[CSE2304]
[progress]

### CSSE, Monash, .au, CSE2304, Tutorial #5, semester 1, 2001

Group B: week 11, 14 May...

Group A: week 12, 21 May...

Class: Your answers must be prepared before the tute.

- Integration is covered in the last lectures of the subject
but you can do some reading ahead.
Integration is the process of finding the area under a curve, f(x),
between x=a and x=b, for some a and b, where a<=b.
The simplest method is the (non-adaptive)
[rectangle rule (click)]:
Divide the interval [a,b] into N subintervals of equal width.
Approximate the integral over each subinterval by a rectangle.

The non-adaptive rectangle rule is not accurate if the
function changes quickly and N is small.

**Give an algorithm for a simple ***adaptive* rectangle rule:
The adaptive version uses more subintervals where
the function changes quickly.
It compares the values of f(x) at the ends of each subinterval.
If the difference in values is less than
a small constant, `epsilon', the function is assumed to
vary slowly in this region, and
the integral over the subinterval
is approximated by a rectangle, as before.
If the difference in values is greater than epsilon,
integrate over the subinterval using a *finer* subdivision.

- Give a
[recursive]
algorithm to generate
all strings consisting of n pairs of matched parentheses e.g.

n=1: ()
n=2: (()), ()()
n=3: ((())), (()()), (())(), ()(()), ()()()
etc.

Note that you can only close, ')', an already open '('.

- Draw an undirected, weighted
[graph]
representing the capital cities of Australia
{Adelaide, Brisbane, Canberra, Darwin, Hobart, Melbourne, Perth, Sydney}
and the air-routes between them flown by jet aircraft
(as far as you know or estimate).

Use
- Kruskal's algorithm
- Prim's algorithm

to find a minimum spanning tree of the cities in *your* graph.

© 2001, L. Allison, Comp. Sci. & SWE,
Monash University, Australia