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import Boolean
import List
)abbrev domain TREE Tree
++ Author:W. H. Burge
++ Date Created:17 Feb 1992
++ Date Last Updated:
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ Examples:
++ References:
++ Description: \spadtype{Tree(S)} is a basic domains of tree structures.
++ Each tree is either empty or else is a {\it node} consisting of a value and
++ a list of (sub)trees.
Tree(S: SetCategory): T==C where
T== RecursiveAggregate(S) with
finiteAggregate
shallowlyMutable
tree: (S,List %) -> %
++ tree(nd,ls) creates a tree with value nd, and children
++ ls.
tree: List S -> %
++ tree(ls) creates a tree from a list of elements of s.
tree: S -> %
++ tree(nd) creates a tree with value nd, and no children
cyclic?: % -> Boolean
++ cyclic?(t) tests if t is a cyclic tree.
cyclicCopy: % -> %
++ cyclicCopy(l) makes a copy of a (possibly) cyclic tree l.
cyclicEntries: % -> List %
++ cyclicEntries(t) returns a list of top-level cycles in tree t.
cyclicEqual?: (%, %) -> Boolean
++ cyclicEqual?(t1, t2) tests of two cyclic trees have
++ the same structure.
cyclicParents: % -> List %
++ cyclicParents(t) returns a list of cycles that are parents of t.
C== add
cycleTreeMax ==> 5
Rep := Union(node:Record(value: S, args: List %),empty:"empty")
empty? t == t case empty
empty() == ["empty"]
children t ==
t case empty => error "cannot take the children of an empty tree"
(t.node.args)@List(%)
setchildren!(t,lt) ==
t case empty => error "cannot set children of an empty tree"
(t.node.args:=lt;t pretend %)
setvalue!(t,s) ==
t case empty => error "cannot set value of an empty tree"
(t.node.value:=s;s)
count(n: S, t: %) ==
t case empty => 0
i := +/[count(n, c) for c in children t]
value t = n => i + 1
i
count(fn: S -> Boolean, t: %): NonNegativeInteger ==
t case empty => 0
i := +/[count(fn, c) for c in children t]
fn value t => i + 1
i
map(fn, t) ==
t case empty => t
tree(fn value t,[map(fn, c) for c in children t])
map!(fn, t) ==
t case empty => t
setvalue!(t, fn value t)
for c in children t repeat map!(fn, c)
t
tree(s,lt) == [[s,lt]]
tree(s: S) == [[s,[]]]
tree(ls: List S) ==
empty? ls => empty()
tree(first ls, [tree s for s in rest ls])
value t ==
t case empty => error "cannot take the value of an empty tree"
t.node.value
child?(t1,t2) ==
empty? t2 => false
"or"/[t1 = t for t in children t2]
distance1(t1: %, t2: %): Integer ==
t1 = t2 => 0
t2 case empty => -1
u := [n for t in children t2 | (n := distance1(t1,t)) >= 0]
positive?(#u) => 1 + "min"/u
-1
distance(t1,t2) ==
n := distance1(t1, t2)
n >= 0 => n
distance1(t2, t1)
node?(t1, t2) ==
t1 = t2 => true
t2 case empty => false
"or"/[node?(t1, t) for t in children t2]
leaf? t ==
t case empty => false
empty? children t
leaves t ==
t case empty => empty()
leaf? t => [value t]
"append"/[leaves c for c in children t]
nodes t == ---buggy
t case empty => empty()
nl := [nodes c for c in children t]
nl = empty() => [t]
cons(t,"append"/nl)
any?(fn, t) == ---bug fixed
t case empty => false
fn value t or "or"/[any?(fn, c) for c in children t]
every?(fn, t) ==
t case empty => true
fn value t and "and"/[every?(fn, c) for c in children t]
member?(n, t) ==
t case empty => false
n = value t or "or"/[member?(n, c) for c in children t]
members t == parts t
parts t == --buggy?
