Selection sorting.
Remember that selection sort works by
finding the smallest element and moving it into the first position,
then finding the second smallest and moving it into the second
position. In general, on the ith iteration, it finds the ith smallest
element and move it to position i. This works as follows:
(selection '(6 4 5 3 9 3))
(cons 3 (selection '(6 4 5 9 3))
(cons 3 (cons 3 (selection '(6 4 5 9))
(cons 3 (cons 3 (cons 4 (selection '(6 5 9))
(cons 3 (cons 3 (cons 4 (cons 5 (selection '(6 9))
(cons 3 (cons 3 (cons 4 (cons 5 (cons 6 (selection '(9))
(cons 3 (cons 3 (cons 4 (cons 5 (cons 6 (cons 9 (selection '())
Which becomes (3 3 4 5 6 9).
The scheme code below does exactly this:
def selection(lis)
if lis.length == 0 then
[]
else # put the smallest element at the front of the current list
[smallest(lis,lis[0])] + selection(myremove(lis,smallest(lis, lis[0])))
# call selection on the list minus the smallest element
end
end
# remove the first occurance of atom A from L
def myremove(lis, a)
lis.delete(a) #delete function returns the value deleted, not the list
lis
end
def smallest(lis, a) # looks for the smallest element in the list
#atom A is the current smallest
if lis.length==0 then
a
elsif lis[0] < a then
smallest(lis[1..lis.length-1], lis[0])
else
return smallest(lis[1..lis.length-1], a)
end
end
x = [6,3,4,9,10,1,7]
print x, "\n"
print smallest(x, 6), "\n"
#print myremove(x, smallest(x,6))
print selection(x), "\n"
Merge Sort
Mergesort divides the list in half everytime,
calling itself on each half. As the recursion unwinds, it merges the
sorted list into a new
sorted list. For example, assume we have the following:
(mergesort '(6 4 5 7 8 9 3 4))
/ \
(mergesort '(6 4 5 7)) (mergesort '(8 9 3 4))
/ \ / \
(mergesort '(6 4)) (mergesort '(5 7)) (mergesort '(8 9)) (mergesort '(3 4))
At the lowest tree leves, the two element get sorted and the recursion unwinds.
(mergesort '(6 4 5 7 8 9 3 4))
/ \
(mergesort '(6 4 5 7)) (mergesort '(8 9 3 4))
/ \ / \
(4 6) (5 7) (8 9)) (3 4))
The next level of recursion:
(mergesort '(6 4 5 7 8 9 3 4))
/ \
(4 5 6 7)) (3 4 8 9)
/ \ / \
(4 6) (5 7) (8 9)) (3 4))
And the final level with merges into the overall list
(3 4 4 5 6 7 8 9)
/ \
(4 5 6 7) (3 4 8 9)
Of course, this is not the actual order that things happen but it give
the overall algorithm flavor.
In scheme, we can write functions to split lists, and merge sorted
lists. Once we have these, the mergesort itself is fairly short.
def mergesort(lis)
if lis.length == 0 then
[]
elsif lis.length == 1 then
lis
elsif lis.length == 2 then
mergelists ([lis[0]], lis[1..lis.length-1])
else
mergelists(mergesort(split(lis)[0]),mergesort(split(lis)[1..lis.length-1][0]))
end
end
def mergelists(lis, m) # assume L and M are sorted lists
if lis.length==0 then
m
elsif m.length==0 then
lis
elsif lis[0] < m[0] then
[lis[0]] + mergelists(lis[1..lis.length-1], m)
else
[m[0]] + mergelists(lis, m[1..m.length-1])
end
end
def sub(lis, start, stop, ctr) # extract elements start to stop into a list
if lis.length== 0 then
lis
elsif ctr < start then
sub(lis[1..lis.length-1], start, stop, ctr+1)
elsif ctr > stop then
[]
else
[lis[0]] + sub(lis[1..lis.length-1],start, stop, ctr+1)
end
end
def split(lis) # split the list in half:, ; returns ((first half)(second half))
len=lis.length
if len == 0 then
[lis]+[lis]
elsif len == 1 then
[lis] + [[]]
else
[sub(lis, 1, len/2, 1)] + [sub(lis, (len/2)+1, len, 1)]
end
end
x = [6,3,4,9,10,1,7]
print x, "\n"
#print split(x)
print mergesort(x), "\n"