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Practice Problem

Practice Problem. Themes Recursion Recurrence Definitions Recursive Relations Induction (prove properties of recursive programs and objects defined recursively ). Induction Examples. Prove: 1 + 3 + 5 + … + (2n-1) = n 2 4n < 2 n ,  n ≥ 5 2 n < n! ,  n ≥ 4. Tiling Dominoes.

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Practice Problem

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  1. Practice Problem • Themes • Recursion • Recurrence Definitions • Recursive Relations • Induction (prove properties of recursive programs and objects defined recursively)

  2. Induction Examples Prove: • 1 + 3 + 5 + … + (2n-1) = n2 • 4n < 2n ,  n ≥ 5 • 2n< n! ,  n ≥4

  3. Tiling Dominoes • Given a set of 1 × 2, how many different ways are there to tile and 2 × n rectangle placing a single domino vertically (2 × 1) or stacking two dominoes horizontally? • Example. For n = 3 there are 3 ways:

  4. Think Recursively • What are the base cases? • How do you solve the 2 × n using solutions to smaller problems?

  5. ACL Program to Count Dominoes • Derive a recurrence relation for D(n) the number of tilings of a 2 × n rectangle. • Write an ACL program, based on your recurrence, to compute D(n). • Produce a table of values of D(n) and compare to 2n and (3/2)n • Formulate a conjecture and use induction to prove your result.

  6. ACL Program • Derive a recurrence relation for D(n) the number of tilings of a 2 × n rectangle. • Write an ACL program, based on your recurrence, to compute D(n). • Produce a table of values of D(n) and compare to 2n and (3/2)n • Formulate a conjecture and use induction to prove your result.

  7. ACL Program to Generate Dominoes • Write a recursive ACL program (use programming mode) to generate all possible tilings of a 2 × n rectangle. • (defunGenDominoes (n) … ) • Represent tilings with “V” for one vertical tile and “HH” for two stacked horizontal tiles. • (concatenate ‘string “HH” “V”)  “HHV” • (length “HHV”)  3 • Return a list of strings, one for each possible tiling • Use auxiliary function mapconcat on next slide

  8. Auxiliary Function • Write a recursive function which concatenates a string s in front of all of the strings in a list L • (mapconcat s L) • (mapconcat “V” ‘(“HH” “VV”))  (“VHH” “VVV”)

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