t case empty => empty()
u := [parts c for c in children t]
u = empty() => [value t]
cons(value t,"append"/u)
---Functions that guard against cycles: =, #, copy-------------
-----> =
equal?: (%, %, %, %, Integer) -> Boolean
t1 = t2 == equal?(t1, t2, t1, t2, 0)
equal?(t1, t2, ot1, ot2, k) ==
k = cycleTreeMax and (cyclic? ot1 or cyclic? ot2) =>
error "use cyclicEqual? to test equality on cyclic trees"
t1 case empty => t2 case empty
t2 case empty => false
value t1 = value t2 and (c1 := children t1) = (c2 := children t2) and
"and"/[equal?(x,y,ot1, ot2,k + 1) for x in c1 for y in c2]
-----> #
treeCount: (%, %, NonNegativeInteger) -> NonNegativeInteger
# t == treeCount(t, t, 0)
treeCount(t, origTree, k) ==
k = cycleTreeMax and cyclic? origTree =>
error "# is not defined on cyclic trees"
t case empty => 0
1 + +/[treeCount(c, origTree, k + 1) for c in children t]
-----> copy
copy1: (%, %, Integer) -> %
copy t == copy1(t, t, 0)
copy1(t, origTree, k) ==
k = cycleTreeMax and cyclic? origTree =>
error "use cyclicCopy to copy a cyclic tree"
t case empty => t
empty? children t => tree value t
tree(value t, [copy1(x, origTree, k + 1) for x in children t])
-----------Functions that allow cycles---------------
--local utility functions:
eqUnion: (List %, List %) -> List %
eqMember?: (%, List %) -> Boolean
eqMemberIndex: (%, List %, Integer) -> Integer
lastNode: List % -> List %
insert: (%, List %) -> List %
-----> coerce to OutputForm
if S has CoercibleTo(OutputForm) then
multipleOverbar: (OutputForm, Integer, List %) -> OutputForm
coerce1: (%, List %, List %) -> OutputForm
coerce(t:%): OutputForm == coerce1(t, empty()$(List %), cyclicParents t)
coerce1(t,parents, pl) ==
t case empty => empty()@List(S)::OutputForm
eqMember?(t, parents) =>
multipleOverbar((".")::OutputForm,eqMemberIndex(t, pl,0),pl)
empty? children t => value t::OutputForm
nodeForm := (value t)::OutputForm
if positive?(k := eqMemberIndex(t, pl, 0)) then
nodeForm := multipleOverbar(nodeForm, k, pl)
prefix(nodeForm,
[coerce1(br,cons(t,parents),pl) for br in children t])
multipleOverbar(x, k, pl) ==
k < 1 => x
#pl = 1 => overbar x
s : String := "abcdefghijklmnopqrstuvwxyz"
c := s.(1 + ((k - 1) rem 26))
overlabel(c::OutputForm, x)
-----> cyclic?
cyclic2?: (%, List %) -> Boolean
cyclic? t == cyclic2?(t, empty()$(List %))
cyclic2?(x,parents) ==
empty? x => false
eqMember?(x, parents) => true
for y in children x repeat
cyclic2?(y,cons(x, parents)) => return true
false
-----> cyclicCopy
cyclicCopy2: (%, List %) -> %
copyCycle2: (%, List %) -> %
copyCycle4: (%, %, %, List %) -> %
cyclicCopy(t) == cyclicCopy2(t, cyclicEntries t)
cyclicCopy2(t, cycles) ==
eqMember?(t, cycles) => copyCycle2(t, cycles)
tree(value t, [cyclicCopy2(c, cycles) for c in children t])
copyCycle2(cycle, cycleList) ==
newCycle := tree(value cycle, nil)
setchildren!(newCycle,
[copyCycle4(c,cycle,newCycle, cycleList) for c in children cycle])
newCycle
copyCycle4(t, cycle, newCycle, cycleList) ==
empty? cycle => empty()
eq?(t, cycle) => newCycle
eqMember?(t, cycleList) => copyCycle2(t, cycleList)
tree(value t,
[copyCycle4(c, cycle, newCycle, cycleList) for c in children t])
-----> cyclicEntries
cyclicEntries3: (%, List %, List %) -> List %
cyclicEntries(t) == cyclicEntries3(t, empty()$(List %), empty()$(List %))
cyclicEntries3(t, parents, cl) ==
empty? t => cl
eqMember?(t, parents) => insert(t, cl)
parents := cons(t, parents)
for y in children t repeat
cl := cyclicEntries3(t, parents, cl)
cl
-----> cyclicEqual?
