CS243: Algorithms and Data Structures - Of the Clux

CS243: Algorithms and Data Structures
Problem sheet 2
Due on Monday 2 February 2009 by 13:05
Solve THREE of the following four problems. State clearly which three problems you have solved.
1. Imagine that you have n music files (songs) F 1 , F 2 , . . . , F n which you want to store on a memory
stick. For every i = 1, 2, . . . , n, the song F i has size s i , or in other words, it requires s i kilobytes
of space. The capacity of the memory stick is C kilobytes. Unfortunately, not all of your songs
can fit on the memory stick because ∑ n
i=1 s i > C.
(a) Suppose that you want to maximize the number of songs held on the memory stick.
Consider the greedy algorithm which selects songs in order of increasing size. Either
prove that this algorithm is correct or provide a counter-example.
(b) Suppose that you want to use as much of the capacity of the memory stick as possible.
Consider the greedy algorithm which selects songs in order of decreasing size. Either
prove that this algorithm is correct or provide a counter-example.
2. Recall that an input to the interval scheduling problem is a sequence s 1 , s 2 , . . . , s n of starting
times, and a sequence f 1 , f 2 , . . . , f n of finishing times of jobs 1, 2, . . . , n, respectively. Consider
the following greedy algorithm for the interval scheduling problem.
Sort the jobs according to their starting times; after this step we have that:
s 1 ≤ s 2 ≤ · · · ≤ s n .
Start with an empty set of selected jobs. For every i = n, n − 1, . . . , 1 (in this order),
add job i to the set of selected jobs unless it is incompatible (that is, it overlaps)
with another job which has been selected before.
Prove that this algorithm is correct, i.e., that it always computes a set of compatible jobs of
maximum size.
3. Suppose that you are consulting for a company which wants to organize an annual party for
its employees. You have agreed with the management that a good party is such that:
• for every participant, there are at least two other participants whom (s)he knows; and
• for every participant, there are at least two other participants whom (s)he doesn’t know.
We assume throughout, that the acquaintance relation is symmetric, that is, any two employees
either both know each other, or they both don’t know each other.
(a) Consider two groups of employees:
• employees 1, 2, and 3 who all know each other; and
• employees 4, 5, and 6 who all know each other;
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and nobody in the first group knows anybody in the other group (and vice versa).
Draw the (undirected) graph in which the vertices correspond to the six employees, and
there is an edge between two of them if and only if they know each other.
Argue that if you invite all six employees to the party then it will be a good party.
(b) Consider the six employees from problem (3a), as well as employees 7 and 8, such that:
• employee 7 knows only employee 1; and
• the only people that employee 8 does not know are employees 6 and 7.
Explain why inviting employee 7 to the party would prevent it from being a good party.
Can a party to which employee 8 is invited be a good party Justify your answer.
What is the size of the biggest possible good party Who should be invited to it
(c) Design an efficient algorithm that—given an (undirected) graph G = (V, E) in which vertices
correspond to employees, and edges join employees who know each other—computes
the size of the largest possible good party, and lists the employees who should be invited
to it.
Argue that your algorithm is correct.
What is the running time of your algorithm
(d) Illustrate the behaviour of your algorithm on the example from problem (3b), that is,
explain how your algorithm computes the list of employees who should be invited to the
biggest possible good party.
4. Solve ONE of the following two problems (4a) or (4b). State clearly whether you have solved
problem (4a) or (4b).
(a) You are consulting for a trucking company that does a large amount of business shipping
packages between Edinburgh and London. The volume is high enough that they have
to send a number of trucks each day between the two locations. Trucks have a fixed
limit W on the maximum amount of weight they are allowed to carry. Boxes arrive at the
Edinburgh station one by one, and each package i has a weight w i . The trucking station
is quite small, so at most one truck can be at the station at any time. Company policy
requires that boxes are shipped in the order they arrive; otherwise, a customer might get
upset upon seeing a box that arrived after his make it to London faster. At the moment,
the company is using a simple greedy algorithm for packing: they pack boxes in the order
they arrive, and whenever the next box does not fit, they send the truck on its way.
But they wonder if they might be using too many trucks, and they want your opinion on
whether the situation can be improved. Here is how they are thinking. Maybe one could
decrease the number of trucks needed by sometimes sending off a truck that was less full,
and in this way allow the next few trucks to be better packed.
Prove that, for a given set of boxes with specified weights, the greedy algorithm currently
in use actually minimizes the number of trucks that are needed. Your proof could follow
the type of analysis that is used for the Interval Scheduling Problem in the Kleinberg
and Tardos book: it should establish the optimality of this greedy packing algorithm by
identifying a measure under which it “stays ahead” of all other solutions.
(b) Let’s consider a long, quiet country road with houses scattered very sparsely along it.
(We can picture the road as a long line segment, with an eastern endpoint and a western
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endpoint.) Further, let’s suppose that the residents of all these houses are avid mobile
phone users. You want to place mobile phone base stations at certain points along the
road, so that every house is within four miles of one of the base stations.
Give an efficient algorithm that achieves this goal using as few base stations as possible.
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