Outline of Class 48: Introduction to Graphs
Held: Wednesday, April 29, 1998
 I hope to have exams graded by Friday (hah!). Given that many of you
were only able to answer three questions, I'll grade the exam out of
80.
 No more homework assignments (yay!).
 Tomorrow at 4:30, three math majors will be discussing internship
experiences. I strongly encourage you to go and learn about what kinds
of cool things Grinnell students are able to do.
 If you signed up for the picnic and are planning to do "FOG Fast", you'll
need to talk to dining services to see if there's an alternative (e.g.,
skipping dinner on Thursday).
 If you haven't done so already, you should begin reading the chapter in
Bailey on graphs (which will be our primary topic for the rest of the
term).
 As you have (hopefully) noted, computer science is not (always) done
for its own sake. Often, we write computer programs to help us
with "real world" problems.
 Sometimes, we answer realworld questions by modeling them
with one or more data structures.
 For example, we might model the consumer's model of a store with
a series of queues (plus some methods for choosing how long
between service and entry in each queue).
 There are a number of problems that require nonlinear data structures,
and which need fewer restrictions than trees require.
 In the telephone system, find the least congested route between
two phones, given connections between switching stations.
 On the web, determine if there is a way to get to one page from
another, just by following normal links.
 While driving, find the shortest path from one city to another.
 As a traveling salescritter who needs to visit a number of cities,
find the shortest path that includes all the cities.
 On the Internet, determine how many connections between
computers you can eliminate and still make sure that
every computer can reach every other computer.
 Determine an ordering of classes so that you always take
prerequisite courses first.
 These, and many other problems, can be modeled by a data structure
known as the graph.
 Graphs are data structures that contain labeled nodes which are
connected by edges.
 Sound familiar? We came up with similar definitions for lists and
trees.
 How do graphs differ from lists and trees?
 The nodes are labeled, so that we can refer to them by name.
 Each node may have multiple edges connected to it.
 Graphs don't have a unique start node.
 Graphs don't necessarily have a unique end node.
 As with lists and trees, we can make the edges unidirectional or
bidirectional.
 If the edges are unidirectional, the graph is called a
directed graph.
 If the edges are bidirectional, the graph is called a
undirected graph.
 Just as each list is a tree (although not a very balanced tree), each
tree is a graph.
 In some uses of graphs, we may associate a numeric weight to each
edge. Graphs with weights on edges are called weighted graphs.
Typically, a weight represents the cost to get from one node to another.
 As we've noted in the past, if you designate a root node in a directed
graph, the directed graph can serve as a more compact representation of
a tree.
 Note that in some graphs it is possible to follow a sequence of edges and
return to the place you started. That path (sequence of edges) is called
a cycle. Graphs with cycles are called cyclic
graphs.
 If there are no cycles in a graph, it is an acyclic graph.
 If you don't say whether or not a graph is cyclic, you are implying
that you will deal with either type.
 When writing recursive graph algorithms that are not restricted to
acyclic graphs, you may need to mark the nodes in the graph
to ensure that you don't repeatedly use the same node.
 We can phrase a number of modeling problems in terms of graphs.
 In the telephone system, find the least congested route between
two phones, given connections between switching stations.
 In a nonnegatively weighted (by congestion) undirected graph, find a
path from node A to node B with lowest maximum edge.
 On the web, determine if there is a way to get to one page from
another, just by following normal links.
 In a directed graph, determine whether node B is reachable from
node A
 While driving, find the shortest path from one city to another.
 In a weighed directed graph (or simply a directed graph) (or even an
undirected graph), find the
shortest path from node A to node B.
 As a traveling salesperson who needs to visit a number of cities,
find the shortest path that includes all the cities.
 In an undirected graph, find a path that includes all nodes.
 On the Internet, determine how many connections between
computers you can eliminate and still make sure that
every computer can reach every other computer.
 In a (possibly weighted) graph, find a minimum spanning tree.
A spanning three is a set of edges such that every node is reachable
from every other node, and the removal of any edge from the tree
eliminates the reachability property. A minimum spanning tree is the
smallest such tree.
 Determine an ordering of classes so that you always take
prerequisite courses first.
 In a directed acyclic graph, find a numbering such that if there
is a path from A to B, then the number assigned to A (the term in
which you take A) is smaller
than the number assigned to B.
 One advantage of modeling problems with graphs is that we can then
solvev more general problems on the graphs an then use those solutions
to solve a wide variety of "realworld" questions.
 That is, we will develop a number of graph algorithms to
solve typical problems.
 What are some of the core graph algorithms/problems? In no
particular order,
 Reachability. Can you get to B from A?
 Shortest path (mincost path). Find the path from B to A with
the minimum cost (determined as some simple function of the edges
traversed in the path).
 Minimum spanning tree. Find the "smallest" subset of the edges in
which all the nodes are connected.
 Traveling salesman. Find the smallest cost path through all the nodes.
 Visit all nodes. Traversal.
 Transitive closure. Determine all pairs of nodes that can reach
each other.
 There are also many variations. For example,
Bailey has a
reachableFrom
algorithm that returns
all the nodes reachable from a partiuclar node.
 We'll consider most of these algorithms in the next few days.
 In order to develop algorithms, we need at least a minimal version
of the
GraphNode
class that will be by the algorithms.
 (Unlike other structures which had a wrapper class, graphs seem most
natural in their "pure node" state.)
 We'll work with nodes in a directed graph without weights
 This should be easy to extend to a weighted graph.
 What methods should a
GraphNode
provide?

addEdge(GraphNode neighbor)
: add
an edge to a neighbor.

removeEdge(GraphNode neighbor)
: remove an edge
to a neighbor.

getNeighbors()
: get the neighbors of a node.
 We might also want to associate some symbol or value with each
node. Presumably, we can do the association when we construct
the node.

getSymbol()
: looks up the symbol.

toString()
: uses the symbol when converting to a string.
 What else might we want? We might want to add a number of edges
en masse, but we'll worry about that later.