hl,False,r,hr) GT -> case forkL_ r hr of UBT5(x0,hx0,be0,x1,hx1) -> case spliceH l hl e x0 hx0 of UBT2(x0_,hx0_) -> UBT5(x0_,hx0_,be0,x1,hx1) -- N.B. forkR t0, according to e1. Neither of the resulting forks will contain an element -- which is "equal" to e1. -- forkR :: AVL a -> UINT -> b -> UBT4(AVL a, UINT, AVL a, UINT) forkR t0 ht0 e1 = forkR_ t0 ht0 where forkR_ E h = UBT4(E,h,E,h) -- Relative heights!! forkR_ (N l e r) h = forkR__ l DECINT2(h) e r DECINT1(h) forkR_ (Z l e r) h = forkR__ l DECINT1(h) e r DECINT1(h) forkR_ (P l e r) h = forkR__ l DECINT1(h) e r DECINT2(h) forkR__ l hl e r hr = case comp e e1 of LT -> case forkR_ r hr of UBT4(x0,hx0,x1,hx1) -> case spliceH l hl e x0 hx0 of UBT2(x0_,hx0_) -> UBT4(x0_,hx0_,x1,hx1) EQ -> UBT4(l,hl,r,hr) -- e1 is dropped. GT -> case forkR_ l hl of UBT4(x0,hx0,x1,hx1) -> case spliceH x1 hx1 e r hr of UBT2(x1_,hx1_) -> UBT4(x0,hx0,x1_,hx1_) ----------------------------------------------------------------------- ----------------------- differenceH Ends Here ------------------------- ----------------------------------------------------------------------- -- | Similar to _differenceH_, but the resulting tree also includes those elements a\_ for which the -- combining comparison returns @Eq (Just a\_)@. -- -- N.B. This function works with relative heights for the first tree and needs no height -- information for the second tree, so it_s OK to initialise the height of the first to zero, -- rather than calculating the absolute height. However, if you do this the height of the resulting -- tree will be incorrect also (it will have the same fixed offset as the first tree). -- -- Complexity: Not sure, but I_d appreciate it if someone could figure it out. differenceMaybeH :: (a -> b -> COrdering (Maybe a)) -> AVL a -> UINT -> AVL b -> UBT2(AVL a,UINT) differenceMaybeH comp = d where -- d :: AVL a -> UINT -> AVL b -> UBT2(AVL a,UINT) d E h0 _ = UBT2(E ,h0) -- Relative heights!! d t0 h0 E = UBT2(t0,h0) d (N l0 e0 r0) h0 (N l1 e1 r1) = d_ l0 DECINT2(h0) e0 r0 DECINT1(h0) l1 e1 r1 d (N l0 e0 r0) h0 (Z l1 e1 r1) = d_ l0 DECINT2(h0) e0 r0 DECINT1(h0) l1 e1 r1 d (N l0 e0 r0) h0 (P l1 e1 r1) = d_ l0 DECINT2(h0) e0 r0 DECINT1(h0) l1 e1 r1 d (Z l0 e0 r0) h0 (N l1 e1 r1) = d_ l0 DECINT1(h0) e0 r0 DECINT1(h0) l1 e1 r1 d (Z l0 e0 r0) h0 (Z l1 e1 r1) = d_ l0 DECINT1(h0) e0 r0 DECINT1(h0) l1 e1 r1 d (Z l0 e0 r0) h0 (P l1 e1 r1) = d_ l0 DECINT1(h0) e0 r0 DECINT1(h0) l1 e1 r1 d (P l0 e0 r0) h0 (N l1 e1 r1) = d_ l0 DECINT1(h0) e0 r0 DECINT2(h0) l1 e1 r1 d (P l0 e0 r0) h0 (Z l1 e1 r1) = d_ l0 DECINT1(h0) e0 r0 DECINT2(h0) l1 e1 