CS 3204 Operating Systems
©William D McQuain, January 2005 1
Computer Science Dept Va Tech March 2007 ©2007 McQuain
Complexity
Data Structures & OO Development II
1
Recursion and Complexity
Recursive algorithms naturally lead to complexity functions that are themselves recursive:
unsigned int Factorial(unsigned int N) {
if ( N <= 1) return 1;
return N * Factorial(N – 1);
}
The complexity function would naturally be expressed as: 2, if 1
() 4(1), if 1
N
TN TN N
≤
=
+
−>
Such a function is called a recurrence relation.
While the equation certainly expresses the complexity function, a non-recursive (or
closed-form) version would be more useful.
Computer Science Dept Va Tech March 2007 ©2007 McQuain
Complexity
Data Structures & OO Development II
2
Solving Recurrence Relations
General techniques exist for solving recurrence relations, and closely resemble techniques
for solving differential equations.
For simple ones, like the one on the preceding slide, examining a few cases will usually
indicate a pattern leading to a reasonable guess as to what the closed form might be:
(1) 2
(2) 4 (1) 4 2
(3)4 (2)442
(4)4 (3)4442
T
TT
TT
TT
=
=+ =+
=+ =++
=+ =+++ () 4 2FN N
=
−
Of course, the guess could be incorrect, but it's easy enough to verify that it satisfies the
original recurrence relation. It's obvious that F(1) = 2. And, a little algebra shows that:
() 4 2 44 6 44( 1)2 4 ( 1)FN N N N FN= −=+ −=+ −−=+ −
So, F(N) satisfies the recurrence relation.
