Discrete Structures: Recursive Algorithms and Sorting Techniques, Study notes of Discrete Structures and Graph Theory

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CS 173: Discrete Structures

Eric Shaffer

Office Hour: Wed. 1-2, 2215 SC

[email protected]

Review

• What does it mean to say f(x) is order g(x)?
• Recursive algorithms:
  • We looked at functions we can compute recursively
  • We described the time complexity of these functions
    • Using what?

Insertion sort… Time for an animation: http://www.cs.oswego.edu/~mohammad/classes/csc241/samples/sort/Sort2-E.html

Create your own recursive algorithm!

How about computing n

2

recursively?

Can we express n

2

in terms of (n-1)

2

What recurrence relation describes the running time?

Tower of Hanoi… The Tower of Hanoi invented by French mathematician, Edouard Lucas, in 1883. TH (A, B, C,n)

1. If n = 1 then Move(A,B)
2. else
3. TH(A,C,B,n-1)
4. Move(A,B)
5. TH(C,B,A,n-1)

Towers of Hanoi efficiency (or lack thereof) Now let’s analyze the running time. T(n) = 2 T(n-1) + C Unroll a bit… = 2 (2 T(n-2) + C) + C = 4 T(n-2) + 3C = 4 (2 T(n-3) + C) + 3C = 8 T(n-3) + 7C … = 2 k T(n-k) + ( k -1)C What is k when this bottoms out at 1?

T(1) = C

n- T(1) + ( n- -1)C = ( n -1) C = O( n ) More general techniques investigated in a couple weeks

Recursive sorting…. MergeSort (list)

1. If |list| = 1 then return list
2. else
3. MergeSort(left half of list)
4. MergeSort(right half of list)
5. Merge(left half of list,right half of list)

Recursive sorting…. MergeSort (list)

1. If |list| = 1 then return list
2. else
3. MergeSort(left half of list)
4. MergeSort(right half of list)
5. Merge(left half of list,right half of list) Animations: http://www.cse.iitk.ac.in/users/dsrkg/cs210/applets/sortingII/mergeSort/mergeSort.html http://vision.bc.edu/~dmartin/teaching/sorting/anim-html/insertion.html

Merge sort Write down the recurrence relation: T(n) = 2 T(n/2) + cn We’ll talk about techniques for solving it later.

When is recursion inefficient? Recursive function for computing the nth Fibonacci number: Iterative function for computing the nth Fibonacci number: Which of these is better?