cyclicEqual4?: (%, %, List %, List %) -> Boolean
cyclicEqual?(t1, t2) ==
cp1 := cyclicParents t1
cp2 := cyclicParents t2
#cp1 ~= #cp2 or null cp1 => t1 = t2
cyclicEqual4?(t1, t2, cp1, cp2)
cyclicEqual4?(t1, t2, cp1, cp2) ==
t1 case empty => t2 case empty
t2 case empty => false
0 ~= (k := eqMemberIndex(t1, cp1, 0)) => eq?(t2, cp2 . k)
value t1 = value t2 and
"and"/[cyclicEqual4?(x,y,cp1,cp2)
for x in children t1 for y in children t2]
-----> cyclicParents t
cyclicParents3: (%, List %, List %) -> List %
cyclicParents t == cyclicParents3(t, empty()$(List %), empty()$(List %))
cyclicParents3(x, parents, pl) ==
empty? x => pl
eqMember?(x, parents) =>
cycleMembers := [y for y in parents while not eq?(x,y)]
eqUnion(cons(x, cycleMembers), pl)
parents := cons(x, parents)
for y in children x repeat
pl := cyclicParents3(y, parents, pl)
pl
insert(x, l) ==
eqMember?(x, l) => l
cons(x, l)
lastNode l ==
empty? l => error "empty tree has no last node"
while not empty? rest l repeat l := rest l
l
eqMember?(y,l) ==
for x in l repeat eq?(x,y) => return true
false
eqMemberIndex(x, l, k) ==
null l => k
k := k + 1
eq?(x, first l) => k
eqMemberIndex(x, rest l, k)
eqUnion(u, v) ==
null u => v
x := first u
newV :=
eqMember?(x, v) => v
cons(x, v)
eqUnion(rest u, newV)
)abbrev category BTCAT BinaryTreeCategory
++ Author:W. H. Burge
++ Date Created:17 Feb 1992
++ Date Last Updated:
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ Examples:
++ References:
++ Description: \spadtype{BinaryTreeCategory(S)} is the category of
++ binary trees: a tree which is either empty or else is a \spadfun{node} consisting
++ of a value and a \spadfun{left} and \spadfun{right}, both binary trees.
BinaryTreeCategory(S: SetCategory): Category == BinaryRecursiveAggregate(S) with
shallowlyMutable
++ Binary trees have updateable components
finiteAggregate
++ Binary trees have a finite number of components
node: (%,S,%) -> %
++ node(left,v,right) creates a binary tree with value \spad{v}, a binary
++ tree \spad{left}, and a binary tree \spad{right}.
add
cycleTreeMax ==> 5
copy t ==
empty? t => empty()
node(copy left t, value t, copy right t)
if % has shallowlyMutable then
map!(f,t) ==
empty? t => t
t.value := f(t.value)
map!(f,left t)
map!(f,right t)
t
if % has finiteAggregate then
treeCount : (%, NonNegativeInteger) -> NonNegativeInteger
#t == treeCount(t,0)
treeCount(t,k) ==
empty? t => k
k := k + 1
k = cycleTreeMax and cyclic? t => error "cyclic binary tree"
k := treeCount(left t,k)
treeCount(right t,k)
)abbrev domain BTREE BinaryTree
++ Description: \spadtype{BinaryTree(S)} is the domain of all
++ binary trees. A binary tree over \spad{S} is either empty or has
++ a \spadfun{value} which is an S and a \spadfun{right}
++ and \spadfun{left} which are both binary trees.
BinaryTree(S: SetCategory): Exports == Implementation where
Exports == BinaryTreeCategory(S) with
binaryTree: S -> %
++ binaryTree(v) is an non-empty binary tree
++ with value v, and left and right empty.
binaryTree: (%,S,%) -> %
++ binaryTree(l,v,r) creates a binary tree with
++ value v with left subtree l and right subtree r.
Implementation == add
Rep := List Tree S
t1 = t2 == (t1::Rep) =$Rep (t2::Rep)
empty()== [] pretend %
node(l,v,r) == cons(tree(v,l:Rep),r:Rep)
binaryTree(l,v,r) == node(l,v,r)
binaryTree(v:S) == node(empty(),v,empty())
empty? t == empty?(t)$Rep
leaf? t == empty? t or empty? left t and empty? right t
right t ==
empty? t => error "binaryTree:no right"
rest t
left t ==
empty? t => error "binaryTree:no left"
children first t
value t==
empty? t => error "binaryTree:no value"
value first t
setvalue! (t,nd)==
empty? t => error "binaryTree:no value to set"
setvalue!(first(t:Rep),nd)
nd
setleft!(t1,t2) ==
empty? t1 => error "binaryTree:no left to set"
setchildren!(first(t1:Rep),t2:Rep)
t1
setright!(t1,t2) ==
empty? t1 => error "binaryTree:no right to set"
setrest!(t1:List Tree S,t2)
)abbrev domain BBTREE BalancedBinaryTree
++ Description: \spadtype{BalancedBinaryTree(S)} is the domain of balanced
++ binary trees (bbtree). A balanced binary tree of \spad{2**k} leaves,
++ for some \spad{k > 0}, is symmetric, that is, the left and right
++ subtree of each interior node have identical shape.
