ICS 253: Discrete Structures I

1. King Fahd University of Petroleum & Minerals Information & Computer Science Department ICS 253: Discrete Structures I Advanced Counting Techniques

2. Reading Assignment • K. H. Rosen, Discrete Mathematics and Its Applications, 6th Ed., McGraw-Hill, 2006. • Chapter 7 (Sections 7.1 and 7.2 up to the end of page 464)

3. Section 7.1: Recurrence Relations • A recurrence relation for the sequence {an} is an equation that expresses an in terms of one or more of the previous terms of the sequence, namely, a0, a1, . . . , an – 1, for all integers n with n  n0, where n0 is a nonnegative integer. • A sequence is called a solution of a recurrence relation if its terms satisfy the recurrence relation.

4. Examples • Q1 pp456: Find the first five terms of the sequence defined by each of these recurrence relations and initial conditions. a) an = 6an – 1, a0 = 2 c) an = an – 1 + 3an – 2, a0 = 1, al = 2 e) an = an – 1 + an – 3, a0 = 1, al = 2, a2 = 0

5. Examples • Q4 pp 456: Show that the sequence {an} is a solution of the recurrence relation an = – 3an – 1 + 4an – 2 if a) an = 0. b) an = 1. c) an = (– 4)n. d) an = 2(– 4)n + 3.

6. Examples • Suppose that a person deposits $10,000 in a savings account at a bank yielding 11% per year with interest compounded annually. How much will be in the account after 30 years?

7. Examples • Find a recurrence relation and give initial conditions for the number of bit strings of length n that do not have two consecutive 0s. How many such bit strings are there of length five?

8. Examples • Q23, pp458: Find a recurrence relation for the number of bit strings of length n that contain a pair of consecutive 0s. • What are the initial conditions? • How many bit strings of length seven contain two consecutive 0s?

9. Examples • A computer system considers a string of decimal digits a valid codeword if it contains an even number of 0 digits. For instance, 1230407869 is valid, whereas 120987045608 is not valid. Let an be the number of valid n-digit codewords. Find a recurrence relation for an.

10. Examples • Find a recurrence relation for Cn, the number of ways to parenthesize the product of n + 1 numbers, x0 . xl . x2 . . . . . xn , to specify the order of multiplication. • What is C2? C3? • This sequence is called the Catalan Numbers.

11. Section 7.2:Solving Linear Recurrence Equations • We will only consider solving linear homogeneous recurrence equations with constant coefficients. • Linear nonhomogeneous recurrence relations with constant coefficients are ommitted. • It will be enough for us, especially those related to estimating the complexity of recursive algorithms.

12. Solving Linear Homogeneous Recurrence Relations with Constant Coefficients • A linear homogeneous recurrence relation of degree k with constant coefficients is a recurrence relation of the form where c1, c2, . . . , ck are real numbers, and ck 0.

13. Solving Linear Homogeneous Recurrence Relations with Constant Coefficients • Theorem: Let c1, c2, . . . , ck be real numbers. Suppose that the characteristic equation has k distinct roots r1, r2, . . . , rk. Then a sequence {an} is a solution of the recurrence relation if and only if for n=0,1, 2, … where 1, 2, . . . , k are constants.

14. Example 1 • Q3, pp 471: Solve these recurrence relations together with the initial conditions given. (d) an = 4an – 1 – 4an – 2 for n  2, a0 = 6, a1 = 8.

15. Example 2 • Q3, pp 471: Solve these recurrence relations together with the initial conditions given. (f) an = 4an – 2 for n  2, a0 = 0, a1 = 4.

16. Solving Linear Homogeneous Recurrence Relations with Constant Coefficients

17. Example