1. Math/CSE 1019: Discrete Mathematics for Computer Science Fall 2011
Suprakash Datta
Office: CSEB 3043
Phone: ext 77875
Course page:
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2. Next:
Recursion
Important in mathematics, programming, thinking about problems
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3. Recursive definitions
In many cases – a natural way of thinking
Ex. 1: f(0) = b, and
 i>0 f(i)= f(i-1)+a
Ex. 2: f(0) = b, and
 i>0 f(i)= f(i-1)*a
Ex. 3: f(0) = 1, f(1) = 1 and
 i>1 f(i)= f(i-1)+f(i-2)
Arithmetic progression
Geometric progression
Fibonacci
series
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4. Full binary trees Each node has 0 successors
2 successors, each of which is a full binary tree
Special case: complete binary tree
The two successors of a node are complete binary trees with the same number of nodes
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5. Height of full binary trees
Recursive definition of height h(T)
If no of nodes n(T) =1, h(T) =0
Else the height is 1 + max(h(T1),h(T2))
where T1, T2 are the two subtrees of T
Theorem: If T is a full binary tree, then
n(T)  2h(T)+1 -1
Proof: By structural induction
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6. Structural induction To prove n(T)  2h(T)+1 -1
Base case: If n(T) = 1, h(T) = 0, so the proposition holds.
Inductive step:
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7. Simple recursive programs
Factorial
Fibonacci
Binary search
Merge sort
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8. Correctness of recursive programs
Merge sort
Inductively assume the two recursive calls correct
Prove the correctness of the merge step
By strong induction the algorithm is correct
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9. Drawbacks of recursion
Repeated computation of the same terms
Wasted storage
Overhead incurred by OS due to stack maintenance
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10. Analyzing recursive programs
Mergesort f(n) = 2f(n/2) + n
What constitutes a “solution”?
How do you solve this recurrence?
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11. Next
More uses of recursion
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12. Recursively defined sets
Basis step: 2  S
Recursive step: If x  S and y  S
then x+ y  S
Defines the set {2k|k=1,2,3,…}
What happens if we change the recursive step to: If x  S and y  S
then x+y  S, x-y  S ?
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13. Recursively defined sets - 2
All possible strings (*) over alphabet 
Basis step:   *
Recursive step: If x   and w  *
then wx  *
Length of a string
l() = 0
l(wx) = l(w) + 1 if x   and w  *
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14. Recursively defined sets - 3
What is this set A defined over  = {0,1}?
Basis step:   A
Recursive step: If x  A then 0x1  A
What is this set B defined over  = {0,1}?
Basis step:   B
Recursive step: If x B
then 0x1  B and 1x0  B
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15. Recursively defined sets - 4
Data structures – full binary trees
Basis step: single vertex
Recursive step: If S, T are full binary trees
then the following is a full binary tree r T S
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