Massachusetts Institute of Technology
Handout 3
6.854J/18.415J: Advanced Algorithms
Wednesday, September 16, 2009
David Karger
Problem Set 2
Due: Wednesday, September 23, 2009.
Collaboration policy:
collaboration is
strongly encouraged
. However, remember that
1. You must write up your own solutions, independently.
2. You must record the name of every collaborator.
3. You must actually participate in solving all the problems. This is diFcult in very large
groups, so you should keep your collaboration groups limited to 3 or 4 people in a given
week.
4.
No bibles. This includes solutions posted to problems in previous years.
Problem 1.
Some counterexamples.
(a)
In class, I stated that single rotations “don’t work” for splay trees. To demon
strate this, consider a degenerate
n
node “linked list shaped” binary tree (i.e.,
where each node’s right child is empty). Suppose the (only) leaf is splayed to the
root by
single
rotations. Show the structure of the tree after this splay. Gener
alizing, argue that there is a sequence of
n/
2 splays that each take at least
n/
2
work.
(b)
Now from the same starting tree, show the ±nal structure after splaying the leaf
with (zigzig) double rotations. Explain how this splay has made much more
progress than single rotations in “improving” the tree.
(c)
Given the theorem about access time in splay trees, it is tempting to conjecture
that splaying does not create trees in which it would take a long time to ±nd an
item. Show that this conjecture is false by showing that for large enough
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 Fall '09
 DavidKarger
 Algorithms

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