CSE 20 DISCRETE MATH Prof. Shachar Lovett Clicker frequency: CA

Todays topics Order relations Section 6.3 in Jenkyns, Stephenson

Order relations

Partial order

Definition: reflexive, transitive, anti-symmetric Equality relation: xRy if x=y Is it a partial order? A. Yes B. No

Partial order

Total order

Lexicographic order

Orders on sequences Which one of the following is the largest one? A. (1,2,5,10) B. (10,5,2,1) C. (5,100,3,3) D. (10,7,100,1000)

Maximal / maximum elements

There could be multiple maximal elements in a set, but the maximal element (if it exists) must be unique Lets prove it

Maximal / maximum elements

Minimal / minimum elements

Maximum/maximal/minimum/minimal

Back to total orders

Sorting Sorting is a basic primitive used by many algorithms Most sorting algorithms are “abstract”, in the sense that they can apply to any partial order Naïve sorting on n elements requires comparing all pairs of elements, which takes ~n 2 comparisons Fast sorting algorithms (for example, quicksort) can do it using only n log n comparisons This can be proven to be tight!

Next class Knights and knaves