Graph Theory and Representation

1. Graph Theory and Representation

2. Graph Algorithms • Graphs and Theorems about Graphs • Graph ADT and implementation • Graph Algorithms • Shortest paths • minimum spanning tree Cutler/Head

3. What can graphs model? • Cost of wiring electronic components together. • Shortest route between two cities. • Finding the shortest distance between all pairs of cities in a road atlas. • Flow of material (liquid flowing through pipes, current through electrical networks, information through communication networks, parts through an assembly line, etc). • State of a machine (FSM). • Used in Operating systems to model resource handling (deadlock problems). • Used in compilers for parsing and optimizing the code. Cutler/Head

4. What is a Graph? • Informally a graph is a set of nodes joined by a set of lines or arrows. 1 2 3 1 3 2 4 4 5 6 5 6 Cutler/Head

5. A directed graph, also called a digraphG is a pair ( V, E ), where the set V is a finite set and E is a binary relation on V . The set V is called the vertex set of G and the elements are called vertices. The set E is called the edge set of G and the elements are edges (also called arcs ). A edge from node a to node b is denoted by the ordered pair ( a, b ). Self loop Isolated node 1 3 7 2 4 5 6 V = { 1, 2, 3, 4, 5, 6, 7 }| V | = 7 E = { (1,2), (2,2), (2,4), (4,5), (4,1), (5,4),(6,3) }| E | = 7 Cutler/Head

6. An undirected graph G = ( V, E ) , but unlike a digraph the edge set E consist of unordered pairs. Our text uses the notation ( a,b ) to refer to a directed edge, and { a, b } for an undirected edge. V = { A, B, C, D, E, F } |V | = 6 A B C E = { {A, B}, {A,E}, {B,E}, {C,F} } |E | = 4 D F E Some texts use (a, b) also for undirected edges. So ( a, b ) and ( b, a ) refers to the same edge. Cutler/Head

7. Degree of a Vertex in an undirected graph is the number of edges incident on it. In a directed graph , the out degree of a vertex is the number of edges leaving it and the in degree is the number of edges entering it. The degree of B is 2. A B C Self-loop D F E The in degree of 2 is 2 andthe out degree of 2 is 3. 1 2 4 5 Cutler/Head

8. Simple Graphs Simple graphs are graphs without multiple edges or self-loops. We will consider only simple graphs. Proposition: If G is an undirected graph then deg(v) = 2 |E | Proposition: If G is a digraph then indeg(v) =  outdeg(v) = |E | v G v G v G Cutler/Head

9. A weighted graph is a graph for which each edge has an associated weight, usually given by a weight functionw: E R. 2 1.2 1 3 1 2 3 2 .2 1.5 5 .5 3 .3 1 4 5 6 4 5 6 .5 Cutler/Head

10. A path is a sequence of vertices such that there is an edge from each vertex to its successor. A path from a vertex to itself is called a cycle. A graph is called cyclic if it contains a cycle; otherwise it is called acyclic A path is simple if each vertex is distinct. A B C 1 3 2 Cycle D F E 4 5 6 Unreachable Cycle If there is path p from u to v then we say v is reachable from u via p. Simple path from 1 to 5 = ( 1, 2, 4, 5 ) or as in our text ((1, 2), (2, 4), (4,5)) Cutler/Head

11. A Complete graph is an undirected/directed graph in which every pair of vertices is adjacent. If (u, v ) is an edge in a graph G, we say that vertex v is adjacent to vertex u. A A B B E D D 4 nodes and (4*3)/2 edges V nodes and V*(V-1)/2 edges Note: if self loops are allowed V(V-1)/2 +Vedges 3 nodes and 3*2 edges V nodes and V*(V-1) edges Note: if self loops are allowed V2 edges Cutler/Head

12. An undirected graph is connected if you can get from any node to any other by following a sequence of edges OR any two nodes are connected by a path.A directed graph is strongly connected if there is a directed path from any node to any other node. • A graph is sparse if | E |  | V | • A graph is dense if | E |  | V |2. 1 2 A B C 4 5 D F E Cutler/Head

13. A bipartite graphis an undirected graph G = (V,E) in which V can be partitioned into 2 sets V1 and V2 such that ( u,v) E implies either uV1 and vV2 OR vV1 and uV2. Cutler/Head

14. A free tree is an acyclic, connected, undirected graph. A forest is an acyclic undirected graph. A rooted tree is a tree with one distinguished node, root. Let G = (V, E ) be an undirected graph. The following statements are equivalent.1. G is a tree2. Any two vertices in G are connected by unique simple path.3. G is connected, but if any edge is removed from E, the resulting graph is disconnected.4. G is connected, and | E | = | V | -15. G is acyclic, and | E | = | V | -16. G is acyclic, but if any edge is added to E, the resulting graph contains a cycle. Cutler/Head

