CISC621 -- Fall 1999
HOMEWORK 1
Handed out: September 2, 1999
Due date: September 16, 1999
Note: Problem 0 is NOT to be handed in! It is included to so
that you can see what a solution to a problem might look like. The solution
to problem 0 can be accessed via the course webpage.
0. [Sample Problem]
- A coloring of the vertices of an undirected graph means that colors
are assigned to the vertices such that adjacent vertices have different
colors. Prove that if a graph has e edges, then that graph may be colored
using at most O(e^1/2) colors.
1. [Individual Problem] Group 1 grades this problem
- You are standing on a road in Death Valley, California. The temperature
is 119F (48.3C), and you have no water. You do know that water is located
somewhere along the road, but you are disoriented from the heat, and do
not know in which direction. You have no vehicle -- you have to walk to
find the water. Assuming that the water is located at distance x from you,
give an algorithm for finding the water where you are sure to walk a
total distance of no more than cx, for some constant c.
-
In writing up your solution, be
sure to state the value of c associated with your solution and remember:
you do not know the value of x.
2. [Individual Problem] Group 2 grades this problem
- Problem 5-2, Friendly graphs
- Do parts (a) and (b) only.
3. [Individual Problem] Group 3 grades this problem
- Consider a set of intervals on the real line I1, I2, ..., In. A coloring
of these intervals is an assigment of colors to the intervals such that
two intervals that overlap must be assigned different colors. Give an O(n
log n) algorithm for finding a minimum coloring of a set of intervals.
- Note: O(n log n) is the time required to sort n values.
B.D. Saunders
September 1, 1999