Problem Set 2

1. Show that $f(n)=2^{n+1}+n$ is $O(2^n)$. Justify your answer with a proof.

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2. Show that $f(n)=n^2+4n+5$ is $\Omega (n^2)$. Justify your answer with a proof.

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3. Show that $f(n)=10nlogn$ is $\Theta (nlogn)$. Justify your answer with a proof.

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4. Prove by induction that $n^3+2n$ is divisible by $3$ for every non-negative integer n.

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5. According to this reference, the Bubble Sort algorithm has a worst case time complexity of $O(n^2)$, but a best case time complexity of $O(n)$. Does this mean that Bubble Sort is more performant than an algorithm like Quicksort with a worst case and best case complexity of $O(nlogn)$? Why don’t we just use Bubble Sort in large systems, all the time?

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6. Suppose I create a data structure called MyCoolStructure which has four operations. Each of the operations runs in O(1) time:

MyCoolStructure mcs;
 
mcs.insert_first(x); // inserts "x" at the front of the structure
mcs.insert_last(x);  // inserts "x" at the end of the structure
mcs.delete_first();  // deletes the element at the front of the structure and returns it to the user
mcs.delete_last();   // deletes the element at the end of the structure and returns it to the user

Write the pseudocode for a function called void swap_ends(MyCoolStructure mcs) which swaps the first and last items in the data structure. What is the Big-$\Theta$ time complexity of this function?