1. Arrange the following functions in increasing order of growth rate with g(n) following f(n) in your list if and only if f(n) = O(g(n))
log nn, n2, nlog n, n log log n, 2log n, log2 n, n√2
2. Suppose that f(n) and g(n) are two positive non-decreasing functions such that f(n) = O(g(n)). Is it true that 2f(n) = O(2g(n) )?
3. Find an upper bound (Big O) on the worst case run time of the following code segment.
void bigOh1(int[] L, int n)
while (n > 0)
find_max(L, n); //finds the max in L[0…n-1] n = n/4;
Carefully examine to see if this is a tight upper bound (Big θ)
4. Find a lower bound (Big Ω) on the best case run time of the following code segment.
string bigOh2(int n)
if(n == 0) return "a";
string str = bigOh2(n-1);
return str + str;
Carefully examine to see if this is a tight lower bound (Big θ)
5. What Mathematicians often keep track of a statistic called their Erdős Number, after the great 20th century mathematician. Paul Erdős himself has a number of zero. Anyone who wrote a mathematical paper with him has a number of one, anyone who wrote a paper with someone who wrote a paper with him has a number of two, and so forth and so on. Supposing that we have a database of all mathematical papers ever written along with their authors:
a. Explain how to represent this data as a graph.
b. Explain how we would compute the Erdős number for a particular researcher.
c. Explain how we would determine all researchers with Erdős number at most two.
6. In class, we discussed finding the shortest path between two vertices in a graph. Suppose instead we are interested in finding the longest simple path in a directed acyclic graph. In particular, I am interested in finding a path (if there is one) that visits all vertices.
Given a DAG, give a linear-time algorithm to determine if there is a simple path that visits all vertices.
