Algorithm Design and Analysis September 15, 2008
Pennsylvania State University CSE 565, Fall 2008
Adam Smith Handout 5
Homework 2 – Due Friday, September 17, 2008
Please refer to the general information handout for the full homework policy and options.
Reminders
•Your solutions are due before the lecture. Late homework will not be accepted.
•Collaboration is permitted, but you must write the solutions by yourself without assistance,
and be ready to explain them orally to a member of the course staff if asked. You must also
identify your collaborators. Getting solutions from outside sources such as the Web or students
not enrolled in the class is strictly forbidden.
•To facilitate grading, please write down your solution to each problem on a separate sheet of
paper. Make sure to include all identifying information and your collaborators on each sheet.
Your solutions to different problems will be graded separately, possibly by different people, and
returned to you independently of each other.
•For problems that require you to provide an algorithm, you must give a precise description of
the algorithm, together with a proof of correctness and an analysis of its running time.
You may use algorithms from class as subroutines. You may also use any facts that we proved
in class or from the book.
Reading Review chapter 4.1-4.5 in Kleinberg Tardos.
Exercises These should not be handed in, but the material they cover may appear on exams:
1. Exercises in Chapter 4.
2. (Caching) In our discussion of optimal caching, we used the fact that any eviction schedule may
be changed to a reduced eviction schedule without affecting its cost. This is stated informally in
the book (Lemma 4.11 on page 134).
Let us make this more precise. If an eviction schedule brings an item dinto the cache in a
step where the dwas not requested, we say the eviction was unreduced. Prove the following
by induction on the number of unreduced items: Let Sbe any eviction schedule which has no
unreduced evictions up to some step j. There exists a reduced schedule ¯
Sthat brings in as most
as many items as Sand that performs the same actions as Sin the first jsteps.
3. Point out the error in the following proof by induction.
Claim 1 In any set of hhorses, all horses are the same color.
Proof: We proceed by induction on the number hof horses. (Base case) If h= 1, then there is
only one horse in the set, and so all the horses in the set are clearly the same color.
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