Fall 2006 CMSC 651: MIDTERM Due: Nov 1
1. (0 points) What is your name?
2. (20 points) Show how to multiply two n×nmatrices over Z5in O(n3/log n) steps in a manner
similar to the one given in class for Z2.
3. (20 points)
(a) Give the algorithm and analysis for taking the union of THREE Binomial Heaps. (Do
not assume that the alg for the union of TWO is known.)
(b) Give the algorithm and analysis for taking the union of THREE Fibonacci Heaps.
4. (20 points) Give a data structure and the psuedocode for maintaining disjoint sets such that,
if the structure has nelements in it, that supports the following three operations in the time
bound given (amortized).
•UNION O(1).
•FIND O(log∗n).
•FINDALL (lists out all the elements of the structure) O(n). (NOTE- the order does not
matter.)
•INSERT O(1)
5. (20 points) In class we did Dijistras algorithm with bounded weights, bound C, with buckets
of size 1,2,22,23,...,2log2C. What goes wrong with the proof if we use buckets of sizes
1,3,32,33,...,3log3C.
6. (20 points) Let G= (V, E ) be a graph. Let Mbe a matching in G. Let M∗be a maximum
matching in G. Show that if M∗has kmore edges then Mthen Mhas at least kdisjoint
augmenting paths.
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