1. ICS 353: Design and Analysis of Algorithms
King Fahd University of Petroleum & Minerals
Information & Computer Science Department
ICS 353: Design and Analysis of Algorithms
Solving Recurrence Relations

2. Reading Assignment Chapter 2 as a whole, but in particular Section 2.8
Chapter 4 from Cormen’s Book.
Section 5: Master Method.

3. Outline
Objective: To discuss techniques for solving recurrence relations.
These techniques will be very important and “handy” tools for analyzing algorithms that are recursive.
Linear Homogenous Recurrences
Characteristic Equations
Inhomogeneous Recurrences
The Master Method
Recurrence Expansion
The Change of Variable Method

4. Linear Homogeneous Equations
Definition: A recurrence relation is called linear homogeneous with constant coefficients if it is of the form:
We restrict our discussion to homogeneous recurrence equations with k=1 or k=2

5. Solution of Linear Homogeneous Equations
When k=1, and hence the solution is
When k=2,
the following steps are followed to solve the recurrence:
Find r1 and r2, the solutions to the characteristic equation
If r1 = r2 = r, then
Otherwise
Determine c1 and c2 from the initial values f(n0) and f(n0+1)

6. Examples

7. Inhomogeneous Equations
Definition: A recurrence relation is called inhomogeneous if it is not a homogeneous recurrence relation. In particular, we will look at
where i 1  i  k  gi(n) is not a constant or g0(n)  0.

8. Solution for Inhomogeneous Recurrences
No general method for solving inhomogeneous recurrences exists. However,
There are cases where a formula for a class of inhomogeneous recurrence relations exists (Master Theorem)
The rest depend on experience and/or trial and error in choosing one of the following techniques
Expansion
Substitution
Change of variable

9. Master Theorem
Theorem: Let a1 and b>1 be constants, let g(n) be a function, and let f(n) be defined on the nonnegative integers by the recurrence
where we interpret n/b to mean either n/b or n/b. Then f(n) can be bounded asymptotically as follows
If >0  g(n)=O(nlogba - ), then f(n) = (nlogba).
If g(n)= (nlogba), then f(n) = (nlogba log n).
If >0  g(n)=(nlogba + ), and if c<1, n0  ag(n/b)  cg(n) n > n0, then f(n) = (g(n)).

10. Examples

11. Gaps in the Master Theorem
The three cases in the theorem do not cover all the possibilities for g(n). There are gaps between cases 1 and 2, and 2 and 3.
For example, can we apply the Master theorem on the following recurrence?

12. The Expansion Method
When expanding few terms of some recurrences, a solution may become more apparent.
Example: Consider the previous recurrence

13. The Change of Variables Method
The idea is to change the domain of the function and define a new recurrence in the new domain whose solution may be easier to obtain or is already known. Once the solution is obtained, we convert the domain of the function back to its original domain.

14. Example