Comprehensive Exam: Algorithms
and
Concrete Mat hematics
Autumn
2003
1.
[lo pts]
Prove a tight asymptotic bound on the behavior of T(71)
=
T(n/3) +T(n/9)
+
n3, where T(n)
5
b for n
<
no, where no and b are some given constants. For simplicity,
solve for n being a power of 9.
Simplest approach is to guess T(n)
=
Q(n3) and prove by induction. First notice that
T(n)
=
fl(n3) just by examining the recurrence.
Now assume T(n)
5
en3 for some
constant c. This constant should be large enough to satisfy initial conditions. Naw the
inductive step is:
The claim follows since the expression in the parenthesis is smaller than
1 for large enough
C.
2.
[lo pts]
Given integers a1
,
az,
.
.
.
,
a,,
give
a randomized algorithm that outputs all
pairs
(i,
j)
such that ai
=
aj. Your algorithm should have expected running time O(n+
K)
where
K is the number of pairs output. Prove the upper bound on the running time.
Use hashing with chaining, with size of the hash table constant factor larger than n.
Moreover, construct each chain as a chain of linked lists, each list holding equal elements.
In other words, if
aj
hashes into position q, look through the linked list and find the
list that corresponds to the value of aj. Then, for each element
ak
of this list, output
(aj,
ah).
The claim follows since the expected number of lists attached to a single hash
cell is constant.
3.
[lo
pts]
Assume that you are given a 'Lbla~k
box" implementation of a comparisons
based data structure that supports "extract minimum" and "insert element". You are
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 Algorithms, Graph Theory, Analysis of algorithms, Kruskal, rightmost hospital

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