consider lower bounds proofs on comparison based algorithms using decision trees. For each problem below, fill...
consider lower bounds proofs on comparison based algorithms using decision trees. For each problem below, fill in the table with:
-a. The number of possible answers to the problem
b. A lower bound based on the height of a decision tree for that
problem. Simplify your answers.
c. Explain why you think it is (or not) possible to create an
algorithm with the same order as the bound that you
found.
Problem: 1- Search for the value X in an ordered array of size N
-a. The number of possible answers to the problem?
-b. A lower bound based on the height of a decision tree for that problem. Simplify your answers.?
-c. Explain why you think it is (or not) possible to create an algorithm with the same order as the bound that you found. ?
2- Search for the value X in an unordered array of size N
-a. The number of possible answers to the problem
-b. A lower bound based on the height of a decision tree for that problem. Simplify your answers.
-c. Explain why you think it is (or not) possible to create an algorithm with the same order as the bound that you found.
3- Sort an array of size N
-a. The number of possible answers to the problem
-b. A lower bound based on the height of a decision tree for that problem. Simplify your answers.
-c. Explain why you think it is (or not) possible to create an algorithm with the same order as the bound that you found.
4- Find the 1 st, 2nd and 3rd largest elements in an unordered array of size N
-a. The number of possible answers to the problem
-b. A lower bound based on the height of a decision tree for that problem. Simplify your answers.
-c. Explain why you think it is (or not) possible to create an algorithm with the same order as the bound that you found.
5- Find the shortest path in a fully-connected graph(with V vertices) that visits every vertex once and returns to the starting vertex.
-a. The number of possible answers to the problem
-b. A lower bound based on the height of a decision tree for that problem. Simplify your answers.
-c. Explain why you think it is (or not) possible to create an algorithm with the same order as the bound that you found.
Homework Answers
1. In this problem, the number of possible answers is 
as
there are N indices at which X could be present in the array.
Hence, any decision tree for this problem must have height at least
,
thus giving a lower bound. This can be achieved by running a binary
search, as the array is ordered.
2. In this problem, the lower bound is the same as part 1. But
it cannot be achieved, as in an unordered array, it takes at least
N comparisons to find an element. This can be proven by as follows:
If an algorithm doesn't look at each index, then there is some
index which is not checked by the algorithm. Consider two array in
which X is at that index, and another in which X is not. The answer
of the algorithm cannot be different, while it should be. Hence, it
needs 
time.
3. In sorting, from each permutation of the input, it must lead
to the same output because as long as the values don't change,
their sorted order is the same. Hence, the number of possible
answers is 
. Thus the height of
the decision tree must be 
. An algorithm can be devised with the same time complexity, as
merge sort takes exactly this time.
4. Any answer will have three different indices, hence the
number of answers is 
. Thus any
decision tree must have height .
This cannot be achieved, as to find the largest element in an
unordered array, it must check all elements. This can be proven by
an argument similar to 2.
5. If every vertex needs to visited once, then there are
possible
paths. Hence, the lower bound is 
again. This may not be achievable unless P=NP, as this problem is
the travelling salesman problem, which is NP-complete.
Comment in case of any doubts.
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