CS173: Discrete Mathematical Structures
Fall 2005
Homework #11
Due 12/04/05, 8a
1. The Master Theorem (Rosen, p 430) states: Let f be an increasing function that
satisfies the recurrence relation
!
f(n)=a"f(n/b)+cnd
whenever
!
n=bk
, where k is a positive integer,
!
a"1
, b is an integer greater than
1, and c and d are real numbers with c positive and d nonnegative. Then
!
f(n)=
O(nd) if a<bd
O(ndlog n) if a=bd
O(nlogba) if a>bd
"
#
$
%
$
.
We will prove the Master Theorem by solving the following parts (some of the
answers can be found in your book. See exercises 29-33 on p 435).
a. Show that if
!
a=bd
and n is a power of b, then
!
f(n)=f(1) "nd+cndlogbn.
b. Use part a. to show that if
!
a=bd
, then f(n) is O(nd log n).
c. Show that if
!
a"bd
and n is a power of b, then
!
f(n)=C1nd+C2nlogba
,
where
!
C1=bdc/(bd"a)
and
!
C2=f(1) +bdc/(a"bd)
.
d. Use part c. to show that if a < bd, then f(n)=O(nd).
e. Use part c. to show that if a > bd, then
!
f(n)=O(nlogba)
.
2. Determine whether the relation R on the set of all real numbers is reflexive,
symmetric, antisymmetric, and/or transitive, where (x,y) ∈ R if and only if
a. x + y = 0.
b. x = ± y.
c. x – y is a rational number.
d. x = 2y.
e. xy ≥ 0.
f. xy = 0.
g. x = 1.
h. x = 1 or y = 1.
3. Let R1 and R2 be the “divides” and “is a multiple of” relations on the set of all
positive integers, respectively. That is, R1 = {(a,b) | a divides b} and R2 = {(a,b) |
a is a multiple of b}. Find:
a. R1 ∪ R2,
