CMPS 101
Winter 2009 Homework Assignment 4
1. (3 Points)
Consider the function defined by the recurrence formula
)(nT
⎣⎦
⎩
⎨
⎧
≥+
<≤
=3)3/(2
316
)( nnnT
n
nT
a. (1 Point) Use the iteration method to write a summation formula for .
)(nT
b. (1 Point) Use the summation in (a) to show that )()( nOnT
=
c. (1 Point) Use the Master Theorem to show that )()( nnT
Θ
=
2. (6 Points)
Use the Master theorem to find asymptotic solutions to the following recurrences.
a. (1 Point)
nnTnT += )4/(7)(
b. (1 Point)
2
)3/(9)( nnTnT +=
c. (1 Point)
2
)5/(6)( nnTnT +=
d. (1 Point)
)log()5/(6)( nnnTnT +=
e. (1 Point)
2
)2/(7)( nnTnT +=
f. (1 Point) (Note: your answer will depend on the parameter a.)
2
)4/()( nnaSnS +=
3. (1 Point) p.75: 4.3-2
The recurrence describes the running time of an algorithm A. A competing
algorithm B has a running time of . What is the largest integer value for a such
that B is asymptotically faster than A ?
2
)2/(7)( nnTnT +=
2
)4/()( nnaSnS +=
4. (1 Points)
Let G be an acyclic graph with n vertices, m edges, and k connected components. Show that
. (Hint: use the fact that
knm −= 1|)(||)(|
−
=
TVTE for any tree T, from the induction handout.)
5. (1 Point) (Appendix B.4 problem 3)
Show that any connected graph G satisfies 1)()( −≥ GVGE . (Hint: use induction on the number
of vertices.)
6. (1 Point) p. 538: 22.2-2
Show the d and
π
values that result from running breadth-first search on the undirected graph of
Figure 22.3, using vertex u as the source.
r s t u
v w x y
1