CMPS 201 Fourth Homework, Fall 04
3 problems, 30 pts, Due Tuesday, October 26
1. (12 pts) For this problem you will show that using groups of three does not lead to a
linear time median-of-medians algorithm.
Recall the median of median algorithm presented in class. Consider a modified algorithm
that divides the input is divided into groups of 3 rather than groups of 5. Let f(n) be
the number of comparisons done by this groups-of-three algorithm assuming that all
partitions are maximally un-even (with the pivot element being the median of medians),
and that the desired key is the sub-array with as many keys as possible. Analyze the
groups-of-three algorithm to obtain a recurence relation T(n) that lower bounds f(n)
(i.e. the value defined by your recurrence should be an underestimate of f(n)). Include
floors and ceilings in your T(n) recurrence and for each term in the recurrence indicate
what part of the algorithm it corresponds to.
Next prove that the function defined by your recurrence is in ฯ(n) (i.e. grows faster
than linearly). This can be done by using an induction to prove that that the solution
to T(n) is at least some function g(n)โฯ(n) (consider using g(n) = cn log n). Although
not mathematically correct, you may drop the floors and/or ceilings in the induction to
simplify the algebra. You may also assume that the algorithm uses heap sort when nis
small (n < 40), so the recurrence relation is only used for large n.
2. (6 pts) Consider the problem of storing ndifferent positive integer keys and an associated
data pointer for each key in a dictionary (perhaps the keys are student IDs and the
associated data is a pointer to the students record).
a. First, assume that all nkeys (and their pointers) are available ahead of time (so
you can do whatever preprocessing you want) and that there will be no insertions
or deletions. Describe a way to store the keys and pointers so that the worst case
lookup time is O(log n) and no extra storage is needed.
b. Next, assume you implement a hash table with mslots and chaining to hold the keys
and data. How many additional pointers are required (including null pointers)?
c. Now assume we want to implement the hash table using open addressing. Assuming
uniform hashing, how big does the table size have to be so that the expected number
of probes for an unsuccessful lookup is at most lg n?
3. (12 pts) Prove that at least l3n
2mโ2 comparisons are required to find both the maximum
and minimum of nelements. (Hint: use an adversary argument which keeps track of
four sets of elements โ those that can be the largest, those that can be the smallest,
those that can be both, and those that can be neither.)