Graphs

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 Chapter 13 -- Graphs

3. Terminology • Def: A graph G consists of two sets • Vertices • Edges • Def: Adjacent • Two vertices are said to be adjacent if they are joined by an edge. Chapter 13 -- Graphs

4. Def: Subgraph • A subgraph consists of a set of a graph’s vertices and a subset of its edges Chapter 13 -- Graphs

5. Def: Path • 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 Chapter 13 -- Graphs

6. A graph is complete if each pair of distinct vertices has an edge between them Chapter 13 -- Graphs

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) Chapter 13 -- Graphs

8. You can label the edges of a graph • When these labels represent numeric values, the graph is called a weighted graph Chapter 13 -- Graphs

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) Chapter 13 -- Graphs

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 • … Chapter 13 -- Graphs

11. Implementing Graphs • 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). Chapter 13 -- Graphs

12. Example: a directed graph. Chapter 13 -- Graphs

13. Example: a weighted graph Chapter 13 -- Graphs

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 Chapter 13 -- Graphs

15. Example: a directed graph Chapter 13 -- Graphs

16. Example: a weighted graph Chapter 13 -- Graphs

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. Chapter 13 -- Graphs

18. Now consider the space requirements Chapter 13 -- Graphs

19. Graph Traversals • Visit all the vertices in the graph. • There are two major ways to do this: • Depth first search • Breadth first search Chapter 13 -- Graphs

20. Example: the order in which the nodes of a tree are visited in a depth-first search and a breadth-first search. Chapter 13 -- Graphs

21. How do you implement them? • Depth-First Search • Typically done with a stack. • Breadth-First Search • Typically done with a Queue Chapter 13 -- Graphs

22. Example: DFS • Start with Node a • Trace the Search Chapter 13 -- Graphs

23. Example: DFS Chapter 13 -- Graphs

24. Example: BFS • Start with Node a • Trace the Search Chapter 13 -- Graphs

25. Example: BFS Chapter 13 -- Graphs

26. Applications of Graphs • Topological Sorting • Given the following graph, place the vertices in order so that the edges all point in one direction Chapter 13 -- Graphs

27. Multiple solutions: • Applications: Prerequisites ( 201,202,236,336,308,…) Chapter 13 -- Graphs

28. Spanning Trees • Given a graph G, construct a tree that has all the vertices of G Chapter 13 -- Graphs

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. Chapter 13 -- Graphs

30. The DFS Spanning Tree Chapter 13 -- Graphs

31. The BFS spanning Tree Chapter 13 -- Graphs

32. Minimum Spanning Trees • Given a weighted undirected graph, compute the spanning tree with the minimum cost Chapter 13 -- Graphs

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. Chapter 13 -- Graphs

34. Trace of Prim’s Chapter 13 -- Graphs

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36. Chapter 13 -- Graphs

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 (0-4-2-1) Chapter 13 -- Graphs

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 Chapter 13 -- Graphs

39. We begin with the vertex set initialized to 0 Chapter 13 -- Graphs

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41. Chapter 13 -- Graphs

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? Chapter 13 -- Graphs

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. Chapter 13 -- Graphs

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 Chapter 13 -- Graphs

45. The four color problem • Given • 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) Chapter 13 -- Graphs

46. Summary • Implementations: • 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 Chapter 13 -- Graphs