r1 d (P l0 e0 r0) h0 (P l1 e1 r1) = d_ l0 DECINT1(h0) e0 r0 DECINT2(h0) l1 e1 r1 d_ l0 hl0 e0 r0 hr0 l1 e1 r1 = case comp e0 e1 of -- e0 < e1, so (l0 < e0 < e1) & (e0 < e1 < r1) Lt -> case forkR r0 hr0 e1 of UBT5( rl0,hrl0,mbe1,rr0,hrr0) -> case forkL e0 l1 of -- (e0 < rl0 < e1) & (e0 < e1 < rr0) UBT5(ll1,_ ,mbe0,lr1,_ ) -> -- (ll1 < e0 < e1) & (e0 < lr1 < e1) -- (l0 + ll1) < e0 < (rl0 + lr1) < e1 < (rr0 + r1) case d rr0 hrr0 r1 of -- right UBT2(r,hr) -> case d rl0 hrl0 lr1 of -- middle UBT2(m,hm) -> case d l0 hl0 ll1 of -- left UBT2(l,hl) -> case (case mbe1 of Just e1_ -> spliceH m hm e1_ r hr -- splice middle right with e1_ Nothing -> joinH m hm r hr) of -- join middle right UBT2(y,hy) -> case mbe0 of Just e0_ -> spliceH l hl e0_ y hy Nothing -> joinH l hl y hy -- e0 = e1 Eq mbe0 -> case d r0 hr0 r1 of -- right UBT2(r,hr) -> case d l0 hl0 l1 of -- left UBT2(l,hl) -> case mbe0 of Just e0_ -> spliceH l hl e0_ r hr -- retain updated e0 Nothing -> joinH l hl r hr -- discard original e0 -- e1 < e0, so (l1 < e1 < e0) & (e1 < e0 < r0) Gt -> case forkL e0 r1 of UBT5( rl1,_ ,mbe0,rr1,_ ) -> case forkR l0 hl0 e1 of -- (e1 < rl1 < e0) & (e1 < e0 < rr1) UBT5(ll0,hll0,mbe1,lr0,hlr0) -> -- (ll0 < e1 < e0) & (e1 < lr0 < e0) -- (ll0 + l1) < e1 < (lr0 + rl1) < e0 < (r0 + rr1) case d r0 hr0 rr1 of -- right UBT2(r,hr) -> case d lr0 hlr0 rl1 of -- middle UBT2(m,hm) -> case d ll0 hll0 l1 of -- left UBT2(l,hl) -> case (case mbe1 of Just e1_ -> spliceH l hl e1_ m hm -- splice left middle with e1_ Nothing -> joinH l hl m hm) of -- join left middle UBT2(x,hx) -> case mbe0 of Just e0_ -> spliceH x hx e0_ r hr Nothing -> joinH x hx r hr -- We need 2 different versions of fork (L & R) to ensure that comparison arguments are used in -- the right order (c e0 e1), and for other algorithmic reasons in this case. -- N.B. forkL returns (Just e0) if t1 does not contain e0 (I.E. If original e0 is an element of the result). -- forkL :: a -> AVL b -> UBT5(AVL b, UINT, Maybe a, AVL b, UINT) forkL e0 t1 = forkL_ t1 L(0) where forkL_ E h = UBT5(E,h,Just e0,E,h) -- Relative heights!! forkL_ (N l e r) h = forkL__ l DECINT2(h) e r DECINT1(h) forkL_ (Z l e r) h = forkL__ l DECINT1(h) e r DECINT1(h) forkL_ (P l e r) h = forkL__ l DECINT1(h) e r DECINT2(h) forkL__ l hl e r hr = case comp e0 e of Lt -> case forkL_ l hl of UBT5(x0,hx0,mbe0,x1,hx1) -> case spliceH x1 hx1 e r hr of UBT2(x1_,hx1_) -> UBT5(x0,hx0,mbe0,x1_,hx1_) Eq mbe0 -> UBT5(l,hl,mbe0,r,hr) Gt -> case forkL_ r hr of UBT5(x0,hx0,mbe0,x1,hx1) -> case spliceH l hl e x0 hx0 of UBT2(x0_,hx0_) -> UBT5(x0_,hx0_,mbe0,x1,hx1) -- N.B. forkR