Graphs. The ultimate data structure. graphs 1

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1 Graphs The ultimate data structure graphs 1

2 Definition of graph Non-linear data structure consisting of nodes & links between them (like trees in this sense) Unlike trees, graph nodes may be completely unordered, or may be linked in any order suited to the particular application graphs 2

3 Undirected graphs Simplest form of graph Set of nodes (vertices -- one node is a vertex) and set of links between them (edges) Either or both sets may be empty Each edge connects two vertices; as the name implies, neither vertex is considered to be a point of origin or a point of destination An edge may also connect a vertex to itself graphs 3

4 Example graph v0 e0 v3 e2 v1 e1 v4 e4 v1 v3 v0 e0 e1 e2 e4 v2 v2 This graph has 5 vertices and 4 edges; a graph is defined by which vertices are connected, and which edges connect them v4 This is the same graph, even though its shape is different graphs 4

5 Application: an undirected state graph Each vertex represents the state of some object or situation Each edge represents the possibility of moving directly from one state to another There exists a path from one state to another if there is an edge connecting each vertex and any vertices between them graphs 5

6 Undirected state graph example graphs 6

7 Directed graphs Each edge has an orientation, or direction One vertex connected by the edge is the source, while the other is the target Directed graphs are diagrammed like undirected graphs, except the edges are represented by arrows instead of lines graphs 7

8 Directed graph example Each arrow represents the direction of the edge; to get from point a to point f, you must pass through c, d and b c (in that order), but to get from f to a requires only going through b (by the shortest route) e a d b f graphs 8

9 More graph terminology Loop: an edge that connects a vertex to itself; in the illustration, edge e3 is a loop Multiple edges: two or more edges connecting to vertices in the same direction e4 and e5 are multiple edges e1 and e2 are not Simple graph: a graph with no loops or multiple edges graphs 9

10 Directed graph application examples Route diagrams for airline routes Course pre- and co-requisites Program flow charts Programming language syntax diagrams graphs 10

11 Modeling with graphs - examples Niche overlap graph in ecology: each species represented by vertex undirected edge connects 2 vertices if the species compete Influence graph: each person in group is represented by vertex directed edge from vertex a to vertex b when a has influence on b 11

12 Modeling with graphs - examples Round-robin tournament graph: each team represented by vertex (a,b) is an edge if team a beats team b Precedence graph: vertices represent statements in a computer program directed edge between vertices means 1st statement must be executed before 2nd 12

13 Terms related to undirected graphs Adjacent: 2 vertices u & v in an undirected graph G are adjacent (neighbors) in G if there is an edge {u,v} Incident, connect: if edge e = {u,v}, e is incident with vertices u & v, and e connects u and v Endpoints: vertices u & v are endpoints of edge e 13

14 Terms related to undirected graphs Degree of vertex in an undirected graph is the number of edges incident with it loops count twice degree of vertex v is denoted deg(v) Example: what are the degrees of the vertices in this graph? 14

15 Terms related to undirected graphs Isolated vertex (like e in previous example) has degree 0, not adjacent to any other vertex Pendant vertex (like d in previous example) - adjacent to exactly one vertex The sum of the degrees of all vertices in a graph is exactly twice the number of edges in the graph 15

16 Handshaking Theorem Let G=(V,E) be an undirected graph with e edges; then 2e = deg(v) v V Note that sum of degrees of vertices is always even; this leads to the theorem: An undirected graph has an even number of vertices of odd degree 16

17 Terminology related to directed graphs Let G be a directed graph, with an edge e=(u,v): u is initial vertex v is terminal, or end vertex u is adjacent TO v v is adjacent FROM u initial & terminal vertex of a loop are the same 17

18 Terminology related to digraphs In-degree of vertex v, denoted deg-(v), is the number of edges with v as the terminal vertex Out-degree of vertex v, denoted deg+(v), is the number of edges with v as the initial vertex A loop has one of each 18

19 Terminology related to digraphs The sum of in-degrees is equal to the sum of outdegrees Both are equal to the number of edges in the graph: Let G=(V,E) be a directed graph; then v V deg ( v ) = deg+ ( v) = E v V The undirected graph that results from ignoring arrows in a directed graph is called the underlying undirected graph 19

20 Classes of simple graphs Complete graphs: complete graph on n vertices, denoted Kn, is the simple graph that contains exactly one edge between each pair of distinct vertices; Examples: 20

