Complexity, etc.

1
Complexity, etc.

• Homework. Comparison to computability.
• Big Oh notation. Sorting.
• Classwork/Homework prepare presentation on
  specific sorts.
• Presentation Day for some.

2
Homework

• CFG and PDA
• Regular expressions

3
Hierarchy

• There is a "proper" hierarchy concerning
  languages FSA and PDA and TM.
• meaning
• If a language is recognized by an FSA, then we
  can define a PDA. If a language is recognized by
  a PDA, we can define a TM.
• AND
• There are languages for which there is not FSA,
  but there is a PDA and similarly for PDA and TM.

4
Trick question

• Define a FSA for the language aibj
• some a's and then some b's.
• Define a FSA for the language aibi
• some a's and then the same number of b's.

5
Pumping lemma for FSA

• If A is a regular language, then there is a
  number p where if s is in A, and the length of s
  is at least p, then we can write s ass xyz,
• length of y is greater than 0
• length of xy is less than p and
• xyiz is also in A.
• General idea finite number of states. Set p to
  be number of states. So if there a string in A of
  length at least p, then the string s must
  re-visit at least 1 state. Call one such state q,
  then the substring from q and then back to q can
  be repeated.

6
Example of language that is not regular

• Alphabet a,b. All strings of the formaibi
• some number of a's followed by the same number of
  b's.
• NOTE the language of strings aibj is regular.

7
Define a CFG for aibj

• Try it!

8
Now, what about aibi

• Write a grammar

9
Answer

• S gt aSb
• S gt ab
• Notice this grammar is not regular.
• It better not be, because an application of the
  pumping lemma indicated that this language could
  not be produced by a FSA.

10
Many similar results

• Pumping lemma for CFL
• Extra credit assignment look up and present to
  class.

11
Complexity

• calculate limit (maximum, minimum, worst case,
  average) amount of resources required to perform
  specific task
• generally specific algorithm
• time
• count of a specific type of operation
• space
• Actual usage in any specific case may be
  different.

12
Comparison

• Computability looks at whether or not task can be
  done using a certain type of machine or grammar
  or definition of functions
• Complexity assumes that something can be done and
  complexity assessment generally looks at specific
  method
• Note there are proofs that say (something like)
  Any method will take at least ..

13
Bubble sort

• Class example
• Sort subset of students by height.
• Designate someone as counter of compare
  operations
• Designate someone as counter of swap operations
• Designate someone as counter of rounds

14
Bubble sort

• Worst case. n elements in set
• Round 1 n-1 compares
• Round 2 n-2 compares
• Round n-1 1 compare
• So, average number of compares per round is
  (n-1)/2
• Total number of compares is (rough)(n-1)
  (n-1)/2

15
Measure

• generally given in terms of the size of the
  problem and given as a bound
• called Big Oh notation
• A function f(n) is O(g(n)), n integer, if there
  is an integer n0 and a constant c such thatf(n)
  lt cg(n) if ngtn0
• Conceptually, if n is big enough, than the
  function f is bounded by a multiple of the
  function g.

16
Common usage of O(n)

• O(c) constant
• O(n) linear
• O(n2) quadratic
• O(np) polynomial
• O(ne) exponential

17
Easy results

• f(n) n IS O(n) and also O(n2) and O(np)Let c
  1, and n0 1, then n lt n2 for all values
  of ngt1
• but
• f(n2) is NOT O(n)

18
Sorts

• There are sorting algorithms that are better than
  bubble sort, that is, time is less than O(n2)
• heap sort
• quick sort
• merge sort
• Assign be able to demonstrate with a deck of
  cards. Sketch proof on complexity.