/usr/share/axiom-20170501/src/algebra/HEAP.spad is in axiom-source 20170501-3.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 | )abbrev domain HEAP Heap
++ Author: Michael Monagan and Stephen Watt
++ Date Created:June 86 and July 87
++ Date Last Updated:Feb 92
++ Description:
++ Heap implemented in a flexible array to allow for insertions
++ Complexity: O(log n) insertion, extraction and O(n) construction
--% Dequeue and Heap data types
Heap(S) : SIG == CODE where
S : OrderedSet
SIG ==> PriorityQueueAggregate S with
heap : List S -> %
++ heap(ls) creates a heap of elements consisting of the
++ elements of ls.
++
++X i:Heap INT := heap [1,6,3,7,5,2,4]
-- Inherited Signatures repeated for examples documentation
bag : List S -> %
++
++X bag([1,2,3,4,5])$Heap(INT)
copy : % -> %
++
++X a:Heap INT:= heap [1,2,3,4,5]
++X copy a
empty? : % -> Boolean
++
++X a:Heap INT:= heap [1,2,3,4,5]
++X empty? a
empty : () -> %
++
++X b:=empty()$(Heap INT)
eq? : (%,%) -> Boolean
++
++X a:Heap INT:= heap [1,2,3,4,5]
++X b:=copy a
++X eq?(a,b)
extract_! : % -> S
++
++X a:Heap INT:= heap [1,2,3,4,5]
++X extract! a
++X a
insert_! : (S,%) -> %
++
++X a:Heap INT:= heap [1,2,3,4,5]
++X insert!(8,a)
++X a
inspect : % -> S
++
++X a:Heap INT:= heap [1,2,3,4,5]
++X inspect a
map : ((S -> S),%) -> %
++
++X a:Heap INT:= heap [1,2,3,4,5]
++X map(x+->x+10,a)
++X a
max : % -> S
++
++X a:Heap INT:= heap [1,2,3,4,5]
++X max a
merge : (%,%) -> %
++
++X a:Heap INT:= heap [1,2,3,4,5]
++X b:Heap INT:= heap [6,7,8,9,10]
++X merge(a,b)
merge! : (%,%) -> %
++
++X a:Heap INT:= heap [1,2,3,4,5]
++X b:Heap INT:= heap [6,7,8,9,10]
++X merge!(a,b)
++X a
++X b
sample : () -> %
++
++X sample()$Heap(INT)
less? : (%,NonNegativeInteger) -> Boolean
++
++X a:Heap INT:= heap [1,2,3,4,5]
++X less?(a,9)
more? : (%,NonNegativeInteger) -> Boolean
++
++X a:Heap INT:= heap [1,2,3,4,5]
++X more?(a,9)
size? : (%,NonNegativeInteger) -> Boolean
++
++X a:Heap INT:= heap [1,2,3,4,5]
++X size?(a,5)
if $ has shallowlyMutable then
map! : ((S -> S),%) -> %
++
++X a:Heap INT:= heap [1,2,3,4,5]
++X map!(x+->x+10,a)
++X a
if S has SetCategory then
latex : % -> String
++
++X a:Heap INT:= heap [1,2,3,4,5]
++X latex a
hash : % -> SingleInteger
++
++X a:Heap INT:= heap [1,2,3,4,5]
++X hash a
coerce : % -> OutputForm
++
++X a:Heap INT:= heap [1,2,3,4,5]
++X coerce a
"=": (%,%) -> Boolean
++
++X a:Heap INT:= heap [1,2,3,4,5]
++X b:Heap INT:= heap [1,2,3,4,5]
++X (a=b)@Boolean
"~=" : (%,%) -> Boolean
++
++X a:Heap INT:= heap [1,2,3,4,5]
++X b:=copy a
++X (a~=b)
if % has finiteAggregate then
every? : ((S -> Boolean),%) -> Boolean
++
++X a:Heap INT:= heap [1,2,3,4,5]
++X every?(x+->(x=4),a)
any? : ((S -> Boolean),%) -> Boolean
++
++X a:Heap INT:= heap [1,2,3,4,5]
++X any?(x+->(x=4),a)
count : ((S -> Boolean),%) -> NonNegativeInteger
++
++X a:Heap INT:= heap [1,2,3,4,5]
++X count(x+->(x>2),a)
_# : % -> NonNegativeInteger
++
++X a:Heap INT:= heap [1,2,3,4,5]
++X #a
parts : % -> List S
++
++X a:Heap INT:= heap [1,2,3,4,5]
++X parts a
members : % -> List S
++
++X a:Heap INT:= heap [1,2,3,4,5]
++X members a
if % has finiteAggregate and S has SetCategory then
member? : (S,%) -> Boolean
++
++X a:Heap INT:= heap [1,2,3,4,5]
++X member?(3,a)
count : (S,%) -> NonNegativeInteger
++
++X a:Heap INT:= heap [1,2,3,4,5]
++X count(4,a)
CODE ==> IndexedFlexibleArray(S,0) add
Rep := IndexedFlexibleArray( S,0)
empty() == empty()$Rep
heap l ==
n := #l
h := empty()
n = 0 => h
for x in l repeat insert_!(x,h)
h
siftUp: (%,Integer,Integer) -> Void
siftUp(r,i,n) ==
-- assertion 0 <= i < n
t := r.i
while (j := 2*i+1) < n repeat
if (k := j+1) < n and r.j < r.k then j := k
if t < r.j then (r.i := r.j; r.j := t; i := j) else leave
extract_! r ==
-- extract the maximum from the heap O(log n)
n := #r :: Integer
n = 0 => error "empty heap"
t := r(0)
r(0) := r(n-1)
delete_!(r,n-1)
n = 1 => t
siftUp(r,0,n-1)
t
insert_!(x,r) ==
-- Williams' insertion algorithm O(log n)
j := (#r) :: Integer
r:=concat_!(r,concat(x,empty()$Rep))
while j > 0 repeat
i := (j-1) quo 2
if r(i) >= x then leave
r(j) := r(i)
j := i
r(j):=x
r
max r == if #r = 0 then error "empty heap" else r.0
inspect r == max r
makeHeap(r:%):% ==
-- Floyd's heap construction algorithm O(n)
n := #r
for k in n quo 2 -1 .. 0 by -1 repeat siftUp(r,k,n)
r
bag l == makeHeap construct(l)$Rep
merge(a,b) == makeHeap concat(a,b)
merge_!(a,b) == makeHeap concat_!(a,b)
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