CMSC 203 / 0201 Fall 2002

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CMSC 203 / 0201Fall 2002

• Week 10 28/30 October and 1 November 2002
• Prof. Marie desJardins

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TOPICS

• Recurrence relations and solutions
• Divide-and-conquer recurrences

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MON 10/28 RECURRENCE RELATIONS (5.1)
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Concepts/Vocabulary

• Recurrence relations
• Solution / solution sequence
• Initial conditions
• Useful examples compound interest, bunny rabbits
  / Fibonacci, Tower of Hanoi, Catalan numbers

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Examples

• Exercise 5.1.4 For each of the following
  sequences, find a recurrence relation satisfied
  by this sequence
• (b) an 2n
• (d) an 5n
• (f) an n2 n
• (g) an n (-1)n

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Examples II

• Save early and often Exercise 5.1.6 A person
  deposits 1000 in an account that yields 9
  interest compounded yearly.
• (a) Set up a recurrence relation for the amount
  in the acount at the end of n years.
• (b) Find an explicit formula for the amount in
  the account at the end of n years.
• (c) How much money will the account contain after
  100 years?

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Examples III

• Exercise 5.1.11 Use mathematical induction to
  verify the formula derived in Example 5 for the
  number of moves required to complete the Tower of
  Hanoi puzzle
• Hn 2n - 1
• Catalan numbers Example 5.1.8 (p. 315) Find a
  recurrence relation for Cn, the number of ways to
  parenthesize the product of n1 numbers,
  x0?x1??xn.
• Cn ?k0n-1 Ck Cn-k-1
• C0 C1 1

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Examples IV

• Exercise 5.23 A ternary string contains only 0s,
  1s, and 2s.
• (a) Find a recurrence relation for the number of
  ternary strings that do not contain two
  consecutive 0s.
• (b) What are the initial conditions?
• (c) How many ternary strings of length six do not
  contain two consecutive 0s?
• Exercise 5.25
• (a) Find a recurrence relation for the number of
  ternary strings that do not contain two
  consecutive 0s or two consecutive 1s.
• (b) What are the initial conditions?
• (c) How many ternary strings of length six do not
  contain two consecutive 0s or two consecutive
  ones?

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WED 10/30 - FRI 11/1DIVIDE-AND-CONQUER (5.3)

• HOMEWORK 7 DUE
• FEEDBACK SESSION TODAY

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Concepts / Vocabulary

• Divide-and-conquer recurrence relations
• f(n) a f(n/b) g(n)

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Examples

• Exercise 5.3.7 Suppose that f(n) f(n/3) 1
  when n is divisible by 3, and f(1) 1. Find
• (a) f(3)
• (b) f(27)
• (c) f(729) f(36)
• Fast multiplication (Example 5.3.3, p. 333)
• ab (22n 2n) A1B1 2n(A1-A0)(B0-B1)
  (2n1)A0B0
• Exercise 5.3.3 Multiply (1110)2 and (1010)2
  using the fast multiplication algorithm.
• Exercise 5.3.4 Express the fast multiplication
  algorithm in pseudocode.

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Examples II

• Exercises 5.3.14-16
• Suppose that there are n 2k teams in an
  elimination tournament, where there are n/2 games
  in the first round, with the n/2 2k-1 winners
  playing in the second round, and so on. Develop a
  recurrence relation for the number of rounds in
  the tournament.
• How many rounds are there when there are 32
  teams?
• Solve the recurrence relation for the number of
  rounds in the tournament.

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Examples III

• Theorem 1 for a recurrence relation f(n) a
  f(n/b) cwhenever bn, a?1, integer bgt1, and
  real cgt0, f(n) O(nlogba) if a gt 1 and f(n)
  O(log n) if a1.