|
| Homework 9General Comments
- For full credit, provide context for each problem, show all calculations,
and justify all answers by providing enough comments to explain your reasoning.
- Homework assignments must be very neatly written or typeset
(e.g. using Word or OpenOffice).
- You can get up to 50% credit on a problem if you get significant outside assistance. Thus, if you are totally stuck on a problem it might be worth getting help. However, you must indicate any assistance/collaboration (See the Homework Assistance section on the Policies page). Failure to do so could result in a failing grade for the course! Note that getting help from the Help Center or me does not count as significant outside assistance, but talking with your classmates or searching on the Internet does!
- If a problem asks for an algorithm,
you should give the most efficient algorithm you can find to ensure full credit.
You should also specify the complexity of the algorithm with justification,
whether or not the problem asks for it.
Details
-
Let \(T(n)\) be defined by \(T(1)=1\) and
\(T(n) \;=\; 4\,T\!\left(\frac{n}{3}\right) + n\),
where \(n = 3^k\), for \(k=0,1,2,\ldots\)
For this assignment, you will show that the
solution to this recurrence relation is
- \(T(n)=4\,n^{\log_3 4}-3n\), or
- \(T(3^k)=4^{\,k+1}-3^{\,k+1}\), or
- \(T(n)=\Theta(n^{\log_3 4})\),
where the first two formulas are equivalent, one expressed in terms
of \(n\) and the other in terms of \(k\)
(Remember, we are assuming \(n = 3^k\)), and the third is a bound.
For each of the following, choose the form of the answer that is easiest to
work with for that given proof type.
-
Prove the given result using Iteration (also called backward substitution).
- Prove the given result using Substitution (prove it by induction).
- Prove the given result using the Master Theorem.
|
|
|