Day 23, Mat385

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Announcements
- The Zooms continue
  A menu of our past zooms...
- Homework 7.2 (aka "6.3"): things are a little crazy here.... I think that I've got the system screwed up. But the idea was to have the section number correspond to the assignment!:) I'll try to right the ship, but, for the moment, I'll grade your 7.2 work!
  Grading 3, 24, 34 (may not be done...:(
  - #24: Ben says: "If you start at 8 you have to go to 1 or at least twice." Can we justify it? I like it, but.... I took the arduous path of "trying all cases", essentially.
    I think that Ryan nailed it, however.
  - #34: a couple of special cases....
Last time:
- We started Section 7.3: Shortest Paths and Minimal Spanning Trees.
Today:
- We'll finish Section 7.3: Shortest Paths and Minimal Spanning Trees, using a handout to practice with, and then
- get started (maybe even finish?) Section 7.4: Traversal Algorithms.
Links:
- Dijkstra's Algorithm (defined recursively): once a node is settled, its job is done. It can be "swallowed up" into a smaller graph, where its use as a shortcut has generally reduced the distance between other pairs of nodes.
  This was fun to program, but probably more trouble than it was worth. A direct approach, as in Bellman-Ford below, would have been simpler to code. But, if I can do something recursively, I probably will...:)
- Bellman-Ford Algorithm: Bellman-Ford finds the shortest distances between the starting node and every other node, and the paths. It starts with paths of length one (the adjacency matrix row for the start node), and then adds in successively longer paths (and can't add more than n-1, or it would be repeating nodes -- which would make no sense).
- Floyd's algorithm: finds the shortest distances between any two nodes, but doesn't contain any path information.
- Graph traversals:
- EulerPath:
- Distilling Free-Form Natural Laws from Experimental Data. A computer rediscovers Newton's laws -- in a day!
  - Supplementary Material On-line
- Graph Terminology Handout
- Tree Terminology Handout
- If you check out this code, you'll see where I compute the power set ("all-subsets") of a set, to cover all the bases. I think that the end result is pretty beautiful, too!
- Recursive in-order, post-order, pre-order in lisp
- Recursive computation of Fibonacci numbers (in lisp)
- Recursive computation of rows from Pascal's triangle (in lisp)
- "String exponention" (in lisp)
- Here is the recursive selection sort algorithm, in lisp:
- Here is a recursive binary search algorithm in lisp:
- Closed-form computation of Fibonacci numbers (in lisp)
- Some examples of the Euclidean algorithm:
  1. Simple and clean recursive Euclidean Algorithm in lisp:
  2. Written as a while loop, using the recurrence relation:
- Here is a recursive bubbleSort algorithm in lisp
  
  (coded up following our author's algorithm, in section 3.3, p. 214, as closely as possible).
Details on why Dijkstra and Bellman-Ford work.

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