21 Classes of Simple Graphs Cycle: Cn, where n 3, consists of n vertices v1, v2,, vn and edges {v1, v2}, {v2, v3},, {vn-1, vn} Examples: 21

22 Classes of Simple Graphs Wheel: a cycle with an additional vertex, which is adjacent to all other vertices; example: 22

23 Classes of Simple Graphs n-cube: denoted Qn, is a graph representing the 2n bit strings of length n: 2 vertices are adjacent if and only if the bit strings they represent differ in exactly one position. Examples: 23

24 Bipartite Graphs A simple graph G is called bipartite if its vertex set V can be partitioned into 2 disjoint non-empty sets V1 and V2 such that: every edge connects a vertex in V1 with a vertex in V2 no edge connects 2 vertices in V1 or in V2 24

25 Example: lining up 1st graders Cate Mary Mary Ann Mary Pat Marie John Mark Terry Tim Jimmy 25

26 Example - cycle C6 is bipartite; can partition its vertex set into 2 distinct sets: V1 = {v1, v3, v5) V2 = {v2, v4, v6} with every edge connecting a vertex in V1 with one in V2 26

27 Examples Graph at left is not bipartite; to divide into 2 sets, one set must include 2 vertices and to be bipartite, those vertices must not be connected But every vertex in this graph is connected to 2 others This graph, on the other hand, is bipartite; the two sets are V1 = {v1, v3, v5} and V2 = {v2, v4, v6} Note that the definition doesn t say a vertex in one set can t connect to more than one vertex in the other - only that each in one must connect to one in the other, and no vertex in a set can connect to a vertex in the same set 27

28 Examples: are they bipartite? 28

29 Complete Bipartite Graphs Denoted Km,n is the graph that has its vertex set partitioned into 2 subsets of m vertices and n vertices There is an edge between 2 vertices if and only if one vertex is in the first subset and the other is in the second subset 29

30 Examples 30

31 Applications of Special Types of Graphs: LAN topology A star is a complete bipartite K1,n graph: A ring is an n-cycle: A redundant network may have both a central hub and a ring, forming a wheel: 31

32 Subgraph Graph obtained by removing vertices and their associated edges from a larger graph; more formally: Subgraph of G=(V,E) is H=(W,F) where W V and F E 32

33 Subgraph Example 33

34 Can combine graphs, forming a union Let G1 = (V1, E1) and G2 = (V2, E2) The union, G1 G2 = (V1 V2, E1 E2) 34

35 Graph implementations All of the following representations can be used to create an instance of a directed graph in which loops are allowed: adjacency list adjacency matrix edge lists edge sets incidence matrices graphs 35

36 Representing Graphs Adjacency list: specifies vertices adjacent with each vertex in a simple graph can be used for directed graph as well - list terminal vertices adjacent from each vertex Vertex Adjacent vertices a c,d b e c a,d d a,c,e e d,b 36

37 Adjacency matrix Square grid of true/false values that represent the edges of a graph If the intersection of a row and column has a true value in it, there exists an edge between the vertices represented by the two indexes Since this is a directed graph, the value will be true only if there is an edge from the vertex indicated by the row index to the vertex indicated by the column index graphs 37

38 Adjacency matrix An adjacency matrix can be stored in a twodimensional array of bools or ints Each vertex is represented by one row and one column For a graph with four vertices, the following declaration could be used: boolean [][] amatrix = new boolean [4][4]; graphs 38

39 Adjacency matrix The graph above could be represented by the adjacency matrix on the right row [0] [1] [2] [3] column [0] [1] [2] [3] f t f f f f f t f f f f f t t t graphs 39

40 Representing Graphs: mathematical POV Adjacency matrix: n x n zero-one matrix with 1 representing adjacency, 0 representing non-adjacency Adjacency matrix of a graph is based on chosen ordering of vertices; for a graph with n vertices, there are n! different adjacency matrices Adjacency matrix of graph with few edges is a sparse matrix - many 0s, few 1s 40

41 Simple graph adjacency matrix With ordering a,b,c,d,e:

42 Using adjacency matrices for multigraphs and pseudographs With multiple edges, matrix is no longer zero-one: if more than one edge exists, list the number of edges A loop is represented with a 1 in position i,i Adjacency matrices for simple graphs, including multigraphs and pseudographs, are symmetric 42

43 Adjacency matrices and digraphs For directed graphs, the adjacency matrix has a 1 in its (i,j)th position if there is an edge from vi to vj Directed multigraphs can be represented by placing the number of edges in in position i,j that exist from vi to vj Adjacency matrices for digraphs are not necessarily symmetric 43

