Recursive Definitions - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

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1

Recursive Definitions and

Structural Induction

2

Recursion

• Recursion means defining something, such as a

function, in terms of itself

• For example, let f ( x ) = x!
• We can define f ( x ) as f ( x ) = x * f( x -1)

4

• – Find f (1), f (2), f (3), and f (4), where f (0) = Recursion example
• a) Let f ( n +1) = f ( n ) +
  • • f (1) = f (0) + 2 = 1 + 2 =
  • • f (2) = f (1) + 2 = 3 + 2 =
  • • f (3) = f (2) + 2 = 5 + 2 =
  • • f (4) = f (3) + 2 = 7 + 2 =
  • • f (1) = 3 * f (0) = 3*1 = b) Let f ( n +1) = 3 f ( n )
  • • f (2) = 3 * f (1) = 3*3 =
  • • f (3) = 3 * f (2) = 3*9 =
  • • f (4) = 3 * f (3) = 3*27 =
• – Find f (1), f (2), f (3), and f (4), where f (0) = Recursion example
  • • f (1) = 2 f (0) = 2^1 = c) Let f ( n +1) = 2 f ( n )
  • • f (2) = 2 f (1) = 2^2 =
  • • f (3) = 2 f (2) = 2^4 =
  • • f (4) = 2 f (3) = 2^16 =
• d) Let f ( n +1) = f ( n ) 2 + f ( n ) +
  • • f (1) = f (0) 2 + f (0) + 1 = 1^2 + 1 + 1 =
  • • f (2) = f (1) 2 + f (0) + 1 = 3^2 + 3 + 1 =
  • • f (3) = f (2) 2 + f (0) + 1 = 13^2 + 13 + 1 =
  • • f (4) = f (3) 2 + f (0) + 1 = 183^2 + 183 + 1 =

5

Fractals

• A fractal is a pattern that uses recursion

• The pattern itself repeats indefinitely

7

Fibonacci sequence in Java

long Fibonacci (int n) {

if ( (n == 1) || (n == 2) ) return 1; else return Fibonacci (n-1) + Fibonacci (n-2);

}

long Fibonacci2 (int n) {

return (long) ((Math.pow((1.0+Math.sqrt(5.0)),n)- Math.pow((1.0-Math.sqrt(5.0)),n)) / (Math.sqrt(5) * Math.pow(2,n)));

}

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Recursion definition

• From “The Hacker’s Dictionary”:

• recursion n. See recursion. See also tail

recursion.

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Defining sets via recursion

• Same two parts:

• Base case (or basis step)
• Recursive step

• Example: the set of positive integers

• Basis step: 1 ∈ S
• Recursive step: if x ∈ S , then x +1 ∈ S

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Defining sets via recursion

• Give recursive definitions for:

a) The set of odd positive integers

 1 ∈ S

 If x ∈ S , then x +2 ∈ S

b) The set of positive integer powers of 3

 3 ∈ S

 If x ∈ S , then 3* x ∈ S

c) The set of polynomials with integer coefficients

 0 ∈ S

 If p ( x ) ∈ S , then p ( x ) + cxn^ ∈ S

• c ∈ Z , n ∈ Z and n ≥ 0

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Defining strings via recursion

• Let Σ = { 0, 1 }

• Thus, Σ* is the set of all binary numbers

• Or all binary strings
• Or all possible computer files

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String length via recursion

• How to define string length recursively?

• Basis step: l (λ) = 0
• Recursive step: l ( wx ) = l ( w ) + 1 if w ∈ Σ* and x ∈ Σ

• Example: l (“aaa”)

• l (“aaa”) = l (“aa”) + 1
• l (“aa”) = l (“a”) + 1
• l (“a”) = l (“”) + 1
• l (“”) = 0
• Result: 3

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Recursion pros

• Easy to program

• Easy to understand

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Recursion cons

• Consider the recursive Fibonacci generator
• How many recursive calls does it make?
  • F (1): 1
  • F (2): 1
  • F (3): 3
  • F (4): 5
  • F (5): 9
  • F (10): 109
  • F (20): 13,
  • F (30): 1,664,
  • F (40): 204,668,
  • F (50): 25,172,538,
  • F (100): 708,449,696,358,523,830,149 ≈ 7 * 10 20
    • At 1 billion recursive calls per second (generous), this would take over 22,000 years
    • But that would also take well over 10 12 Gb of memory!

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Rooted trees

• Recursive definition:
  • Basis step: A single vertex r is a rooted tree
  • Recursive step:
    • Let T 1 , T 2 , …, T (^) n be rooted trees
    • Form a new tree with a new root r that contains an edge to the root of each of the trees T 1 , T 2 , …, T (^) n

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(Extended) Binary trees

• Recursive definition
  • Basis step: The empty set is an extended binary tree
  • Recursive step:
    • Let T 1 , and T 2 be extended binary trees
    • Form a new tree with a new root r
    • Form a new tree such that T 1 is the left subtree, and T 2 is the right subtree