1. David Evans http://www.cs.virginia.edu/evans
Lecture 13: Quicksorting
CS200: Computer Science
University of Virginia
Computer Science
David Evans

2. Menu Improving insertsort quicksort 14 February 2003
CS 200 Spring 2003

3. Last time: insertsort insertsort is (n2) (define (insertsort cf lst)
(if (null? lst)
null
(insertel cf
(car lst)
(insertsort cf
(cdr lst)))))
(define (insertel cf el lst)
(if (null? lst)
(list el)
(if (cf el (car lst))
(cons el lst)
(cons
(car lst)
(insertel cf el
(cdr lst))))))
insertsort is (n2)
14 February 2003
CS 200 Spring 2003

4. Can we do better? (insertel < 88 (list 1 2 3 5 6 23 63 77 89 90))
Suppose we had procedures
(first-half lst)
(second-half lst)
that quickly divided the list in two halves?
14 February 2003
CS 200 Spring 2003

5. Insert Halves (define (insertelh cf el lst) (if (null? lst) (list el)
(let ((fh (first-half lst))
(sh (second-half lst)))
(if (cf el (car fh))
(append (cons el fh) sh)
(if (null? sh)
(append fh (list el))
(if (cf el (car sh))
(append (insertelh cf el fh) sh)
(append fh (insertelh cf el sh))))))))
14 February 2003
CS 200 Spring 2003

6. Evaluating insertelh Every time we call insertelh, the size
> (insertelh < 3 (list ))
|(insertelh #<procedure:traced-<> 3 ( ))
| (< 3 1)
| #f
| (< 3 5)
| #t
| (insertelh #<procedure:traced-<> 3 (1 2 4))
| |(< 3 1)
| |#f
| |(< 3 4)
| |#t
| |(insertelh #<procedure:traced-<> 3 (1 2))
| | (< 3 1)
| | #f
| | (< 3 2)
| | (insertelh #<procedure:traced-<> 3 (2))
| | |(< 3 2)
| | |#f
| | (2 3)
| |(1 2 3)
| ( )
|( )
( )
(define (insertelh cf el lst)
(if (null? lst)
(list el)
(let ((fh (first-half lst))
(sh (second-half lst)))
(if (cf el (car fh))
(append (cons el fh) sh)
(if (null? sh)
(append fh (list el))
(if (cf el (car sh))
(append (insertelh cf el fh) sh)
(append fh (insertelh cf el sh))))))))
Every time we call insertelh, the size
of the list is approximately halved!
14 February 2003
CS 200 Spring 2003

7. How much work is insertelh?
Assume first-half and second-half are  (1)
Each time we call
insertelh, the size of
lst halves. So, doubling
the size of the list only
increases the number of
calls by 1.
(define (insertelh cf el lst)
(if (null? lst)
(list el)
(let ((fh (first-half lst))
(sh (second-half lst)))
(if (cf el (car fh))
(append (cons el fh) sh)
(if (null? sh)
(append fh (list el))
(if (cf el (car sh))
(append (insertelh cf el fh) sh)
(append fh
(insertelh cf el sh))))))))
List Size Number of insertelh applications
1 1
2 2
4 3
8 4
16 5
14 February 2003
CS 200 Spring 2003

8. How much work is insertelh?
Assume first-half and second-half are  (1)
Each time we call
insertelh, the size of
lst halves. So, doubling
the size of the list only
increases the number of
calls by 1.
insertelh is  (log2 n)
log2 a = b
means
2b = a
List Size Number of insertelh applications
1 1
2 2
4 3
8 4
16 5
14 February 2003
CS 200 Spring 2003

9. insertsorth would be (n log2 n)
Same as insertsort, except uses insertelh
(define (insertsorth cf lst)
(if (null? lst)
null
(insertelh cf
(car lst)
(insertsorth cf
(cdr lst)))))
(define (insertelh cf el lst)
(if (null? lst)
(list el)
(let ((fh (first-half lst))
(sh (second-half lst)))
(if (cf el (car fh))
(append (cons el fh) sh)
(if (null? sh)
(append fh (list el))
(if (cf el (car sh))
(append (insertelh cf el fh) sh)
(append fh
(insertelh cf el sh))))))))
insertsorth would be (n log2 n)
if we have fast first-half/second-half
14 February 2003
CS 200 Spring 2003

