Day 26, Mat385

Last time

Next time

Announcements
- The Zooms continue
  A menu of our past zooms...
- Homework:
  - 7.3: #1, 11, 27
  - 7.4: #5, 12, 30
  - 8.1: #1, 10, 12
- Yoseph asked me what all this Boolean stuff is good for, and there are some interesting examples in this article (e.g. J-K Flip-Flops).
- I hope that you're working on your projects!
  I know that some of you didn't get your logins for the wiki. When you're ready email me and I'll activate it. That way we'll know that it's just been sent out.
Last time:
- We finished Section 8.1: Boolean algebras, and then
- We hit Section 8.2: Logic Networks, in which we saw three different ways of thinking about the same problem: a truth table, a Boolean expression, and a logic network.
Today:
- We'll finish Section 8.2: Logic Networks.
- Then it's on to Section 8.3: Minimization.
  I've made a handout with some of the tables we'll be using, so that you can play along.
Links:
- Shannon's 1938 paper A symbolic analysis of relay and switching circuits (here's a local copy)
- George Boole's biography
- 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 my implementation of 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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