/
defaults.spad
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/
defaults.spad
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)abbrev package REPSQ RepeatedSquaring
++ Repeated Squaring
++ Description:
++ Implements exponentiation by repeated squaring
++ RelatedOperations: expt
-- the following package is only instantiated over %
-- thus shouldn't be cached. We prevent it
-- from being cached by declaring it to be mutableDomains
)bo PUSH('RepeatedSquaring, $mutableDomains)
RepeatedSquaring(S) : Exports == Implementation where
S : SetCategory with
"*":(%,%)->%
++ x*y returns the product of x and y
Exports == with
expt : (S, PositiveInteger) -> S
++ expt(r, i) computes r^i by repeated squaring
Implementation == add
x : S
n : PositiveInteger
expt(x, n) ==
(n = 1) => x
odd?(n)$Integer=> x*expt(x*x, qcoerce(shift(n, -1))@PositiveInteger)
expt(x*x, qcoerce(shift(n, -1))@PositiveInteger)
)abbrev package REPDB RepeatedDoubling
++ Repeated Doubling
++ Integer multiplication by repeated doubling.
++ Description:
++ Implements multiplication by repeated addition
++ RelatedOperations: *
-- the following package is only instantiated over %
-- thus shouldn't be cached. We prevent it
-- from being cached by declaring it to be mutableDomains
)bo PUSH('RepeatedDoubling, $mutableDomains)
RepeatedDoubling(S) : Exports ==Implementation where
S : SetCategory with
"+":(%,%)->%
++ x+y returns the sum of x and y
Exports == with
double : (PositiveInteger, S) -> S
++ double(i, r) multiplies r by i using repeated doubling.
Implementation == add
x : S
n : PositiveInteger
double(n, x) ==
(n = 1) => x
odd?(n)$Integer =>
x + double(qcoerce(shift(n, -1))@PositiveInteger, x + x)
double(qcoerce(shift(n, -1))@PositiveInteger, x + x)
)abbrev package FLASORT FiniteLinearAggregateSort
++ FiniteLinearAggregateSort
++ Sort package (in-place) for shallowlyMutable Finite Linear Aggregates
++ Author: Michael Monagan Sep/88
++ RelatedOperations: sort
++ Description:
++ This package exports 3 sorting algorithms which work over
++ FiniteLinearAggregates.
-- the following package is only instantiated over %
-- thus shouldn't be cached. We prevent it
-- from being cached by declaring it to be mutableDomains
)bo PUSH('FiniteLinearAggregateSort, $mutableDomains)
FiniteLinearAggregateSort(S, V) : Exports == Implementation where
S : Type
V : Join(FiniteLinearAggregate(S), shallowlyMutable)
B ==> Boolean
I ==> Integer
Exports ==> with
quickSort : ((S, S) -> B, V) -> V
++ quickSort(f, agg) sorts the aggregate agg with the ordering function
++ f using the quicksort algorithm.
heapSort : ((S, S) -> B, V) -> V
++ heapSort(f, agg) sorts the aggregate agg with the ordering function
++ f using the heapsort algorithm.
shellSort : ((S, S) -> B, V) -> V
++ shellSort(f, agg) sorts the aggregate agg with the ordering function
++ f using the shellSort algorithm.
Implementation ==> add
siftUp : ((S, S) -> B, V, I, I) -> Void
partition : ((S, S) -> B, V, I, I, I) -> I
QuickSort : ((S, S) -> B, V, I, I) -> V
quickSort(l, r) == QuickSort(l, r, minIndex r, maxIndex r)
siftUp(l, r, i, n) ==
t := qelt(r, i)
while (j := 2*i+1) < n repeat
if (k := j+1) < n and l(qelt(r, j), qelt(r, k)) then j := k
if l(t, qelt(r, j)) then
qsetelt!(r, i, qelt(r, j))
qsetelt!(r, j, t)
i := j
else break
heapSort(l, r) ==
not zero? minIndex r => error "not implemented"
n := (#r)::I
for k in shift(n, -1) - 1 .. 0 by -1 repeat siftUp(l, r, k, n)
for k in n-1 .. 1 by -1 repeat
swap!(r, 0, k)
siftUp(l, r, 0, k)
r
partition(l, r, i, j, k) ==
-- partition r[i..j] such that r.s <= r.k <= r.t
x := qelt(r, k)
while i <= j repeat
t := qelt(r, i)
while l(t, x) repeat
i := i + 1
t := qelt(r, i)
s := qelt(r, j)
while l(x, s) repeat
j := j - 1
s := qelt(r, j)
if i <= j then
qsetelt!(r, i, s)
qsetelt!(r, j, t)
i := i + 1
j := j - 1
j + 1
QuickSort(l, r, i, j) ==
repeat
n := j - i
if (n = 1) and l(qelt(r, j), qelt(r, i)) then swap!(r, i, j)
n < 2 => return r
-- for the moment split at the middle item
k := partition(l, r, i, j, i + shift(n, -1))
if k - i < j - k then
QuickSort(l, r, i, k - 1)
i := k
else
QuickSort(l, r, k, j)
j := k - 1
shellSort(l, r) ==
m := minIndex r
n := maxIndex r
-- use Knuths gap sequence: 1, 4, 13, 40, 121, ...
g := 1
while g <= (n-m) repeat g := 3*g+1
g := g quo 3
while g > 0 repeat
for i in m+g..n repeat
j := i-g
while j >= m and l(qelt(r, j+g), qelt(r, j)) repeat
swap!(r, j, j+g)
j := j-g
g := g quo 3
r
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
-- - Redistributions of source code must retain the above copyright
-- notice, this list of conditions and the following disclaimer.
--
-- - Redistributions in binary form must reproduce the above copyright
-- notice, this list of conditions and the following disclaimer in
-- the documentation and/or other materials provided with the
-- distribution.
--
-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the
-- names of its contributors may be used to endorse or promote products
-- derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.