10. Is there a fast first-half procedure?
No!
To produce the first half of a list length n, we need to cdr down the first n/2 elements
So:
first-half is  (n)
insertelh calls first-half every time…so
insertelh is  (n) *  (log2 n) =  (n log2 n)
insertsorth is  (n) *  (n log2 n) =  (n2 log2 n)
Yikes! We’ve done all this work, and its still worse than our simple bubblesort!
14 February 2003
CS 200 Spring 2003

11. 14 February 2003
CS 200 Spring 2003

12. The Great Lambda Tree of Ultimate Knowledge and Infinite Power
14 February 2003
CS 200 Spring 2003

13. el Sorted Binary Trees A tree containing A tree containing
left
right
A tree containing
all elements x such
that (cf x el) is true
A tree containing
all elements x such
that (cf x el) is false
14 February 2003
CS 200 Spring 2003

14. 3 5 2 8 4 7 1 Tree Example cf: < null null 14 February 2003
CS 200 Spring 2003

15. Representing Trees (define (make-tree left el right)
(list left el right))
(define (get-left tree)
(first tree))
(define (get-element tree)
(second tree))
(define (get-right tree)
(third tree))
left and right are trees
(null is a tree)
tree must be a non-null tree
tree must be a non-null tree
tree must be a non-null tree
14 February 2003
CS 200 Spring 2003

16. 5 2 8 1 Trees as Lists (make-tree (make-tree (make-tree null 1 null) 2
(define (make-tree left el right)
(list left el right))
(define (get-left tree)
(first tree))
(define (get-element tree)
(second tree))
(define (get-right tree)
(third tree)) 2 8 1
(make-tree (make-tree (make-tree null 1 null) 2
null) 5
(make-tree null 8 null))
14 February 2003
CS 200 Spring 2003

17. insertel-tree (define (insertel-tree cf el tree) (if (null? tree)
(make-tree null el null)
(if (cf el (get-element tree))
(make-tree
(insertel-tree cf el (get-left tree))
(get-element tree)
(get-right tree))
(get-left tree)
(insertel-tree cf el (get-right tree))))))
If the tree is null, make a new tree with el as its element and no left or
right trees.
Otherwise, decide
if el should be in
the left or right subtree.
insert it into that
subtree, but leave the
other subtree unchanged.
14 February 2003
CS 200 Spring 2003

18. How much work is insertel-tree?
Each time we call
insertel-tree, the size of
the tree. So, doubling
the size of the tree only
increases the number of
calls by 1!
(define (insertel-tree cf el tree)
(if (null? tree)
(make-tree null el null)
(if (cf el (get-element tree))
(make-tree
(insertel-tree cf el (get-left tree))
(get-element tree)
(get-right tree))
(get-left tree)
(insertel-tree cf el (get-right tree))))))
insertel-tree is
 (log2 n)
log2 a = b
means 2b = a
14 February 2003
CS 200 Spring 2003

19. insertsort-tree (define (insertsort cf lst) (if (null? lst) null
(insertel cf (car lst)
(insertsort cf (cdr lst)))))
(define (insertsort-worker cf lst)
(if (null? lst) null
(insertel-tree cf (car lst)
(insertsort-worker cf (cdr lst)))))
No change…but insertsort-worker evaluates to a tree not a list!
(((() 1 ()) 2 ()) 5 (() 8 ()))
14 February 2003
CS 200 Spring 2003

20. extract-elements
We need to make a list of all the tree elements, from left to right.
(define (extract-elements tree)
(if (null? tree)
null
(append (extract-elements (get-left tree))
(cons (get-element tree)
(extract-elements (get-right tree))))))
14 February 2003
CS 200 Spring 2003

21. How much work is insertsort-tree?
(define (insertsort-tree cf lst)
(define (insertsort-worker cf lst)
(if (null? lst)
null
(insertel-tree cf
(car lst)
(insertsort-worker cf (cdr lst)))))
(extract-elements (insertsort-worker cf lst)))
(n) applications
of insertel-tree
each is (log n)
(n log2 n)
14 February 2003
CS 200 Spring 2003

