Graphs

OverviewWhat is a graph? Some terminologyTypes of graphImplementing graphs (briefly)Some graph algorithms

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Questions we can’t answer with linear structures or treesWhat if:The city wants to clean each street without driving down

any of them twice?

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Questions we can’t answer with linear structures or treesWhat if:The city wants to clean each street without driving down any

of them twice?A trucking company wants to make deliveries to a set of

locations without visiting the same location twice?

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Questions we can’t answer with linear structures or treesWhat if:The city wants to clean each street without driving down

any of them twice?A trucking company wants to make deliveries to a set of

locations without visiting the same location twice?We want to find the (shortest) path between two people on

Facebook, going from friend to friend?

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Questions we can’t answer with linear structures or treesWhat if:The city wants to clean each street without driving down

any of them twice?A trucking company wants to make deliveries to a set of

locations without visiting the same location twice?We want to find the (shortest) path between two people on

Facebook, going from friend to friend?We want to find the shortest airplane trip connecting a set

of cities? The cheapest? Graphs p. 6/18

Some terminologyConsider Figures 13.1 and 13.2 (pp. 378, 389).

List the vertices.List the edges.Give two pairs of adjacent vertices.What are the neighbors of vertex D in each graph?What would we need to do to make Figure 13.1 a complete

graph? Explain.If there are n vertices, how many edges are needed to make a

complete graph? Explain.Graphs p. 7/18

Terminology (continued)Still with regard to Figures 13.1 and 13.2:Give two paths from A to D in Figure 13.1, one of which

involves visiting the same vertex at least twice.Are the two graphs connected? Why or why not?Is there a cycle in Figure 13.1? If yes, list the vertices on

the cycle in order. If no, why not?Name a type of graph that we have seen that has no cycles

and explain why.

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Types of graphFind an example of an undirected graph in your

book. Explain.Find an example of a directed graph (aka

“digraph”). Explain.Find an example of a weighted graph. Explain.

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Undirected graphs: towards implementationGiven the undirected graph with edges((A,B), (B,C), (C, D), (A, C))Draw the graph.For each vertex, give its adjacency list. Draw the adjacency matrix (2D array) corresponding

to the graph.

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Directed graphs: towards implementationGiven the directed graph with edges((A,B), (B,C), (C, D), (A, C))Draw the graph.For each vertex, give its adjacency list. Draw the adjacency matrix (2D array) corresponding

to the graph.

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Back to those questions we wanted to askWhat are the vertices and edges if:The city wants to clean each street without driving down

any of them twice?A trucking company wants to make each delivery without

visiting the same location twice?We want to find if the (shortest) path between two people

on Facebook, going from friend to friend?We want to find the shortest airplane trip connecting a

sets of cities? The cheapest?Graphs p. 12/18

GraphADTIf we had a GraphADT, it might have operations like:public Vector<Vertex> findEulerCircult(Vector<Vertex>);

public Vector<Vertex> travelingSalesman();

public Vector<Vertex> findPath(Vertex a, Vertex b);

public Vector<Vertex> findShortestPath(Vertex a, Vertex b);

Each of which solves one of the problems above.

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Finding an Euler CircuitThe Euler Circuit problem: cross all edges without repeats.Algorithm:

If (some vertex has an odd number of edges) return null.Start with any vertex.Follow edges in any direction EXCEPT never “burn bridges.”

Try this out with the following graph:((A, B), (B, D), (D, C), (C, A), (D, F), (F, E), (E, C), (F, H), (H, G), (G,

E), (E, N), (N, C), (D, I), (I, J), (J, F), (I, L), (L, J), (I, K), (K, L), (L, M), (M, J))

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The Traveling Salesman ProblemTraveling Salesman (a famous problem in computer

science): given a weighted graph, find the “cheapest” path that visits each vertex exactly once and returns back to the start.

Big Oh?

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The Traveling Salesman ProblemTraveling Salesman (a famous problem in computer

science): given a weighted graph, find the “cheapest” path that visits each vertex exactly once and returns back to the start.

Big Oh? O(n!) -- even worse than 2n

An NP-complete problem.

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The Traveling Salesman Problem (simplified)A simpler version: find any path that visits each vertex

once and returns to the start (a Hamiltonian circuit). Some graphs have noneSome have more than oneOnly algorithm: trial and error

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Coming AttractionsNext time we’ll look at a new data structure called a

heap that allows us to model situations like hospital emergency rooms (where you insert according to some priority but delete like a queue)

Homework: read Chapter 11 Heaps and Priority Queues (or the equivalent in the earlier edition).

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