44 Edge lists Linked lists are used to represent vertices Each vertex is a linked list; nodes in each list contain numbers indicating the target vertices for that particular source vertex In other words, if list A contains a link to vertex B, there is an edge from A to B graphs 44

45 Edge lists For n vertices, there must be n lists; often the lists are stored as an array of list head references A vertex that is not a source would be represented in the array by a NULL pointer graphs 45

46 Edge lists [0] [1] [2] [3] 1 3 null 1 The graph above could be repesented by the array of edge lists on the right null null 2 3 graphs 46 null

47 Edge Sets Requires a previously programmed implementation of a set ADT Properties of a set: contains an unordered collection of entries each item is unique operations include insertion, deletion, and checking for the presence of a specific item To represent a graph with 4 vertices, can declare an array of 4 sets of ints graphs 47

48 Incidence Matrices In an adjacency matrix, both rows and columns represent vertices In an incidence matrix, rows represent vertices while columns represent edges A 1 in any matrix position means the edge in that column is incident with the vertex in that row Multiple edges are represented identical columns depicting the edges 48

49 Incidence Matrix Example e1 e2 e3 e4 e5 e6 e7 e8 e9 a b c d e

50 Choosing an implementation Adjacency matrix is easier to implement & use than any of the edge-based solutions Frequency of certain operations may dictate another solution graphs 50

51 Choosing an implementation Both adding and removing edges and checking for the presence of a particular edge are O(N) operations in the worst case for edge lists; may be as low as O(logN) for edge sets, depending on representation Edge lists are very efficient for operations that execute one time for each edge with a particular vertex graphs 51

52 Disadvantages of adjacency matrix Operations on all edges for a vertex require stepping through entire row (whether or not an entry is an edge) -- this is O(N) If each vertex has very few edges, an adjacency matrix is mostly wasted space graphs 52

53 Simple graph ADT implementation: labeled graph Operations: constructor add/removeedge size: returns number of vertices in graph isedge: returns true if edge exists between specified source & target clone get/setlabel neighbors: returns array of neighbors of a specified vertex graphs 53

54 Simple graph ADT implementation: labeled graph Data components: 2-dimensional int array representing edges array of vertex labels: each vertex includes data consisting of an informational label graphs 54

55 Labeled graph class definition public class Graph implements Cloneable { private boolean[ ][ ] edges; private Object[ ] labels; public Graph(int n) // n: # of vertices { edges = new boolean[n][n]; // All values initially false labels = new Object[n]; // All values initially null } graphs 55

56 Labeled graph class definition public void addedge(int source, int target) { edges[source][target] = true; } public void removeedge(int source, int target) { edges[source][target] = false; } public boolean isedge(int source, int target) { return edges[source][target]; } graphs 56

57 Labeled graph class definition public Object getlabel(int vertex) { return labels[vertex]; } public void setlabel(int vertex, Object newlabel) { labels[vertex] = newlabel; } public int size( ) { return labels.length; } graphs 57

58 Labeled graph class definition public int[ ] neighbors(int vertex) { int i, count=0; int[ ] answer; // First count how many edges have the vertex as their source for (i = 0; i < labels.length; i++) { if (edges[vertex][i]) count++; } // Allocate the array for the answer answer = new int[count]; graphs 58

59 Labeled graph class definition // Fill the array for the answer count = 0; for (i = 0; i < labels.length; i++) { if (edges[vertex][i]) answer[count++] = i; } return answer; } graphs 59

60 Labeled graph class definition public Object clone( ) { // Clone a Graph object. Graph answer; try { answer = (Graph) super.clone( ); } catch (CloneNotSupportedException e) { // This would indicate an internal error in the Java runtime system // because super.clone always works for an Object. throw new InternalError(e.toString( )); } answer.edges = (boolean [ ][ ]) edges.clone( ); answer.labels = (Object [ ]) labels.clone( ); return answer; } graphs 60

61 Graph traversals Traversal: visiting all nodes in a data structure Can t apply tree traversal methods to graphs, since nodes in a graph don t have the same child/parent relationships Graph traversal methods utilize subsidiary data structures (queues and stacks) to keep track of vertices to be visited graphs2 61

62 Graph traversals Purpose of traversal algorithm: starting at a given vertex, process the information at that vertex, then move along an edge to process a neighbor When traversal is complete, all vertices that can be reached from the start vertex (but not necessarily all vertices in the graph) have been visited graphs2 62