22. Growth of time to sort random list
bubblesort
n log2 n
insertsort-tree
14 February 2003
CS 200 Spring 2003

23. Comparing sorts > (testgrowth bubblesort)
n = 250, time = 110
n = 500, time = 371
n = 1000, time = 2363
n = 2000, time = 8162
n = 4000, time = 31757
( )
> (testgrowth insertsort)
n = 250, time = 40
n = 500, time = 180
n = 1000, time = 571
n = 2000, time = 2644
n = 4000, time = 11537
( )
> (testgrowth insertsorth)
n = 250, time = 251
n = 500, time = 1262
n = 1000, time = 4025
n = 2000, time = 16454
n = 4000, time = 66137
( )
> (testgrowth insertsort-tree)
n = 250, time = 30
n = 500, time = 250
n = 1000, time = 150
n = 2000, time = 301
n = 4000, time = 1001
( )
14 February 2003
CS 200 Spring 2003

24. Can we do better? Making all those trees is a lot of work
Can we divide the problem in two halves, without making trees?
14 February 2003
CS 200 Spring 2003

25. Quicksort C. A. R. (Tony) Hoare, 1962 Divide the problem into:
Sorting all elements in the list where
(cf (car list) el)
is true (it is < the first element)
(not (cf (car list) el)
is true (it is >= the first element)
Will this do better?
14 February 2003
CS 200 Spring 2003

26. Quicksort (define (quicksort cf lst) (if (null? lst) lst (append
(filter (lambda (el) (cf el (car lst)))
(cdr lst)))
(list (car lst))
(filter (lambda (el) (not (cf el (car lst))))
(cdr lst))))))
14 February 2003
CS 200 Spring 2003

27. filter (define (filter f lst) (insertl (lambda (el rest)
(if (f el) (cons el rest) rest))
lst
null))
14 February 2003
CS 200 Spring 2003

28. How much work is quicksort?
(define (quicksort cf lst)
(if (null? lst) lst
(append
(quicksort cf
(filter (lambda (el) (cf el (car lst)))
(cdr lst)))
(list (car lst))
(filter (lambda (el) (not (cf el (car lst))))
(cdr lst))))))
What if the input list is sorted?
Worst Case: (n2)
What if the input list is random?
Expected: (n log2 n)
14 February 2003
CS 200 Spring 2003

29. Comparing sorts Both are (n log2 n)
> (testgrowth insertsort-tree)
n = 250, time = 20
n = 500, time = 80
n = 1000, time = 151
n = 2000, time = 470
n = 4000, time = 882
n = 8000, time = 1872
n = 16000, time = 9654
n = 32000, time = 31896
n = 64000, time = 63562
n = , time =
( )
> (testgrowth quicksort)
n = 250, time = 20
n = 500, time = 80
n = 1000, time = 91
n = 2000, time = 170
n = 4000, time = 461
n = 8000, time = 941
n = 16000, time = 2153
n = 32000, time = 5047
n = 64000, time = 16634
n = , time = 35813
( )
Both are (n log2 n)
Absolute time of quicksort much faster
14 February 2003
CS 200 Spring 2003

30. Good enough for VISA? (n log2 n) How long to sort 800M items?
n = , time = 35813
36 seconds to sort with quicksort
(n log2 n)
How long to sort 800M items?
> (log 4)
> (* (log ))
> (/ (* (log )) 36)
> (/ (* (log )) )
> (/ (* (log )) )
seconds ~ 4.5 days
14 February 2003
CS 200 Spring 2003

31. Quicksorting 800M items? > (log 4) 1.3862943611198906
> (/ (* (log )) 36)
> (/ (* (log )) )
> (/ (* (log )) )
> (/ (* (log )) )
> (/ (* (log )) )
> (/ (/ (/ (/ (* (log )) ) 60) 60) 24)
4.5 days with quicksort
is a lot better than
20,000 years with bubblesort…
but still not good enough for VISA.
14 February 2003
CS 200 Spring 2003

32. Later in the course…
So far, we have been talking amount the work a procedure requires
In a few weeks, we will learn how to talk about the amount of work a problem requires
That is, how much work is the best possible sorting procedure?
For the general case, you can’t do better than (n log2 n)
VISA’s problem is simpler, so they can do much better: (n)
14 February 2003
CS 200 Spring 2003

33. Charge Monday: Tyson’s essay Wednesday: Tim Koogle Friday: Exam Review
Remember to send me a question for him
That is your “ticket” to come to class Weds
Friday: Exam Review
Ask any questions you want before picking up exam 1.
14 February 2003
CS 200 Spring 2003