March 14

The agenda for today :

• Work through & discuss the homework assignment.
• Describe what we're doing over the next few weeks: sorting (bubble, quick, heap, radix)
• Describe next two assignments : practice, and a midterm project (sorting numerical experiment)
• Introduce sorting algorithms

Coming attractions (and things to read about)

• loop invariants (or "how do we know that it works?")
• reversing a linked list (homework)
• doubly linked lists
• min/max heap data structure
• more practice with recursion
• Some sorting algorithms:
  • insertion sort O(n n)
  • quicksort O(n log(n))
  • radix sort O(n) (but only for non-comparison bin-possible values)
• properties of sorting algorithms: stability, distributed, indexed_array vs linked_list, memory, ...

Questions about anything before we start?

homework

First let's do a live coding walkthrough of the homework. I've posted a few related programs over in the answers folder.

sorting

OK, time to dig into sorting algorithms. Please start reading about these : the various approaches, their properties, time and memory efficiency, and other properties.

• wikipedia: sorting algorithm
• visualization of sorting algorithms
• 15 sorting algorithms in 6 minutes

Here are a few classics worth understanding.

bubble sort

• how does it work?
• O() ?
• homework : code this ; discuss
• musical animation
• animation
• folk dancers

quick sort

• how does it work?
• O() ?
• homework : code this ; discuss
• folk dancers

heap sort

• ... will do this next week in more detail (so don't do it for a midterm project)
• basic ideas : binary tree with smallest on top ... but other parts only partially sorted.
• after removing top, "promote" up the tree
• can store as a regular array !
• visualization

         0
     1            2
  3    4      5      6   
 7,8  9,10  11,12  13,14

left child at (2*i+1) right child at (2*i+2)

radix sort : O(n) (but ...)

• not comparisons - instead just "put it in the right box"
• very fast, but only for data with relatively few unique values

group exercises in class

• exercise 1 : in english, describe one of these sorting algorithm

Come back together & compare notes.

• exercise 2 : in python, working on a list of numbers like [2, 1, 10, 12], implement one of those.

induction, invariants, and loops

We probably won't get to this today, but here's where we're going next :

I would like to discuss a bit more about how to understand when an algorithm is working correctly ... and how that is related to "proof by induction".

*  What is "proof by induction" ?

I think this is clearest with an example.

* Prove that sum(range(1, 1+n)) == n*(n+1)/2 by induction.
  (Let's see if we can understand the math proof of this.)

The on to the code version of the calculation :

* write some code to calculate that sum with 
  an "accumulator loop"

And finally, let's add an explicit "loop invariant" assertion which is (a) True at the start of each loop, and (b) at the end of the loop, is the claim that our result is correct.

I'll use the attached files to discuss this.

attachments