CS 170 Fall 2008 - Solutions to Midterm 1
October 14, 2008
1 1. True: nlog n≤n2.
2. False: limn→∞
n2
nlog n=∞.
3. True: 2clog2n= (2log2n)c=nc.
4. False (not always True): For f(n) = clog2n=O(log n), we have
2f(n)=ncwhich is not O(n3) for c > 3.
5. True: log100 50000 <2.5 because 1002.5= 100000 >50000; hence by
Master’s theorem, T(n) = Θ(nlog100 50000) = o(n2.5).
6. False.
7. If we apply Master’s theorem, a=b= 3 and c= 1; since logba=c,
we have T(n) = nlogn.
8. True: gcd(3,8) = 1 and in fact 3−1= 3.
9. True: We cannot have 4x= 1 mod 8, since then 4x= 8k+ 1, and 1
would be a multiple of 4, a contradiction.
10. True: There are approximately N/ ln(N) prime numbers ≤N. Thus
the probability that an n-bit number is prime is approximately N/ ln(N)
N=
1
ln(N)for N= 2n. That would be ≈ln 2
n= Θ( 1
n).
11. True.
12. True: Let ube the vertex with lowest post order number that is not
a sink. Then there exists some edge (u, v ). If vertex vis visited while
exploring u, then post[v]<post[u]; hence that cannot happen. This
means vis already visited once we begin to explore u, but then the
edge (u, v) would be a backedge, and the graph would have a cycle,
contradicting the fact that the graph is a DAG.
13. False: The graph with vertex set V={1,2,3}and edge set E=
{(1,2),(2,1),(1,3)}is a counterexample. We can have pre[1] =
1,pre[2] = 2,post[2] = 3,pre[3] = 4,post[3] = 5,post[1] = 6,
and then vertex 2 has lowest post order but the strongly connected
component {1,2}is not a sink strongly connected component, since
it has the outgoing edge (1,3) to the strongly connected component
{3}.
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