++ In general, the left and right subtree of a given node can differ
++ by at most leaf node.
BalancedBinaryTree(S: SetCategory): Exports == Implementation where
Exports == BinaryTreeCategory(S) with
finiteAggregate
shallowlyMutable
-- BUG: applies wrong fnct for balancedBinaryTree(0,[1,2,3,4])
-- balancedBinaryTree: (S, List S) -> %
-- ++ balancedBinaryTree(s, ls) creates a balanced binary tree with
-- ++ s at the interior nodes and elements of ls at the
-- ++ leaves.
balancedBinaryTree: (NonNegativeInteger, S) -> %
++ balancedBinaryTree(n, s) creates a balanced binary tree with
++ n nodes each with value s.
setleaves!: (%, List S) -> %
++ setleaves!(t, ls) sets the leaves of t in left-to-right order
++ to the elements of ls.
mapUp!: (%, (S,S) -> S) -> S
++ mapUp!(t,f) traverses balanced binary tree t in an "endorder"
++ (left then right then node) fashion returning t with the value
++ at each successive interior node of t replaced by
++ f(l,r) where l and r are the values at the immediate
++ left and right nodes.
mapUp!: (%, %, (S,S,S,S) -> S) -> %
++ mapUp!(t,t1,f) traverses t in an "endorder" (left then right then node)
++ fashion returning t with the value at each successive interior
++ node of t replaced by
++ f(l,r,l1,r1) where l and r are the values at the immediate
++ left and right nodes. Values l1 and r1 are values at the
++ corresponding nodes of a balanced binary tree t1, of identical
++ shape at t.
mapDown!: (%,S,(S,S) -> S) -> %
++ mapDown!(t,p,f) returns t after traversing t in "preorder"
++ (node then left then right) fashion replacing the successive
++ interior nodes as follows. The root value x is
++ replaced by q := f(p,x). The mapDown!(l,q,f) and
++ mapDown!(r,q,f) are evaluated for the left and right subtrees
++ l and r of t.
mapDown!: (%,S, (S,S,S) -> List S) -> %
++ mapDown!(t,p,f) returns t after traversing t in "preorder"
++ (node then left then right) fashion replacing the successive
++ interior nodes as follows. Let l and r denote the left and
++ right subtrees of t. The root value x of t is replaced by p.
++ Then f(value l, value r, p), where l and r denote the left
++ and right subtrees of t, is evaluated producing two values
++ pl and pr. Then \spad{mapDown!(l,pl,f)} and \spad{mapDown!(l,pr,f)}
++ are evaluated.
Implementation == BinaryTree(S) add
Rep := BinaryTree(S)
leaf? x ==
empty? x => false
empty? left x and empty? right x
-- balancedBinaryTree(x: S, u: List S) ==
-- n := #u
-- n = 0 => empty()
-- setleaves!(balancedBinaryTree(n, x), u)
setleaves!(t, u) ==
n := #u
n = 0 =>
empty? t => t
error "the tree and list must have the same number of elements"
n = 1 =>
setvalue!(t,first u)
t
m := n quo 2
acc := empty()$(List S)
for i in 1..m repeat
acc := [first u,:acc]
u := rest u
setleaves!(left t, reverse! acc)
setleaves!(right t, u)
t
balancedBinaryTree(n: NonNegativeInteger, val: S) ==
n = 0 => empty()
n = 1 => node(empty(),val,empty())
m := n quo 2
node(balancedBinaryTree(m, val), val,
balancedBinaryTree((n - m) pretend NonNegativeInteger, val))
mapUp!(x,fn) ==
empty? x => error "mapUp! called on a null tree"
leaf? x => x.value
x.value := fn(mapUp!(x.left,fn),mapUp!(x.right,fn))
mapUp!(x,y,fn) ==
empty? x => error "mapUp! is called on a null tree"
leaf? x =>
leaf? y => x
error "balanced binary trees are incompatible"
leaf? y => error "balanced binary trees are incompatible"
mapUp!(x.left,y.left,fn)
mapUp!(x.right,y.right,fn)
x.value := fn(x.left.value,x.right.value,y.left.value,y.right.value)
x
mapDown!(x: %, p: S, fn: (S,S) -> S ) ==
empty? x => x
x.value := fn(p, x.value)
mapDown!(x.left, x.value, fn)
mapDown!(x.right, x.value, fn)
x
mapDown!(x: %, p: S, fn: (S,S,S) -> List S) ==
empty? x => x
x.value := p
leaf? x => x
u := fn(x.left.value, x.right.value, p)
mapDown!(x.left, u.1, fn)
mapDown!(x.right, u.2, fn)
x
)abbrev domain BSTREE BinarySearchTree
++ Description: BinarySearchTree(S) is the domain of
++ a binary trees where elements are ordered across the tree.
