Computability

1
Computability

• Sort homework. Formal definitions of time
  complexity. Big 0.
• Homework Exercises. Searching. Shuffling.

2
Reprise

• Bubble sort
• phases. Compare to neighbor and swap, if
  necessary. Only do next phase if there was a
  change.
• Merge
• divide in half. Sort each half and then merge
  (merge step easy/quick)

3
Reports

• Insertion
• Heap
• Quicksort

4
My summaries

• Insertion
• Going i 2 to n, Put the ith element in its
  proper place. Simplest to write.
• Heap
• put data into a heap (max heap, binary tree)
  parent greater than each of child nodes. Remove
  root (the biggest). Re-build tree and repeat
• Quicksort
• choose pivot. Place items in sections less than
  pivot and greater than or equal to pivot
  (variation less than, equal, and greater than
  pivot). Repeat for each section. Depends on good
  choice of pivots
• Random. Median. Median of median. Use knowledge
  of data.

5
Cards

• Do sorts using cards.

6
Median

• There are ways to get median or estimate of
  median or median of medians that do not involve
  sorting the whole list.
• Extra credit opportunity!

7
O(f(n))

• defines an upper bound. AKA called asymptotic
  notation.
• Given two functions, f and gN ?R
• then we say f(n) O(g(n)) or F(n) is O(gn)) g(n)
  is an asymptotic bound for f(n)
• IF there exists positive integers c and n0 such
  that
• f(n) lt cg(n) for ngt n0
• Informally, for high enough values n, and a
  coefficient, g(n) is upper bound to f(n).

8
Examples

• if f(n) 4n3100n21000, then f(n) O(n3)
• Try n0 1000, c5.
• Don't need to pick the lowest values.
• Note it also is true that
• f(n) O(n4). f(n) is . Note the equal sign is
  used but it is problematic.
• But f(n) is not O(n2). f(n) will be more than
  cn2, no matter what choice of c, at some point.

9
Exercises

• Find c and n0 and determine g(n) for f(n)
• n4 1000000
• 10n3100n100
• Generalize about polynomials

10
logarithms

• Recall logbn is the value e such that ben
• Now
• logbn logan / logab
• Proof Show logbn logab logan. That is, a
  raised to expression on right is equal to n
• Call X logbn, Y logab
• aYX (aY)X by rules of exponentsaY is equal
  to b. bX is equal to n.

11
logarithms for O() notation

• Can use log (or ln) without mentioning base
  because the difference is just a coefficient and
  Big Oh notation allows/uses a coefficient.

12
Note

• f(n) log(n) is O(n)
• nlog(n) is O(n2)
• These are each strict bounds.

13
small o

• Given functions f and g, then f(n) o(g(n))
• if that for any cgt0, a number n0 exists such that
  f(n) ltcg(n) for ngtn0.
• f(n) is asymptotically less.
• Another way to say this is
• lim (f(n)/g(n)) 0 as n ? infinity

14
Examples

• n log(n) o(n2)
• n2 o(n3)
• sqrt(n) o(n)

15
Time complexity

• let tN ? R (t a function from naturals to
  positive reals)
• Time complexity class TIME(t(n)) is all languages
  that are decidable by an O(t(n)) time Turing
  machine.

16
Homework

• Prove or disprove 100n O(n) n3
  O(n) 100n o(n) en o(3n)
• Develop a way to shuffle cards. (Can do research)
• Think of how to search a sorted for a particular
  value.
• Extra credit research median algorithms to
  present to class.