CSE115/ENGR160 Discrete Mathematics 04/19/12 Ming-Hsuan Yang UC Merced 1

8.1 Recurrence relations Many counting problems can be solved with recurrence relations Example: The number of bacteria doubles every 2 hours. If a colony begins with 5 bacteria, how many will be present in n hours? Let a n =2a n-1 where n is a positive integer with a 0 =5 2

Recurrence relations A recurrence relation for the sequence {a n } is an equation that expresses an in terms of 1 or more of the previous terms of the sequence, i.e., a 0, a 1, …, a n-1, for all integers n with n≥n 0 where n 0 is a nonnegative integer A sequence is called a solution of a recurrence relation if its terms satisfy the recurrence relation 3

Recursion and recurrence A recursive algorithm provides the solution of a problem of size n in terms of the solutions of one or more instances of the same problem of smaller size When we analyze the complexity of a recursive algorithm, we obtain a recurrence relation that expresses the number of 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 instance of smaller size 4

Example Let {a n } be a sequence that satisfies the recurrence relation a n =a n-1 – a n-2 for n=2, 3, 4, … and suppose that a 0 =3 and a 1 =5, what are a 2 and a 3 ? Using the recurrence relation, a 2 =a 1 -a 0 =5-3=2 and a 3 =a 2 -a 1 =2-5=-3 5

Example Determine whether the sequence {a n }, where a n =3n for every nonnegative integer n, is a solution of the recurrence relation a n =2a n-1 – a n-2 for n=2, 3, 4, … Suppose a n =3n for every nonnegative integer n. Then for n ≥2, we have 2a n-1 -a n-2 =2(3(n- 1))-3(n-2)=3n=a n. Thus, {a n } where a n =3n is a solution for the recurrence relation 6

Modeling with recurrence relations Compound interest: 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 it be in the account after 30 years? Let P n denote the amount in the account after n years. The amount after n years equals the amount in the amount after n-1 years plus interest for the n-th year, we see the sequence {P n } has the recurrence relation P n =P n P n-1 =(1.11)P n-1 7

Modeling with recurrence relations The initial condition P 0 =10,000, thus P 1 =(1.11)P 0 P 2 =(1.11)P 1 =(1.11) 2 P 0 P 3 =(1.11)P 2 =(1.11) 3 P 0 … P n =(1.11)P n-1 =(1.11) n P 0 We can use mathematical induction to establish its validity 8

Modeling with recurrence relations We can use mathematical induction to establish its validity Assume P n =(1.11) n 10,000. Then from the recurrence relation and the induction hypothesis P n+1 =(1.11)P n =(1.11)(1.11) n 10,000=(1.11) n+1 10, 000 n=30, P 30 =(1.11) 30 10,000=228,

8.2 Solving linear recurrence relations 10

From mathematical induction 11

Linear homogenous recurrence relations with constant coefficients 12 characteristic equation

Theorem 1 13

Example 14

Fibonacci numbers 15

16

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Recurrence relations Play an important role in many aspects of algorithms and complexity Can be used to – analyze the complexity of divide-and-conquer algorithms (e.g., merge sort) – Solve dynamic programming problems (e.g., scheduling tasks, shortest-path, hidden Markov model) – Fractal 18

8.5 Inclusion-exclusion The principle of inclusion-exclusion: For two sets A and B, the number of elements in the union is defined by |A ⋃B|=|A|+|B|-|A⋂B| Example: How many positive integers not exceeding 1000 are divisible by 7 or 11? 19

Principle of inclusion-exclusion Consider union of n sets, where n is a positive integer Let n=3 20

Principle of inclusion-exclusion Let A 1, A 2, …, A n be finite sets. Then Proof: Prove it by showing that an element in the union is counted exactly once by the right-hand side of the equation Suppose that a is a member of exactly r of the sets A 1, A 2, …, A n where 1 ≤r ≤n This element is counted C(r,1) times by ∑|A i | 21

Principle of inclusion-exclusion It is counted C(r,2) times by ∑|A i ⋂ A j | In general, it is counted C(r,m) times by the summation involving m of the sets A i. Thus, this element is counted exactly C(r,1)-C(r,2)+C(r,3)-…+(-1) r+1 C(r,r) Recall, C(r,0)-C(r,1)+C(r,2)-C(r,3)-…+(-1) r C(r,r)=0 Thus, C(r,1)-C(r,2)+C(r,3)-…+(-1) r+1 C(r,r)=C(r,0)=1 Thus, this element a is counted exactly once by the right hand side 22

Principle of inclusion-exclusion Gives a formula for the number of elements in the union of n sets for every positive integer n There are terms in this formula for the number of elements in the intersection of every nonempty subset of the collection of the n sets. Hence there are 2 n -1 terms in the formula Example: 15 terms 23

Example For the union of 4 sets, there are 15 different terms, one for each nonempty subset of {A 1, A 2, A 3, A 4 } 24