CSE115/ENGR160 Discrete Mathematics 04/19/12

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CSE115/ENGR160 Discrete Mathematics04/19/12

• Ming-Hsuan Yang
• UC Merced

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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 an2an-1 where n is a positive integer with
  a05

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Recurrence relations

• A recurrence relation for the sequence an is an
  equation that expresses an in terms of 1 or more
  of the previous terms of the sequence, i.e., a0,
  a1, , an-1, for all integers n with nn0 where
  n0 is a nonnegative integer
• A sequence is called a solution of a recurrence
  relation if its terms satisfy the recurrence
  relation

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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

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Example

• Let an be a sequence that satisfies the
  recurrence relation anan-1 an-2 for n2, 3, 4,
  and suppose that a03 and a15, what are a2 and
  a3?
• Using the recurrence relation, a2a1-a05-32 and
  a3a2-a12-5-3

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Example

• Determine whether the sequence an, where an3n
  for every nonnegative integer n, is a solution of
  the recurrence relation an2an-1 an-2 for n2,
  3, 4,
• Suppose an3n for every nonnegative integer n.
  Then for n2, we have 2an-1-an-22(3(n-1))-3(n-2)
  3nan.
• Thus, an where an3n is a solution for the
  recurrence relation

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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 Pn 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 Pn has
  the recurrence relation
• PnPn-10.11Pn-1(1.11)Pn-1

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Modeling with recurrence relations

• The initial condition P010,000, thus
• P1(1.11)P0
• P2(1.11)P1(1.11)2P0
• P3(1.11)P2(1.11)3P0
• Pn(1.11)Pn-1(1.11)nP0
• We can use mathematical induction to establish
  its validity

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Modeling with recurrence relations

• We can use mathematical induction to establish
  its validity
• Assume Pn(1.11)n10,000. Then from the recurrence
  relation and the induction hypothesis
• Pn1(1.11)Pn(1.11)(1.11)n10,000(1.11)n110,000
• n30, P30(1.11)3010,000228,922.97

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8.2 Solving linear recurrence relations

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From mathematical induction

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Linear homogenous recurrence relations with
constant coefficients

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characteristic equation
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Theorem 1

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Example

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Fibonacci numbers

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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

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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?BAB-A?B
• Example How many positive integers not exceeding
  1000 are divisible by 7 or 11?

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Principle of inclusion-exclusion

• Consider union of n sets, where n is a positive
  integer
• Let n3

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Principle of inclusion-exclusion

• Let A1, A2, , An 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 A1, A2, , An where 1rn
• This element is counted C(r,1) times by ?Ai

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Principle of inclusion-exclusion

• It is counted C(r,2) times by ?Ai? Aj
• In general, it is counted C(r,m) times by the
  summation involving m of the sets Ai. Thus, this
  element is counted exactly
• C(r,1)-C(r,2)C(r,3)-(-1)r1C(r,r)
• Recall , C(r,0)-C(r,1)C(r,2)-
  C(r,3)-(-1)rC(r,r)0
• Thus, C(r,1)-C(r,2)C(r,3)-(-1)r1C(r,r)C(r,0)
  1
• Thus, this element a is counted exactly once by
  the right hand side

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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 2n-1 terms in the formula
• Example 15 terms

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Example

• For the union of 4 sets, there are 15 different
  terms, one for each nonempty subset of A1, A2,
  A3, A4