CSC 301.01, Class 02: Shortest path

Overview

• Preliminaries
  • Notes and news
  • Upcoming work
  • Extra credit
  • Questions
• Formalizing the problem
• Selecting possible approaches
• Sample solutions
• Analyzing the sample solutions
• Hints
• A more common solution

News / Etc.

• I will once again take quick attendance.
• Take a card!
• Thank you! to the students who started assignment 1 early and have been sending along questions.
• If you have not yet signed the department’s academic honesty policy, please do so now.
• The Web site is still under development. Expect some significant changes to the topics, but not to the timing of homework assignments and examinations.
• I will check in on daily note-taking responsibilities on Wednesday.
• We now have a Piazza site (thanks AV).
• Be responsible for your classroom. Make sure that your work area is as good or better than you found it. At the end of class
  • Pick up and discard your trash.
  • Return your playing card to me.
  • Push in your chair.

Upcoming work

• Assignment 1, due 10:30 pm Wednesday

Extra credit (Academic)

• Community time: Healing from Hate, Tuesday, 11am, JRC 101.
• CS Table, Tuesday noon, topic tbd, Whale room.
• CS Extras, Thursday 4:15pm, Contracts. Snacks at 4pm.

Extra credit (Peer)

• Try out for Neverland players, today at 4:30 p.m. in The Wall (one of our two black-box theatres). (Act out stories written by middle-school students.)

Good things to do

• Ag days Thursday.

Questions

Can you fix the link to the eboard?

Tonight, I hope

Are the points in two-space?

Yes.

What is a diagonal sweep?

Top left to bottom right. Kinda like horizontal, but 45 degrees off.

Can we do something utterly stupid for 2d, or would you like us to have

a rationale for our heuristic?

Some rationale would be nice, but something stupid is also okay if you can’t come up with a good rationale.

What should add return?

Option 1: Nothing

Option 2: The modified heap

Shouldn’t it be called add!?

Yes.

How do we change the size of the heap?

(vector-set! heap 0 (+ 1 (vector-ref heap 0)))

What happens when we want to grow the size of the underlying vector of

elements?

You should be able to do something similar at position 1.

Can we work with others?

Yes.

How do I compare the priority of two elements in a vector?

(if (higher-priority? (vector-ref vec i) (vector-ref vec j)) ...)

This assumes you’ve written something like (let ([higher-priority? (vector-ref heap 2)] [vec (vector-ref heap 1)])

Formalizing the problem

• Question: Given a non-net-neutral Internet and a starting node on the Internet and an ending node, find the cheapest way to get from the starting node to the ending node.
• Question: Given two cities and the cost of flying between any cites, find the cheapest set of flights between the first city and the last.

Formalize: What is the input? What is the output? What do we expect of the output?

Input:

• N, a series of nodes (distinct names or values)
• E, a cost function (an element of NxN to Z)
• s, an element of N (the starting point)
• t, an element of N (ending point)

Goal

• Find a sequence of nodes n_a_0 … n_a_k s.t. n_a_0 = s, n_a_k = t, and [white board]

Selecting possible approaches

Talk to your group for a minute. Be prepared to present two or three ways we might start approaching the problem.

• Greed: Always choose the lowest cost next edge.
• Exhuastive search: List every path.
• Random: Pick any.
• Try sizes of paths in increasing order (0 then 1 then 2 then 3 …)
• Divide and conquer.
• Just try something.

Question

• What do we do if there isn’t an edge/connection? E(a,b) = infinite.

Some sample solutions

I will assign approaches to groups. Groups will have five minutes to think about them and then present.

Analyzing the sample solutions

We will then think about each.

Greed

Greed does not work. Consider a situation in which the smallest edge leads you to a dead end (a node with only infinite cost edges). Boom!

Fixing it:

• Add backtracking. Sounds hard.
• Remove any nodes that are dead ends and are not the target node.

Exhaustive search

1. Make a list of every possible sequence of nodes. How?
2. Find the cost of each sequence of nodes using the summation.
3. Choose the least.

Correct?

Yes

How expensive is this?

O(

N

*#sequence of nodess)

How many sequences of nodes are there?

Claim 1:

N

!

Claim 2: Infinite. There might be cycles.

How do you fix the problem statement?

Next class!

Random

Once in a while, we might get the best sequence of nodes.

A hint

What if we wanted the shortest sequence of nodes from s to *every* node.

A more standard solution