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Graphs & Graph Algorithms

B Smith: Fa05: Rate: 3. Should be more interactive! The adjacency. Time: about 60 min. B Smith: Fa07: Rate:3, time: 75 minutes. B Smith: Decide what will be consistently done with self-directed edges. Use 1’s or 0’s?. Graphs & Graph Algorithms. B Smith:

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Graphs & Graph Algorithms

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  1. B Smith: Fa05: Rate: 3. Should be more interactive! The adjacency. Time: about 60 min B Smith: Fa07: Rate:3, time: 75 minutes B Smith: Decide what will be consistently done with self-directed edges. Use 1’s or 0’s? Graphs & Graph Algorithms B Smith: Sp06: 90 minutes. Rate:3. Heavy on definitins. Redo in entirety for sharing/distribution B Smith: Might be easier to simply consider undirected graphs first B Smith: See lecture23.ppt under graphs for great coverage

  2. Outline • What is a Graph? • Applications of Graphs • Categorization • Terminology • Representation • Searching • Depth First Search (DFS) • Breadth First Search (BFS)

  3. 3 1 Vertex (Node) 2 Edge 4 5 What is a Graph? A graph is a data structure which consists of a set of vertices, plus a set of edgesthat connect (some of) them. G = ( V, E ) Where, V- set of vertices E - set of edges V = {1, 2, 3, 4, 5} E = { (1,2), (1,3), (1,4), (2,3), (3,5), (4,5) }

  4. Computer Resistor/Inductor/… City Applications • Computer Networks • Electrical Circuits • Road Map

  5. Connected Components A B

  6. Connected Components Application: Minesweeper • Challenge: Implement the game of Minesweeper • Critical subroutine: • User selects a cell and program reveals how many adjacent mines • If zero, reveal all adjacent cells • If any newly revealed cells have zero adjacent mines, repeat

  7. Connected Components

  8. High School Dating http://www-personal.umich.edu/~mejn/networks/

  9. TB or SARS Contagions http://www-personal.umich.edu/~mejn/networks/

  10. HS Friendships http://www-personal.umich.edu/~mejn/networks/

  11. The internet http://www-personal.umich.edu/~mejn/networks/

  12. Graphs and Mazes

  13. Graph categorization • A Directed Graph or Digraph is a graph where each edge has a direction • The edges in a digraph are called Arcs or Directed Edges

  14. 1 6 4 2 5 3 Graph categorization • Digraph - Example G = (V, E) V = {1, 2, 3, 4, 5, 6} E = {(1,4), (2,1), (2,3), (3,2), (4,3), (4,5), (4,6), (5,3), (6,1), (6,5)} (1, 4) = 1→4 where 1 is the tail and 4 is the head

  15. Graph categorization • An Undirected Graph is a graph where the edges have no directions • The edges in an undirected graph are called Undirected Edges

  16. 3 1 2 4 5 Graph categorization • Undirected Graph - Example G = (V, E) V = {1, 2, 3, 4, 5} E = {(1,2), (1,3), (1,4), (2,3), (3,5), (4,5)}

  17. 40 3 1 20 60 10 2 70 4 50 5 Graph categorization • A Weighted Graph is a graph where all the edges are assigned weights

  18. 1 2 3 Graph categorization • If the same pair of vertices have more than one edge, that graph is called a Multigraph

  19. 3 1 2 4 5 Graph Terminology • Adjacent vertices: If (i,j) is an edge of the graph, then the nodes i and j are adjacent • An edge (i,j) is Incident tovertices i and j Vertices 2 and 5 are not adjacent

  20. 3 1 2 4 5 B Smith: are they permitted in Digraphs (only?) Graph Terminology • Loop or self edges:An edge ( i,i ) is called a self edge or a loop • In graphs, loops are not permitted ( 1,1 ) and ( 4,4 ) are self edges

  21. G A F D B E C Graph Terminology • Path: A sequence of edges in the graph • There can be more than one path between two vertices • Vertex A is reachable from B if there is a path from A to B • Paths from B to D - B, A, D • B, C, D

  22. 3 1 2 4 5 Graph Terminology • Simple Path:A path whereall the vertices are distinct 1,4,5,3 is a simple path. But 1,4,5,4 is not a simple path.

