Algorithm Design

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Chapter 4 Greedy Algorithms
The Problem. Given a set of n streams, each specified by its number of
bits bi and its time duration ti, as well as the link parameter r, determine
whether there exists a valid schedule.
Example. Suppose we have n = 3 streams, with
(hi, q) = (2000, 1), (b2, t2) = (6000, 2), (b3, t3) = (2000, 1),
and suppose the link’s parameter is r = 5000. Then the schedule that runs
the streams in the order 1, 2, 3, is valid, since the constraint (.) is satisfied:
t = 1: the whole first stream has been sent, and 2000 < 5000.1
t = 2: half of the second stream has also been sent,
and 2000+ 5000 5000- 2
Similar calcalations hold for t = 3 and t = 4.
(a) Consider the following claim:
(b)
Claim: There exists a valid schedule if and only if each stream i satisfies
b i < rt i.
Decide whether you think the claim is true or false, and give a proof
of either the claim or its negation.
Give an algorithm that takes a set of n streams, each specified by its
number of bits bi and its time duration ti, as well as the link parameter
r, and determines whether there exists a valid schedule. The rtmning
time of your algorithm should be polynomial in n.
A small business--say, a photocopying service with a single large
machine--faces the following scheduling problem. Each morning they
get a set of jobs from customers. They want to do the jobs on their single
machine in an order that keeps their customers happiest. Customer i’s
job will take ti time to complete. Given a schedule (i.e., an ordering of the
jobs), let Ci denote the finishing time of job i. For example, if job j is the
first to be donel we would have Ci = tj; and ff job j is done right after job
i, we would have Ci = Q + ti. Each customer i also has a given weight wg
~sents his or her importance to the business. The happiness of
customer i is expected to be dependent o~ the finishing time of i’s job.
So the company decides that they want to order the jobs to mJnimlze the
weighted sum of the completion times, ~,n i=1 wiCi"
Design an efficient algorithm to solve this problem. That is, you are
given a set of n jobs with a processing time ti and a weight w~ for each
job. You want to order the jobs so as to minimize the weighted sum of
the completion times, ~P=I wiCi-
Example. Suppose there are two jobs: the first takes time q = ! and has
weight wl = !0, while the second job takes time t 2 = 3 and has weight
14.
Exercises
w2 = 2. Then doing job 1 first would yield a weighted completion time
of 10.1 + 2.4 = 18, while doing the second job first would yield the larger
weighted completion time of 10.4 + 2.3 = 46.
You’re working with a group of security consultants who are helping to
monitor a large computer system. There’s particular interest in keeping
track of processes that are labeled "sensitive." Each such process has a
designated start time and finish time, and it rtms continuously between
these times; the consultants have a list of the planned start and finish
times of al! sensitive processes that will be run that day.
As a simple first step, they’ve written a program called s~ca~;us_check
that, when invoked, runs for a few seconds and records various pieces
of logging information about all the sensitive processes running on the
system at that moment. (We’ll model each invocation of status_check
as lasting for only this single point in time.) What they’d like to do is to
run status_check as few times as possible during the day, but enough
that for each sensitive process P, status_check is invoked at least once
during the execution of process P.
(a) Give an efficient algorithm that, given the start and finish times of
all the sensitive processes, finds as small a set of times as possible
at which to invoke s~;a~cus_check, subject to the requirement
that s~a~cus_check is invoked at least once during each sensitive
process P.
(b) WtKle you were designing your algorithm, the security consultants
were engaging in a little back-of-the-envelope reasoning. "Suppose
we can find a set of k sensitive processes with the property that no
two are ever running at the same time. Then clearly your algorithm
will need to invoke s~ca~;us_check at least k times: no one invocation
of s~a~cus_check can handle more than one of these processes."
This is true, of course, and after some further discussion, you al!
begin wondering whether something stronger is true as well, a kind
of converse to the above argument. Suppose that k* is the largest
value of k such that one can find a set of k sensitive processes with
no two ever running at the same time. Is it the ~ase that there must
be a set of k* times at which you can run s~a~;us_check so that some
invocation occurs during the execution of each sensitive process? (In
other words, the kind of argument in the previous paragraph is really
the only thing forcing you to need a lot of invocations of
check.) Decide whether you think this claim is true or false, and give
a proof or a counterexample.
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