1. Graphs
Chapter 13

2. Preview
Graphs are an important mathematical concept that have significant applications not only in computer science, but also in many other fields.
You can view a graph as a mathematical construct, a data structure, or an abstract data type.
This chapter provides an introduction to graphs and it presents the major operations and applications of graphs that are relevant to the computer scientist
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3. Terminology Def: A graph G consists of two sets Def: Adjacent Vertices
Edges
Def: Adjacent
Two vertices are said to be adjacent if they are joined by an edge.
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4. Def: Subgraph
A subgraph consists of a set of a graph’s vertices and a subset of its edges
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5. Def: Path Def: Cycle Def: Connected
A path between two vertices is a sequence of edges that begins at one vertex and ends at another.
Simple path: does not pass through the same vertex more than once.
Def: Cycle
A cycle is a path that begins and ends at the same vertex.
Def: Connected
A graph is connected if each pair of distinct vertices has a path between them
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6. A graph is complete if each pair of distinct vertices has an edge between them
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7. Since a graph has a set of edges it cannot have duplicate edges.
However, a multigraph does allow multiple edges
A graph also does not allow self-edges (loops)
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8. You can label the edges of a graph
When these labels represent numeric values, the graph is called a weighted graph
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9. All of the previous examples were of undirected graphs, because the edges did not indicate a direction. If there is direction, then the graph is called a directed graph (or digraph)
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10. Graphs as ADTs Basic Functions Create, Destroy, Determine empty
Determine number of vertices and edges
Is there an edge between I and J?
Insert a vertex, or Insert an edge
Delete a vertex, or Delete an Edge …
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11. Implementing Graphs Adjacency Matrix
The two most common implementations of a graph are the Adjacency Matrix and the Adjacency List
Adjacency Matrix
Given a graph G with n vertices
Matix is n x n of Booleans
Entries – True if an edge exists between I and J (false otherwise).
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12. Example: a directed graph.
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13. Example: a weighted graph
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14. An Adjacency List Given a graph G with n vertices N linked lists
The ith list has node for vertex j iff there is an edge between i and j
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15. Example: a directed graph
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16. Example: a weighted graph
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17. Which implementation is better?
Let’s look at the two most common operations
Determine if there is an edge (i, j)
Find all vertices adjacent to a given vertex i.
The adjacency matrix supports the first one somewhat more efficiently than the list
The second operation is supported more efficiently by the adjacency list than the matrix.
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18. Now consider the space requirements
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19. Graph Traversals Visit all the vertices in the graph.
There are two major ways to do this:
Depth first search
Breadth first search
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20. Example: the order in which the nodes of a tree are visited in a depth-first search and a breadth-first search.
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21. How do you implement them?
Depth-First Search
Typically done with a stack.
Breadth-First Search
Typically done with a Queue
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22. Example: DFS Start with Node a Trace the Search CS 302
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23. Example: DFS
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24. Example: BFS Start with Node a Trace the Search CS 302
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25. Example: BFS
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26. Applications of Graphs
Topological Sorting
Given the following graph, place the vertices in order so that the edges all point in one direction
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27. Multiple solutions:
Applications: Prerequisites ( 201,202,236,336,308,…)
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28. Spanning Trees
Given a graph G, construct a tree that has all the vertices of G
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29. A few observations about undirected graphs
A connected undirected graph that has n vertices must have at least n-1 edges
A connected undirected graph that has n vertices and exactly n-1 edges cannot contain a cycle.
A connected undirected graph that has n vertices and more than n-1 edges must contain at least 1 cycle.
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30. The DFS Spanning Tree
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31. The BFS spanning Tree
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32. Minimum Spanning Trees
Given a weighted undirected graph, compute the spanning tree with the minimum cost
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33. Prim’s Algorithm Start with two sets
Vertices not visited (not part of the tree)
Vertices visited (part of the tree – initialized to some start vertex)
Find the shortest edge from the visited set to the not visited set.
Add the edge to the MST
Move the vertex on the other end from not visited to visited.
Continue until all vertices are visited.
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34. Trace of Prim’s
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37. Shortest Paths Consider once again a map of airline routes.
The shortest path from 0 to 1 in the following graph is not the edge 0-1 (weight 8) but the path ( )
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38. The algorithm to calculate the answer is attributed to Dijkstra
Need two arrays
Weight – Weight[v] = weight of the shortest path from 0 to v
Matrix – Matrix[v][u] = weight of edge from v to u
Then compare the Weight[u] with Weight[v]+Matrix[v][u]. Put the smallest in Weight[u]
Then we need a set of vertices we have visited
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39. We begin with the vertex set initialized to 0
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42. Circuits Eulerian Circuits
Probably the first application of graphs occurred in the early 1700s when Euler proposed the Königsberg bridge problem.
Can you start and end at the same point and only cross each bridge once?
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43. Some Difficult Problems
The Traveling Salesperson Problem
Def: A Hamiltonian circuit – pass through every vertex once.
Given
a weighted graph
an origin city
Problem:
the salesperson must begin at an origin city and visit all others (exactly once) and return to the origin
and do it with the least cost.
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44. The three utilities problem
Given
A graph (3 houses, 3 utilities)
Problem
Connect each house with each utility so the edges do not cross
This is a planarity question (minimum crossing number = 0) for K3,3
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45. The four color problem Given Problem
A planar graph
Problem
Can you color the vertices so no two adjacent vertices have the same color?
The question was posed more than a century ago (1852) and was only solved in the 1976 (with the help of a computer)
Trivia: Nature July 17, 1879 carried a proof of this problem (which was proved false in 1890)
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46. Summary Implementations: Graph searching Trees Problems looked at
Adjacency Matrix and Adjacency List
Graph searching
uses stacks and Queues
Trees
Connected, undirected, acyclic graphs
Problems looked at
Shortest Paths, Eulerian circuits, Hamiltonian Circuits
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