15. Graph ADT • What kinds of methods should be supplied? • The next slides assume an undirected graph • For simplicity the implementation of the interface in the next slide assumes that the vertices of a graph are numbered 0,…,numVertices-1. Cutler/Head

16. A possible interface for edges public interface Edge { public int getFrom ();//returns first node public int getTo (); //returns second node public Object getInfo ();//returns the data } Cutler/Head

17. A possible interface for undirected graphs public interface Graph { public int numberOfVertices(); public void addUEdge (int to, int from); public void addUEdge (int to, int from, Object info); public Iterator getAdjacent(int vertex); public Edge getEdge (int v, int w); public boolean isEdge(int v, int w); public Iterator getEdges (); public void addVertex (); } Cutler/Head

18. Implementation of a Graph. • Adjacency-list representation of a graph G = ( V, E ) consists of an array ADJ of |V | lists, one for each vertex in V. For each uV , ADJ [ u ] points to all its adjacent vertices. 2 5 1 1 2 2 1 5 3 4 3 3 2 4 5 4 4 2 5 3 5 4 1 2 Cutler/Head

19. Adjacency-list representation for a directed graph. 2 5 1 1 2 2 5 3 4 3 3 4 5 4 4 5 5 5 Variation: Can keep a second list of edges coming into a vertex. Cutler/Head

20. Representing graph as an adjacency list protected ArrayList adjacencyList; private int numVertices = 0; //constructor public GraphAdjLists (int size) { numVertices = size; //creates a graph with n nodes, no edges adjacencyList= new ArrayList(size); for (int i = 0; i < numVertices; ++i) { adjacencyList.add(new LinkedList()); } } ArrayList and LinkedList are the Java implementations of a growable array, and a linked list Cutler/Head

21. Implementing a graph as an adjacency list public void addUEdge (int i, int j) { ((LinkedList)adjacencyList.get(i)).add(new Integer(j)); if (i!=j){ ((LinkedList)adjacencyList.get(j)).add(new Integer(i)); } } public Iterator getAdjacent(int i) { return ((LinkedList)adjacencyList.get(i)).iterator(); } Cutler/Head

22. Adjacency lists • Advantage: • Saves space for sparse graphs. Most graphs are sparse. • “Visit” edges that start at v • Must traverse linked list of v • Size of linked list of v is degree(v) • (degree(v)) • Disadvantage: • Check for existence of an edge (v, u) • Must traverse linked list of v • Size of linked list of v is degree(v) • (degree(v)) Cutler/Head

23. Adjacency List • Storage • We need V pointers to linked lists • For a directed graph the number of nodes (or edges) contained (referenced) in all the linked lists is (out-degree (v)) = | E |. So we need ( V + E ) • For an undirected graph the number of nodes is(degree (v)) = 2 | E | Also ( V + E ) v V v V Cutler/Head

24. Adjacency-matrix-representation of a graph G = ( V, E) is a |V | x |V | matrix A = ( aij ) such that aij = 1 (or some Object) if (i, j ) E and 0 (or null) otherwise. 0 1 2 3 4 0 1 2 0 1 0 0 1 0 1 2 3 4 1 0 1 1 1 4 3 0 1 0 1 0 0 1 1 0 1 1 1 0 1 0 Cutler/Head

25. Adjacency Matrix Representation for a Directed Graph 0 1 2 3 4 0 1 0 0 1 0 1 0 1 2 3 4 0 0 1 1 1 2 0 0 0 1 0 4 0 0 0 0 1 3 0 0 0 0 0 Cutler/Head

26. Implementation of graph in an adjacency matrix protected boolean[ ][ ] adjacent; private int numVertices; public GraphMatrix (int size) { numVertices = size; adjacent = new boolean [size][size]; initialize matrix to false } public void addUEdge (int i, int j) { adjacent [i] [j]=true; adjacent [j] [i]=true; } Cutler/Head

27. Adjacency Matrix Representation • Advantage: • Saves space on pointers for dense graphs, and on • small unweighted graphs using 1 bit per edge. • Check for existence of an edge (v, u) • (adjacency [i] [j]) == true?) • So (1) • Disadvantage: • “visit” all the edges that start at v • Row v of the matrix must be traversed. • So (|V|). Cutler/Head

28. Adjacency Matrix Representation • Storage • ( | V |2) ( We usually just write, (V2) ) • For undirected graphs you can save storage (only 1/2(V2)) by noticing the adjacency matrix of an undirected graph is symmetric. • Need to update code to work with new representation. • Gain in space is offset by increase in the time required by the methods. Cutler/Head