  23. 3 1 2 4 5 Graph Terminology • Length :Sum of the lengths of the edges on the path. Length of the path 1,4,5,3 is 3

  24. 3 1 2 4 5 Graph Terminology • Circuit: A path whose first and last vertices are the same The path 3,2,1,4,5,3 is a circuit.

  25. 3 1 2 4 5 GraphTerminology • Cycle: A circuit where all the vertices are distinct except for the first (and the last) vertex 1,4,5,3,1 is a cycle 1,4,5,4,1 is not a cycle

  26. 3 1 2 4 5 GraphTerminology • Hamiltonian Cycle: A Cycle that contains all the vertices of the graph 1,4,5,3,2,1 is a Hamiltonian Cycle

  27. 3 1 2 4 5 GraphTerminology • Degree of a Vertex : In an undirected graph, the no. of edges incident to the vertex • In-degree: The no. of edges entering the vertex in a digraph • Out-Degree: The no. of edges leaving the vertex in a digraph In-degree of 1 is 3 Out-degree of 1 is 1

  28. 3 1 1 2 3 4 2 5 H=(U,F) G=(V,E) Graph Terminology • A Subgraph of graph G=(V,E) is a graph H=(U,F) such that U ЄV and FЄE

  29. 3 3 1 1 2 2 4 4 5 5 Connected Unconnected Graph Terminology • A graph is said to be Connected if there is at least one path from every vertex to every other vertex in the graph

  30. 3 1 3 1 2 2 4 4 5 5 Tree Forest Graph Terminology • Tree: A connected undirected graph that contains no cycles • Forest: A graph that does not contain a cycle

  31. 3 3 1 1 2 2 4 4 5 5 Spanning Tree Graph Graph Terminology • The Spanning Tree of a Graph G is a subgraph of G that is a tree and contains all the vertices of G

  32. 1 if (i,j) Є E • 0 otherwise aij = Representation of Graphs • Adjacency Matrix (A) • The Adjacency Matrix A=(ai,j) of a graph G=(V,E) with n nodes is an nXn matrix • Each element of A is either 0 or 1, depending on the adjacency of the nodes

  33. 3 1 2 3 4 1 5 2 B Smith: Discuss the fact that what we do with the diagonal could depend on the problem, but using either 1’s or 0’s is a possibility Representation of Graphs • Eg: Find the adjacency matrices of the following graphs B Smith: Use (r,c) notation to disambiguate

  34. 9 3 1 5 4 2 Representation of Graphs • Adjacency Matrix of a Weighted Graph • The weight of the edge can be shown in the matrix when the vertices are adjacent • A nil value (0 or ∞) depending on the problem is used when they are not adjacent • Eg. To find the minimum distance between nodes...

  35. 5 5 1 3 1 1 2 1 3 1 2 3 2 4 4 5 5 Representation of Graphs • Adjacency List • An Adjacency list is an array of lists, each list showing the vertices a given vertex is adjacent to

  36. 5 5 4 3 2 2 1 2 3 9 1 9 3 1 5 4 2 Representation of Graphs • Adjacency List of a Weighted Graph • The weight is included in the list

  37. Searching Graphs • Why do we need to search graphs? • To find paths • To look for connectivity • Two strategies • Depth-First Search (DFS) • Breadth-First Search (BFS)

  38. Depth-First Search • Start from an arbitrary node • Visit (Explore) an unvisited adjacent edge • If the node visited is a dead end, go back tothe previous node (Backtrack) • Stop when no unvisited nodes are found and no backtracking can be done • Implemented using a Stack Explore if possible, Backtrack otherwise…

  39. B F H A C G D E Depth-First Search A , F , C , B , G , E

  40. Breadth-First Search • Start from an arbitrary node • Visit all the adjacent nodes (distance=1) • Visit the nodes adjacent to the visited nodes (distance=2, 3 etc.) • Stop when a dead end is met • Implemented using a Queue Explore all nodes at distance d…

  41. B F H A C G D E Breadth-First Search A , C , E , B , F , G

  42. Summary • What is a graph? • Graph terminology • Graph representation • Searching • Depth-First Search • Breadth-First Search

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