Algorithm Design

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Chapter 7 Network Flow
Thus the hospital relaxes the requirements as follows. They add
a new parameter c > 0, and the system now should try to return to
each doctor j a list L~ with the following properties.
(A*) L~ contains at most c days that do not appear on the list Lj.
L~,
(B) (Same as before) If we consider the whole set of lists L~ .....
it causes exactly Pi doctors to be present on day i, for i = 1, 2 ..... n.
Describe a polynomial-time algorithm that implements this revised
system. It should take the numbers Pl, Pz ..... Pn, the lists
L1 ..... Lk, and the parameter c > 0, and do one of the following two
things.
- Return lists L1, , , L. ’ satisfying properties (A*) and (B); or
2 .... Lk ’ that
- Report (correctly) that there is no set of lists Lv L2 ..... Lk
satisfies both properties (A*) and (B).
20. Your friends are involved in a large-scale atmospheric science experiment.
They need to get good measurements on a set S of n different
conditions in the atmosphere (such as the ozone level at various places),
and they have a set of m balloons that they plan to send up to make these
measurements. Each balloon can make at most two measurements.
Unfortunately, not all balloons are capable of measuring all conditions,
so for each balloon i = 1 ..... m, they have a set Si of conditions
that balloon i can measure. Finally, to make the results more reliable, they
plan to take each measurement from at least k different balloons. (Note
that a single balloon should not measure the same condition twice.) They
are having trouble figuring out which conditions to measure on which
balloon.
Example. Suppose that k = 2, there are n = 4 conditions labeled q, c2, c3, c4,
and there are rn = 4 balloons that can measure conditions, subject to
the limitation that S~ = Sz = {q, c2, c3}, and $3 = $4 = {q, q, c4}..Then one
possible way to make sure that each condition is measured at least k = 2
times is to have
¯ balloon I measure conditions
¯ balloon 2 measure conditions cz, cs,
¯ balloon 3 measure conditions q, c4, and
¯ balloon 4 measure conditions c~, c4.
(a) Give a polynomial-time algorithm that takes the input to an instance
of this problem (the n conditions, the sets Si for each of the ra
balloons, and the parameter k) and decides whether there is a way to
measure each condition by k different balloons, while each balloon
only measures at most two conditions.
21.
Exercises
You show your friends a solution computed by your algorithm from
(a), and to your surprise they reply, "This won’t do at all--one of the
conditions is only being measured by balloons from a single subcontractor."
You hadn’t heard anything about subcontractors before; it
turns out there’s an extra wrinkle they forgot to mention ....
Each of the balloons is produced by one of three different sub.
contractors involved in the experiment. A requirement of the experiment
is that there be no condition for which all k measurements
come from balloons produced by a single subcontractor.
For example, suppose balloon 1 comes from the first subcontractor,
balloons 2 and 3 come from the second subcontractor, and
balloon 4 comes from the third subcontractor. Then our previous solution
no longer works, as both of the measurements for condition
c~ were done by balloons from the second subcontractor. However,_
we could use balloons 1 and 2 to each measure conditions c1, c2, and
use balloons 3 and 4 to each measure conditions c3, c4.
Explain how to modify your polynomial-time algorithm for part
(a) into a new algorithm that decides whether there exists a solution
satisfying all the conditions from (a), plus the new requirement about
subcontractors.
You’re helping to organize a class on campus that has decided to give all
its students wireless laptops for the semester. Thus there is a collection
of n wireless laptops; there is also have a collection of n wireless access
points, to which a laptop can connect when it is in range.
The laptops are currently scattered across campus; laptop e is within
range of a set S~ of access points. We will assume that each laptop is within
range of at least one access point (so the sets S~ are nonempty); we will
also assume that every access point p has at least one laptop within range
of it.
To make sure that all the wireless connectivity software is working
correctly, you need to try having laptops make contact with access points
in such a way that each laptop and each access point is involved in at
least one connection. Thus we will say that a test set T is a collection of
ordered pairs of the form (e, p), for a laptop e and access point p, with
the properties that
(i) If (LP) ~ T, then e is within range ofp (i.e., p ~ Se).
(ii) Each laptop appears in at least one ordered pair in T.
(i~) Each access point appears in at least one ordered pair in T.
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