Homework 4
CSE 450/598 Spring 2010 Arizona State University
Due: Thursday 2/18 before 10:30
1.
(4.10)
Let
G
= (
V
,
E
)
be an undirected graph with a nonnegative edge cost function
c
.
Assume you are given a minimumcost spanning tree
T
in
G
. Now assume that a new edge
is added to
G
, connecting two nodes
v
and
w
with cost
c
.
(a) Give an efficient algorithm to test if
T
remains the minimumcost spanning tree with
the new edge added to
G
. Make your algorithm run in time
O
(

E

)
. Can you do it
in
O
(

V

)
time? Please note any assumptions you make about what dada structure is
used to represent the tree
T
and the graph
G
.
(b) Suppose
T
is no longer the minimumcost spanning tree. Give a lineartime algorithm
(time
O
(

E

)
) to update the tree
T
to the new minimumcost spanning tree.
2.
(4.17)
Consider the following variation on the Interval Scheduling Problem. You have a
processor that can operate 24 hours a day, every day. People submit requests to run
daily
jobs
on the processor. Each such job comes with a
start time
and an
end time
. If the job
is accepted to run on the processor, it must run continuously, every day, for the perod
between its start and end times. (Note that certain jobs can begin before midnight and
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 Spring '08
 
 Graph Theory, web site, Planar graph, Vertextransitive graph

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