1. Spring 2010CS 2251 Graphs Chapter 10

2. Spring 2010CS 2252 Chapter Objectives To become familiar with graph terminology and the different types of graphs To study a Graph ADT and different implementations of the Graph ADT To learn the breadth-first and depth-first search traversal algorithms To learn some algorithms involving weighted graphs To study some applications of graphs and graph algorithms

3. Spring 2010CS 2253 More General Trees

4. Spring 2010CS 2254 Other Linked Information

5. Spring 2010CS 2255 The internet http://isiosf.isi.it/~jramas co/fig/wired.png

6. Spring 2010CS 2256 Graph Terminology A graph is a data structure that consists of a set of vertices and a set of edges between pairs of vertices Edges represent paths or connections between the vertices The set of vertices and the set of edges must both be finite and neither one be empty A tree is a special case of a graph

7. Spring 2010CS 2257 Visual Representation of Graphs Vertices are represented as points or labeled circles and edges are represented as lines joining the vertices The physical layout of the vertices and their labeling are not relevant

8. Spring 2010CS 2258 Directed Graphs The edges of a graph are directed if the existence of an edge from A to B does not necessarily guarantee that there is a path in both directions A graph with directed edges is called a directed graph

9. Spring 2010CS 2259 Graph Vocabulary A vertex is adjacent to another vertex if there is an edge to it from that other vertex A path is a sequence of vertices in which each successive vertex is adjacent to its predecessor In a simple path, the vertices and edges are distinct except that the first and last vertex may be the same

10. Spring 2010CS 22510 Paths and Cycles A cycle is a simple path in which only the first and final vertices are the same

11. Spring 2010CS 22511 Weighted Graphs The edges in a graph may have values associated with them known as their weights –Weight are usually assumed to be positive A graph with weighted edges is known as a weighted graph

12. Spring 2010CS 22512 Connected vs. Unconnected A graph is connected if there is a path between any pair of vertices in the graph If a graph is not connected, it may still consist of connected endpoints

13. Spring 2010CS 22513 The Graph ADT and Edge Class Java does not provide a Graph ADT In making our own, we need to be able to do the following Create a graph with the specified number of vertices Iterate through all of the vertices in the graph Iterate through the vertices that are adjacent to a specified vertex Determine whether an edge exists between two vertices Determine the weight of an edge between two vertices Insert an edge into the graph

14. Spring 2010CS 22514 The Graph ADT and Edge Class

15. Spring 2010CS 22515 Implementing the Graph ADT Because graph algorithms have been studied and implemented throughout the history of computer science, many of the original publications of graph algorithms and their implementations did not use an object-oriented approach and did not even use abstract data types Two representations of graphs are most common –Edges are represented by an array of lists called adjacency lists, where each list stores the vertices adjacent to a particular vertex –Edges are represented by a two dimensional array, called an adjacency matrix

16. Spring 2010CS 22516 Adjacency List An adjacency list representation of a graph uses an array of lists One list for each vertex

17. Spring 2010CS 22517 Adjacency ListExample 1

18. Spring 2010CS 22518 Adjacency List Example 2

19. Spring 2010CS 22519 Adjacency Matrix Uses a two-dimensional array to represent a graph For an unweighted graph, the entries can be Boolean values For a weighted graph, the matrix would contain the weights

20. Spring 2010CS 22520 Graph Class Hierarchy

21. Spring 2010CS 22521 Class AbstractGraph

22. Spring 2010CS 22522 The ListGraph Class

23. Spring 2010CS 22523 Traversals of Graphs Most graph algorithms involve visiting each vertex in a systematic order Most common traversal algorithms are the breadth first and depth first search

24. Spring 2010CS 22524 Breadth-First Search In a breadth-first search, we visit the start first, then all nodes that are adjacent to it next, then all nodes that can be reached by a path from the start node containing two edges, three edges, and so on –There is no special start vertex in a graph Must visit all nodes for which the shortest path from the start node is length k before we visit any node for which the shortest path from the start node is length k+1

25. Spring 2010CS 22525 Algorithm for Breadth-First Search

26. Spring 2010CS 22526 Example of a Breadth-First Search

27. Spring 2010CS 22527 Trace of Breadth-First Search

28. Spring 2010CS 22528 ??? Breadth-First Search

29. Spring 2010CS 22529 Depth-First Search In depth-first search, you start at a vertex, visit it, and choose one adjacent vertex to visit; then, choose a vertex adjacent to that vertex to visit, and so on until you go no further; then back up and see whether a new vertex can be found

30. Spring 2010CS 22530 Depth-First Search Algorithm

31. Spring 2010CS 22531 Depth-First Search

32. Spring 2010CS 22532 Depth-First Search

33. Spring 2010CS 22533 Implementing Depth-First Search

34. Spring 2010CS 22534 Shortest Path Through a Maze

35. Spring 2010CS 22535 Shortest Path Through a Maze

36. Spring 2010CS 22536 Directed Acyclic Graph Directed graphs with no cycles are called directed acyclic graphs (DAG for short) They come up in a number of kinds of problems –scheduling –trees are DAGs

37. Spring 2010CS 22537 DAG Examples

38. Spring 2010CS 22538 Topological Sort of a Graph A topological sort of a DAG is an ordering of the vertices of the graph such that if (u, v) is an edge, u comes before v in the sort For the graph below –0, 1, 2, 3, 4, 5, 6, 7, 8 is a valid topological sort –0, 3, 1, 4, 6, 2, 5, 7, 8 is a valid topological sort –0, 1, 5, 3, 4, 2, 6, 7, 8 is not a topologicl sort

39. Spring 2010CS 22539 Topological Sort of a Graph If there is an edge (u, v) in a graph, the finish time of u must be after that of v in a depth-first search The the reverse of finish order of a depth- first search is a topological sort Algorithm for topological sort 1.Perform a depth-first search 2.List the vertices in the reverse of their finish order

40. Spring 2010CS 22540 Other Graph Algorithms Shortest Path for weighted graphs –Breadth-first search finds the shortest path if all edges have the same weight Minimum spanning tree –Subset of the edges such that there is exactly one path between any pair of vertices

41. Spring 2010CS 22541 Shortest Path in Weighted Graph Finding the shortest path from a vertex to all other vertices Uses the following data –S is the set of vertices for which shortest distance has been computer –V-S is the remaining vertices –d is an array containing the shortest distance from s to each vertex –p is an array of predecessors for each vertex

42. Spring 2010CS 22542 Shortest Path Algorithm Developed by Edsger Dijkstra

43. Spring 2010CS 22543 Practice

44. Spring 2010CS 22544 Spanning Tree A minimum spanning tree is a subset of the edges of a graph such that there is only one edge between each vertex, and all of the vertices are connected The cost of a spanning tree is the sum of the weights of the edges We want to find the minimum spanning tree or the spanning tree with the smallest cost –Solution formulated by R.C. Prim is very similar to Dijkstra’s algorithm –Kruskal's algorithm is also commonly used

45. Spring 2010CS 22545 Prim’s Algorithm

46. Spring 2010CS 22546 Practice