1. Lecture 14 Sort Quicksort Shortest Paths
COMP 201
Formal Specification using ASML.
(Examples)

2. SORT Algorithm (simple specification
of the one-swap-at-a-time sorting algorithm)

3. Sorting 4 1 5 2 3 1 2 3 4 5

4. var A as Seq of Integer
swap()
choose i in {0..length(A)-1}, j in {0..length(A)-1} where i < j and
A(i) > A(j)
A(j) := A(i)
A(i) := A(j)
sort()
step until fixpoint
Main()
step A := [-4,6,9,0, 2,-12,7,3,5,6]
step WriteLine(“Sequence A : ")
step sort()
step WriteLine("after sorting: " + A)

5. Hoare’s quicksort
Quicksort was discovered by Tony Hoare (published in 1962).
Here is the outline
Pick one item from the array--call it the pivot
Partition the items in the array around the pivot so all elements to the left are smaller than the pivot and all elements to the right are greater than the pivot
Use recursion to sort the two partitions
partition: items > pivot
pivot
parition 1: items <= pivot

6. Example
Initial array 4 1 3 8 2 11 9 5

7. Here is Hoare's quicksort using sequence comprehensions:
qsort(s as Seq of Integer) as Seq of Integer
if s = [] then return []
else pivot = Head(s) rest = Tail(s)
return qsort([y | y ∈ rest where y < pivot]) [pivot] + qsort([y | y ∈ rest where y ≥ pivot])
A sample main program sorts the Sequence [7, 8, 2, 42] and prints the result:
Main()
WriteLine(qsort([7, 8, 2, 42]))
Partition 2:
items > pivot
parition 1:
items <= pivot
pivot

8. Shortest Paths Algorithm
Specification of Shortest Paths from a given node s.
The nodes of the graph are given as a set N.
The distances between adjacent nodes are given by a map D, where D(n,m)=infinity denotes that the two nodes are not adjacent.

9. What is the shortest distance from SeaTac to Redmond?11 13
SeaTac
Seattle 11 5 5 9 13 9 5
Bellevue
Redmond 5

10. Graph declaration structure Node s as String infinity = 9999
N = {SeaTac, Seattle, Bellevue, Redmond}
D = {(SeaTac, SeaTac) -> 0,
(SeaTac, Seattle) -> 11,
(SeaTac, Bellevue) -> 13,
(SeaTac, Redmond) -> infinity, // to be calculated
(Seattle, SeaTac) -> 11,
(Seattle, Seattle) -> 0,
(Seattle, Bellevue) -> 5,
(Seattle, Redmond) -> 9,
(Bellevue, SeaTac) -> 13,
(Bellevue, Seattle) -> 5,
(Bellevue, Bellevue) -> 0,
(Bellevue, Redmond) -> 5,
(Redmond, SeaTac) -> infinity, // to be calculated
(Redmond, Seattle) -> 9,
(Redmond, Bellevue) -> 5,
(Redmond, Redmond) -> 0}
structure Node
s as String
infinity = 9999
SeaTac = Node("SeaTac")
Seattle = Node("Seattle“)
Bellevue = Node("Bellevue")
Redmond = Node("Redmond")

11. shortest( s as Node,. N as Set of Node,
shortest( s as Node, N as Set of Node, D as Map of (Node, Node) to Integer) as Map of Node to Integer
var S = {s -> 0} merge {n -> infinity | n in N where n ne s}
step until fixpoint
forall n in N where n ne s
S(n) := min({S(m) + D(m,n) | m in N})
step return S
min(s as Set of Integer) as Integer
require s ne {}
return any x | x in s where forall y in s holds x lte y

12. S(n) := min({S(m) + D(m,n) | m in N})?

13. The main program Main() // … Graph specification …
shortestPathsFromSeaTac = shortest(SeaTac, N, D)
WriteLine("The shortest distance from SeaTac to Redmond is "
shortestPathsFromSeaTac(Redmond) + " miles.")
The shortest distance from SeaTac to Redmond is 18 miles.