Recurrence Relations

1
Recurrence Relations

• Reading Material
• Chapter 2 as a whole, but in particular Section
  2.8
• Chapter 4 from Cormens Book.

2
Recurrence Relations

• 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 Theorem
• Recurrence Expansion
• The Substitution Method
• The Change of Variable Method

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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 k1 or k2

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Solution of Linear Homogeneous Equations

• When k1, and hence
  the solution is
• When k2,
• 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(n01)

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Examples

6
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.

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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

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Master Theorem

• Theorem Let a?1 and bgt1 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 ??gt0 ? g(n)O(nlogba - ?), then f(n)
  ?(nlogba).
• If g(n) ?(nlogba), then f(n) ?(nlogba log n).
• If ??gt0 ? g(n)?(nlogba ?), and if ?clt1, n0?? ?
  ag(n/b) ? cg(n) ?n gt n0, then f(n) ?(g(n)).

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Examples

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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?

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The Expansion Method

• When expanding few terms of some recurrences, a
  solution may become more apparent.
• Example Consider the previous recurrence

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The Substitution Method

• If we can have a good guess of what the answer
  may be, then we can use mathematical induction to
  prove that it is correct.
• Example Consider the following recurrence
• This recurrence seems similar to
  (when n is a power of 2, they are
  exactly equal.) By applying case 2 of the master
  theorem, the solution to the second recurrence is
  ?(nlog n). Therefore, we can guess that we can
  find c,d gt 0 ? dnlog n ? f(n)2f(?n/2?) n ?
  cnlog n for all values of n.

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Example (Cont.)

• We show by induction that f(n) ? cnlog n . The
  other inequality is left as an exercise.
• Induction Step
• Initial Step

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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.

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Example