t0, according to e1. Returns Nothing if t0 does not contain e1. -- forkR :: AVL a -> UINT -> b -> UBT5(AVL a, UINT, Maybe a, AVL a, UINT) forkR t0 ht0 e1 = forkR_ t0 ht0 where forkR_ E h = UBT5(E,h,Nothing,E,h) -- Relative heights!! forkR_ (N l e r) h = forkR__ l DECINT2(h) e r DECINT1(h) forkR_ (Z l e r) h = forkR__ l DECINT1(h) e r DECINT1(h) forkR_ (P l e r) h = forkR__ l DECINT1(h) e r DECINT2(h) forkR__ l hl e r hr = case comp e e1 of Lt -> case forkR_ r hr of UBT5(x0,hx0,mbe1,x1,hx1) -> case spliceH l hl e x0 hx0 of UBT2(x0_,hx0_) -> UBT5(x0_,hx0_,mbe1,x1,hx1) Eq mbe1 -> UBT5(l,hl,mbe1,r,hr) Gt -> case forkR_ l hl of UBT5(x0,hx0,mbe1,x1,hx1) -> case spliceH x1 hx1 e r hr of UBT2(x1_,hx1_) -> UBT5(x0,hx0,mbe1,x1_,hx1_) ----------------------------------------------------------------------- --------------------- differenceMaybeH Ends Here ---------------------- ----------------------------------------------------------------------- -- | The symmetric difference is the set of elements which occur in one set or the other but /not both/. -- -- Complexity: Not sure, but I_d appreciate it if someone could figure it out. symDifferenceH :: (e -> e -> Ordering) -> AVL e -> UINT -> AVL e -> UINT -> UBT2(AVL e,UINT) symDifferenceH c = u where -- u :: AVL e -> UINT -> AVL e -> UINT -> UBT2(AVL e,UINT) u E _ t1 h1 = UBT2(t1,h1) u t0 h0 E _ = UBT2(t0,h0) u (N l0 e0 r0) h0 (N l1 e1 r1) h1 = u_ l0 DECINT2(h0) e0 r0 DECINT1(h0) l1 DECINT2(h1) e1 r1 DECINT1(h1) u (N l0 e0 r0) h0 (Z l1 e1 r1) h1 = u_ l0 DECINT2(h0) e0 r0 DECINT1(h0) l1 DECINT1(h1) e1 r1 DECINT1(h1) u (N l0 e0 r0) h0 (P l1 e1 r1) h1 = u_ l0 DECINT2(h0) e0 r0 DECINT1(h0) l1 DECINT1(h1) e1 r1 DECINT2(h1) u (Z l0 e0 r0) h0 (N l1 e1 r1) h1 = u_ l0 DECINT1(h0) e0 r0 DECINT1(h0) l1 DECINT2(h1) e1 r1 DECINT1(h1) u (Z l0 e0 r0) h0 (Z l1 e1 r1) h1 = u_ l0 DECINT1(h0) e0 r0 DECINT1(h0) l1 DECINT1(h1) e1 r1 DECINT1(h1) u (Z l0 e0 r0) h0 (P l1 e1 r1) h1 = u_ l0 DECINT1(h0) e0 r0 DECINT1(h0) l1 DECINT1(h1) e1 r1 DECINT2(h1) u (P l0 e0 r0) h0 (N l1 e1 r1) h1 = u_ l0 DECINT1(h0) e0 r0 DECINT2(h0) l1 DECINT2(h1) e1 r1 DECINT1(h1) u (P l0 e0 r0) h0 (Z l1 e1 r1) h1 = u_ l0 DECINT1(h0) e0 r0 DECINT2(h0) l1 DECINT1(h1) e1 r1 DECINT1(h1) u (P l0 e0 r0) h0 (P l1 e1 r1) h1 = u_ l0 DECINT1(h0) e0 r0 DECINT2(h0) l1 DECINT1(h1) e1 r1 DECINT2(h1) u_ l0 hl0 e0 r0 hr0 l1 hl1 e1 r1 hr1 = case c e0 e1 of -- e0 < e1, so (l0 < e0 < e1) & (e0 < e1 < r1) LT -> case forkR r0 hr0 e1 of UBT5(rl0,hrl0,be1,rr0,hrr0) -> case forkL e0 l1 hl1 of -- (e0 < rl0 < e1) & (e0 < e1 < rr0) UBT5(ll1,hll1,be0,lr1,hlr1) -> -- (ll1 < e0 < e1) & (e0 < lr1 < e1) -- (l0 + ll1) < e0 < (rl0 + lr1) < e1 < (rr0 + r1) case u l0 hl0 ll1 hll1 of UBT2(l,hl) -> case u rl0 hrl0 lr1 hlr1 of UBT2(m,hm) -> case u rr0 hrr0 r1 hr1 of UBT2(r,hr) -> case (if be1 then spliceH m hm e1 r hr else joinH m hm r hr ) of UBT2(t,ht) -> if be0 then spliceH l hl e0 t ht else joinH l hl t ht -- e0 = e1 EQ -> case u l0 hl0 l1 hl1 of UBT2(l,hl) -> case u r0 hr0 r1 hr1 of UBT2(r,hr) -> joinH l hl r hr -- e1 < e0, so (l1 < e1 < e0) & (e1 < e0 < r0) GT -> case forkL e0 r1 hr1 of UBT5(rl1,hrl1,be0,rr1,hrr1) -> case forkR l0 hl0 e1 of -- (e1 < rl1 < e0) & (e1 < e0 < rr1) UBT5(ll0,hll0,be1,lr0,hlr0) -> -- (ll0 < e1 < e0) & (e1 < lr0 < e0) -- (ll0 + l1) < e1 < (lr0 + rl1) < e0 < (r0 + rr1) case u ll0 hll0 l1 hl1 of UBT2(l,hl) -> case u lr0 hlr0 rl1 hrl1 of UBT2(m,hm) -> case u r0 hr0 rr1 hrr1 of UBT2(r,hr) -> case (if be1 then spliceH l hl e1 m hm else joinH l hl m hm ) of UBT2(t,ht) -> if be0 then spliceH t ht e0 r hr else joinH t ht r hr -- We need 2 different versions of fork (L & R) to ensure that comparison arguments are used in -- the right order (c e0 e1) -- forkL :: e -> AVL e -> UINT -> UBT5(AVL e,UINT,Bool,AVL e,UINT) forkL e0 t1 ht1 = forkL_ t1 ht1 where forkL_ E _ = UBT5(E, L(0), True, E, L(0)) forkL_ (N l e r) h = forkL__ l DECINT2(h) e r DECINT1(h) forkL_ (Z l e r) h = forkL__ l DECINT1(h) e r DECINT1(h) forkL_ (P l e r) h = forkL__ l DECINT1(h) e r DECINT2(h) forkL__ l hl e r hr = case c e0 e of LT -> case forkL_ l hl of UBT5(l0,hl0,be0,l1,hl1) -> case spliceH l1 hl1 e r hr of UBT2(l1_,hl1_) -> UBT5(l0,hl0,be0,l1_,hl1_) EQ -> UBT5(l,hl,False,r,hr) GT -> case forkL_ r hr of UBT5(l0,hl0,be0,l1,hl1) -> case spliceH l hl e l0 hl0 of UBT2(l0_,hl0_) -> UBT5(l0_,hl0_,be0,l1,hl1) -- forkR :: AVL e -> UINT -> e -> UBT5(AVL e,UINT,Bool,AVL e,UINT) forkR t0 ht0 e1 = forkR_ t0 ht0 where forkR_ E _ = UBT5(E, L(0), True, E, L(0)) forkR_ (N l e r) h = forkR__ l DECINT2(h) e r DECINT1(h) forkR_ (Z l e r) h = forkR__ l DECINT1(h) e r DECINT1(h) forkR_ (P l e r) h = forkR__ l DECINT1(h) e r DECINT2(h) forkR__ l hl e r hr = case c e e1 of LT -> case forkR_ r hr of UBT5(l0,hl0,be1,l1,hl1) -> case spliceH l hl e l0 hl0 of UBT2(l0_,hl0_) -> UBT5(l0_,hl0_,be1,l1,hl1) EQ -> UBT5(l,hl,False,r,hr) GT -> case forkR_ l hl of UBT5(l0,hl0,be1,l1,hl1) -> case spliceH l1 hl1 e r hr of UBT2(l1_,hl1_) -> UBT5(l0,hl0,be1,l1_,hl1_) ----------------------------------------------------------------------- ----------------------- symDifferenceH Ends Here ---------------------- -----------------------------------------------------------------------