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--Copyright (C) 2007-2009, Gabriel Dos Reis.
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)abbrev domain SET Set
++ Author: Michael Monagan; revised by Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: May 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A set over a domain D models the usual mathematical notion of a finite set
++ of elements from D.
++ Sets are unordered collections of distinct elements
++ (that is, order and duplication does not matter).
++ The notation \spad{set [a,b,c]} can be used to create
++ a set and the usual operations such as union and intersection are available
++ to form new sets.
++ In our implementation, \Language{} maintains the entries in
++ sorted order. Specifically, the parts function returns the entries
++ as a list in ascending order and
++ the extract operation returns the maximum entry.
++ Given two sets s and t where \spad{#s = m} and \spad{#t = n},
++ the complexity of
++ \spad{s = t} is \spad{O(min(n,m))}
++ \spad{s < t} is \spad{O(max(n,m))}
++ \spad{union(s,t)}, \spad{intersect(s,t)}, \spad{minus(s,t)}, \spad{symmetricDifference(s,t)} is \spad{O(max(n,m))}
++ \spad{member(x,t)} is \spad{O(n log n)}
++ \spad{insert(x,t)} and \spad{remove(x,t)} is \spad{O(n)}
Set(S:SetCategory): FiniteSetAggregate S == add
Rep := FlexibleArray(S)
# s == _#$Rep s
brace() == empty()
set() == empty()
empty() == empty()$Rep
copy s == copy(s)$Rep
parts s == parts(s)$Rep
inspect s == (empty? s => error "Empty set"; s(maxIndex s))
extract! s ==
x := inspect s
delete!(s, maxIndex s)
x
find(f, s) == find(f, s)$Rep
map(f, s) == map!(f,copy s)
map!(f,s) ==
map!(f,s)$Rep
removeDuplicates! s
reduce(f, s) == reduce(f, s)$Rep
reduce(f, s, x) == reduce(f, s, x)$Rep
reduce(f, s, x, y) == reduce(f, s, x, y)$Rep
if S has ConvertibleTo InputForm then
convert(x:%):InputForm ==
convert [convert('set)@InputForm,
convert(parts x)@InputForm]
if S has OrderedSet then
s = t == s =$Rep t
max s == inspect s
min s == (empty? s => error "Empty set"; s(minIndex s))
construct l ==
zero?(n := #l) => empty()
a := new(n, first l)
for i in minIndex(a).. for x in l repeat a.i := x
removeDuplicates! sort! a
insert!(x, s) ==
n := inc maxIndex s
k := minIndex s
while k < n and x > s.k repeat k := inc k
k < n and s.k = x => s
insert!(x, s, k)
member?(x, s) == -- binary search
empty? s => false
t := maxIndex s
b := minIndex s
while b < t repeat
m := (b+t) quo 2
if x > s.m then b := m+1 else t := m
x = s.t
remove!(x:S, s:%) ==
n := inc maxIndex s
k := minIndex s
while k < n and x > s.k repeat k := inc k
k < n and x = s.k => delete!(s, k)
s
-- the set operations are implemented as variations of merging
intersect(s, t) ==
m := maxIndex s
n := maxIndex t
i := minIndex s
j := minIndex t
r := empty()
while i <= m and j <= n repeat
s.i = t.j => (concat!(r, s.i); i := i+1; j := j+1)
if s.i < t.j then i := i+1 else j := j+1
r
difference(s:%, t:%) ==
m := maxIndex s
n := maxIndex t
i := minIndex s
j := minIndex t
r := empty()
while i <= m and j <= n repeat
s.i = t.j => (i := i+1; j := j+1)
s.i < t.j => (concat!(r, s.i); i := i+1)
j := j+1
while i <= m repeat (concat!(r, s.i); i := i+1)
r
symmetricDifference(s, t) ==
m := maxIndex s
n := maxIndex t
i := minIndex s
j := minIndex t
r := empty()
while i <= m and j <= n repeat
s.i < t.j => (concat!(r, s.i); i := i+1)
s.i > t.j => (concat!(r, t.j); j := j+1)
i := i+1; j := j+1
while i <= m repeat (concat!(r, s.i); i := i+1)
while j <= n repeat (concat!(r, t.j); j := j+1)
r
subset?(s, t) ==
m := maxIndex s
n := maxIndex t
m > n => false
i := minIndex s
j := minIndex t
while i <= m and j <= n repeat
s.i = t.j => (i := i+1; j := j+1)
s.i > t.j => j := j+1
return false
i > m
union(s:%, t:%) ==
m := maxIndex s
n := maxIndex t
i := minIndex s
j := minIndex t
r := empty()
while i <= m and j <= n repeat
s.i = t.j => (concat!(r, s.i); i := i+1; j := j+1)
s.i < t.j => (concat!(r, s.i); i := i+1)
(concat!(r, t.j); j := j+1)
while i <= m repeat (concat!(r, s.i); i := i+1)
while j <= n repeat (concat!(r, t.j); j := j+1)
r
else
insert!(x, s) ==
for k in minIndex s .. maxIndex s repeat
s.k = x => return s
insert!(x, s, inc maxIndex s)
remove!(x:S, s:%) ==
n := inc maxIndex s
k := minIndex s
while k < n repeat
x = s.k => return delete!(s, k)
k := inc k
s
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