Recurrence Relations
Chapter 7
p
(7.1 Introduction)
ctober 30, 2009
October 30, 2009
So far …
•
ets
Sets
• Logic, Proof
• Functions, Sequences, Relations
•
lgorithms
Algorithms
• (Number theory)
• Counting Methods
Pigeonhole Principle
… Pigeonhole Principle
jinah@cs.kaist.ac.kr
CS204 Discrete Math (Fall 2009)
2
sequence
{ C
n
}
1,
1,
2,
5,
14, 42, 132, 429,
1430, 4862, 19796
…
C
n
…
•
A special type of function in which the domain
consists of a set of consecutive integers.
– D = {0,1,2,3,4,5,6,7,8,9,10}
–C
0
= 1, C
1
=1, C
2
= 2, … , C
10
= 19796,
… , C
n
= ?
n0123 4 5 6 7
8
9
1
0
4
2
32
29
430
862
6796
jinah@cs.kaist.ac.kr
CS204 Discrete Math (Fall 2009)
3
C
n
1
1
2
5
14
42
132
429
1430 4862 16796
Recurrence Relations
• Advanced Counting Technique
ecursion
• Recursion
• Recursive algorithms
i
d
t
h
l
t
i
f
b
l
f
i
i
t
– Provide the solution of a problem of size n in terms
of the solutions of one or more instances of the
ame problem of smaller size
same problem of smaller size
– To analyze its complexity
•
btain a recurrence relation hat expresses the number of
Obta
a ecu e ce e at o
t at e p esses t e u be o
operations required to solve a problem of size n in terms of the
number of operations required to solve the problem for one or
more instances of smaller size.