63 Depth-first search After processing the initial vertex, a depth-first search moves along an edge to one of the vertex s neighbors Once the second vertex is processed, the search moves along an edge to one of its neighbors The process continues to move forward as long as vertices can be reached in this manner graphs2 63

64 Depth-first search The search always proceeds forward until no further forward moves can be made Once forward motion is no longer possible at the current node, the search backs up to the previous vertex and visits any previously unvisited neighbors Eventually the search backs up to the original starting node, having visited all nodes that were reachable from this point graphs2 64

65 Depth-first search example v0 v1 v2 v3 v4 v5 graphs2 65

66 Depth-first printing routine public static void depthfirstprint(graph g, int start) { boolean[ ] marked = new boolean [g.size( )]; depthfirstrecurse(g, start, marked); } // The part of a stack is played here by the system stack; // the depthfirstrecurse method stacks up activation records // with each call graphs 66

67 Recursive method for depth-first printing public static void depthfirstrecurse(graph g, int v, boolean[ ] marked) { int[ ] connections = g.neighbors(v); int i, nextneighbor; marked[v] = true; System.out.println(g.getLabel(v)); // Traverse all the neighbors, looking for unmarked vertices: for (i = 0; i < connections.length; i++) { nextneighbor = connections[i]; if (!marked[nextneighbor]) depthfirstrecurse(g, nextneighbor, marked); } } graphs 67

68 Breadth-first search Uses queue to keep track of vertices which might still have unprocessed neighbors Initial vertex is processed, marked, and placed in queue Repeats the following until queue is empty: remove vertex from front of queue for each unmarked neighbor of the current vertex, mark the neighbor, then place the neighbor on the queue graphs2 68

69 Breadth-first search example v0 v2 v4 v2 v0 v1 v4 Queue of visited nodes v1 v2 v3 v4 v5 graphs2 69

70 Graph Isomorphism The simple graphs G1=(V1,E1) and G2=(V2,E2) are isomorphic if there is a one-to-one and onto function f from V1 to V2 with vertices a and b adjacent in G1 if and only if f(a) and f(b) are adjacent in G2 for all vertices a and b in G1 Function f is called an isomorphism 70

71 Determining Isomorphism The number of vertices, number of edges, and degrees of the vertices are invariants under isomorphism If any of these quantities differ in two simple graphs, the graphs are not isomorphic If these quantities are the same, the graphs may or may not be isomorphic 71

72 Determining Isomorphism To show that a function f from the vertex set of graph G to the vertex set of graph H is an isomorphism, we need to show that f preserves edges Adjacency matrices can help with this: we can show that the adjacency matrix of G is the same as the adjacency matrix of H when rows and columns are arranged according to f 72

73 Example Consider the two graphs below: Both have six vertices and seven edges; Both have four vertices of degree 2 and two vertices of degree 3; The subgraphs of G and H consisting of vertices of degree 2 and the edges connecting them are isomorphic 73

74 Example continued Need to define a function f and then determine whether or not it is an isomorphism: deg(u1) in G is 2; u1 is not adjacent to any other degree 2 vertex, so the image of u1 must be either v4 or v6 in H can arbitrarily set f(u1) = v6 (if that doesn t work, we can try v4) 74

75 Example continued Continuing definition of f: Since u2 is adjacent to u1, the possible images of u2 are v3 and v5 We set f(u2) = v3 (again, arbitrarily) Continuing in this fashion, using adjacency and degree of vertices as guides, we derive f for all vertices in G, as shown on next slide 75

76 Example continued f(u1) = v6 f(u2) = v3 f(u3) = v4 f(u4) = v5 f(u5) = v1 f(u6) = v2 Next, we set up adjacency matrices for G and H, arranging the matrix for H so that the images of G s vertices line up with their corresponding vertices 76

77 Example continued u1 u2 u3 u4 u5 u6 u u u u u u AG v6 v3 v4 v5 v1 v2 v v v v v v AH 77

78 Example concluded Since AG = AH, it follows that f preserves edges, and we conclude that f is an isomorphism; thus G and H are isomorphic If f were not an isomorphism, we would not have proved that G and H are not isomorphic, since another function might show isomorphism 78

Graphs. The ultimate data structure. graphs 1

Graphs. The ultimate data structure. graphs 1 Graphs The ultimate data structure graphs 1 Definition of graph Non-linear data structure consisting of nodes & links between them (like trees in this sense) Unlike trees, graph nodes may be completely