Big Data Analytics

. Input is a sorted array a[1 : n] of arbitrary real numbers. The array could only be of one of the
following two types: 1) Type I: All the elements in the array are distinct; or 2) Type II: The
array has √
n copies of one element, the other elements being distinct. Present a Monte Carlo
algorithm that determines the type of the array in O(

n log n) time. Show that the output of
your algorithm will be correct with high probability.(Fact: (1 − x)
1/x ≤ 1/e for any 1 > x > 0.)
2. Show that the maximum of n given elements can be found in O(1) time using n
1+
common
CRCW PRAM processors, where  is any constant > 0.
3. Present an O(

n) time algorithm for the selection problem. You can use up to √
n CREW PRAM
processors.
4. What happens to the I/O complexity of the sorting algorithm we discussed in class if we choose
k to be cM
B
for some integer c > 1?
5. Present an efficient implementation of Djikstra’s algorithm for the single source shortest paths
problem on an out-of-core computing model with a single disk. What is the I/O complexity of
your implementation?


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