++ A binary search tree is either empty or has
++ a value which is an S, and a
++ right and left which are both BinaryTree(S)
++ Elements are ordered across the tree.
BinarySearchTree(S: OrderedSet): Exports == Implementation where
Exports == BinaryTreeCategory(S) with
shallowlyMutable
finiteAggregate
binarySearchTree: List S -> %
++ binarySearchTree(l) \undocumented
insert!: (S,%) -> %
++ insert!(x,b) inserts element x as leaves into binary search tree b.
insertRoot!: (S,%) -> %
++ insertRoot!(x,b) inserts element x as a root of binary search tree b.
split: (S,%) -> Record(less: %, greater: %)
++ split(x,b) splits binary tree b into two trees, one with elements greater
++ than x, the other with elements less than x.
Implementation == BinaryTree(S) add
Rep := BinaryTree(S)
binarySearchTree(u:List S) ==
null u => empty()
tree := binaryTree(first u)
for x in rest u repeat insert!(x,tree)
tree
insert!(x,t) ==
empty? t => binaryTree(x)
x >= value t =>
setright!(t,insert!(x,right t))
t
setleft!(t,insert!(x,left t))
t
split(x,t) ==
empty? t => [empty(),empty()]
x > value t =>
a := split(x,right t)
[node(left t, value t, a.less), a.greater]
a := split(x,left t)
[a.less, node(a.greater, value t, right t)]
insertRoot!(x,t) ==
a := split(x,t)
node(a.less, x, a.greater)
)abbrev domain BTOURN BinaryTournament
++ Description: \spadtype{BinaryTournament(S)} is the domain of
++ binary trees where elements are ordered down the tree.
++ A binary search tree is either empty or is a node containing a
++ \spadfun{value} of type \spad{S}, and a \spadfun{right}
++ and a \spadfun{left} which are both \spadtype{BinaryTree(S)}
BinaryTournament(S: OrderedSet): Exports == Implementation where
Exports == BinaryTreeCategory(S) with
shallowlyMutable
binaryTournament: List S -> %
++ binaryTournament(ls) creates a binary tournament with the
++ elements of ls as values at the nodes.
insert!: (S,%) -> %
++ insert!(x,b) inserts element x as leaves into binary tournament b.
Implementation == BinaryTree(S) add
Rep := BinaryTree(S)
binaryTournament(u:List S) ==
null u => empty()
tree := binaryTree(first u)
for x in rest u repeat insert!(x,tree)
tree
insert!(x,t) ==
empty? t => binaryTree(x)
x > value t =>
setleft!(t,copy t)
setvalue!(t,x)
setright!(t,empty())
setright!(t,insert!(x,right t))
t
)abbrev domain PENDTREE PendantTree
++ A PendantTree(S)is either a leaf? and is an S or has
++ a left and a right both PendantTree(S)'s
PendantTree(S: SetCategory): T == C where
T == Join(BinaryRecursiveAggregate(S),CoercibleTo Tree S) with
ptree : S->%
++ ptree(s) is a leaf? pendant tree
ptree:(%, %)->%
++ ptree(x,y) \undocumented
C == add
Rep := Tree S
import Tree S
coerce (t:%):Tree S == t pretend Tree S
ptree(n) == tree(n,[])$Rep pretend %
ptree(l,r) == tree(value(r:Rep)$Rep,cons(l,children(r:Rep)$Rep)):%
leaf? t == empty?(children(t)$Rep)
t1=t2 == (t1:Rep) = (t2:Rep)
left b ==
leaf? b => error "ptree:no left"
first(children(b)$Rep)
right b ==
leaf? b => error "ptree:no right"
tree(value(b)$Rep,rest (children(b)$Rep))
value b ==
leaf? b => value(b)$Rep
error "the pendant tree has no value"
coerce(b:%): OutputForm ==
leaf? b => value(b)$Rep :: OutputForm
paren blankSeparate [left b::OutputForm,right b ::OutputForm]
|