Part 10. Graphs

1. Part 10. Graphs CS 200 Algorithms and Data Structures

2. Outline • Introduction • Terminology • Shortest Paths • Implementing Graphs • Graph Traversals • Topological Sorting • Spanning Trees • Minimum Spanning Trees • Circuits

3. Graphs What can this represent? • A computer network • Abstraction of a map • Social network A collection of nodes and edges

4. Directed Graphs A collection of nodes and directed edges Sometimes we want to represent directionality: • Unidirectional network connections • One way streets • The web

6. Example • “Follow the money” example (the international science and engineering visualization challenge, 2009, Science magazine and National Science Foundation) • http://www.sciencemag.org/site/special/vis2009/show/ • Reference The Scaling Laws of Human Travel, D. Brockmann, L. Hufnagel and T. Geisel, Nature 439, 462 (2006) • http://rocs.northwestern.edu/index_assets/brockmann2006nature.pdf

7. Outline • Introduction • Terminology • Implementing Graphs • Shortest Paths • Properties • Graph Traversals • Topological Sorting • Spanning Trees • Minimum Spanning Trees • Circuits

8. Edges Vertices/ Nodes Graph Terminology G=(V, E) Two vertices are adjacent if they are connected by an edge. An edge is incident on two vertices Degree of a vertex: number of edges incident on it Vertices Edges u e v Graph terminology: 14.1 in W&M, 9.1 in Rosen

9. subgraph Graph Terminology A subgraphof a graph G = (V,E) is a graph (V’,E’) such that V’ is a subset of V and, E’ is a subset of E

10. Paths • Path: a sequence of edges • (e1, e2, e3) is a path of length 3 from v1 to v4 • In a simple graph a path can be represented as a sequence of vertices v1 e1 v2 v4 e2 e3 v3

11. Graph Terminology Self loop (loop): an edge that connects a vertex to itself Simple graph: no self loops and no two edges connect the same vertices Multigraph: may have multiple edges connecting the same vertices Pseudograph: multigraph with self-loops

12. Directed Graphs Indegree: number of incoming edges Outdegree: number of outgoing edges w v

13. a b c d e f g Connected Components • An undirected graph is called connected if there is a path between every pair of vertices of the graph. • A connected component of a graph G is a connected subgraph of G that is not a proper subgraph of another connected subgraph of G. G={{a,b,c,d,e,f,g},E} G1={{a,b,c},E1} G2={{d,e,f,g}, E2}

14. The degree of a vertex • The degree of a vertex in an undirected graph • the number of edges incident with it • except that a loop at a vertex contributes twice to the degree of that vertex.

15. Example d c b a g e f deg(a) = 2 deg(b) = deg(f) = 4 deg(d) = 1 deg(e) = 3 deg(g) = 0

16. Complete Graphs • Simple graph that contains exactly one edge between each pair of distinct vertices. Complete Graph

17. Cycles The cycle Cn, n≥ 3, consists of n vertices v1, v2, …, vnand edges {v1, v2}, {v2, v3},…., and {vn-1, vn}.

18. Wheels • We obtain the wheel Wn when we add an additional vertex to the cycle Cn, for n≤ 3, and connect this new vertex to each of the nvertices in Cn, by new edges

19. n-Cubes (n-dimensional hypercube) Hypercube 110 111 101 100 001 010 011 000

20. Bipartite Graphs • A simple graph G is called bipartite, if its vertex set V can be partitioned into two disjoint sets V1 and V2 such that every edge in the graph connects a vertex in V1 and a vertex in V2.

21. Outline • Introduction • Terminology • Implementing Graphs • Shortest Paths • Properties • Graph Traversals • Topological Sorting • Spanning Trees • Minimum Spanning Trees • Circuits

22. Shortest Path Algorithms (Dijkstra’s Algorithm) • Graph G(V,E) with non-negative weights (“distances”) • Compute shortest distances from vertex s to every other vertex in the graph

23. Shortest Path Algorithms (Dijkstra’s Algorithm) • Algorithm • Maintain array d (minimum distance estimates) • Init: d[s]=0, d[v]=vV-s • Priority queue of vertices not yet visited • select minimum distance vertex, visit v, update neighbors

24. Shortest Path Algorithms (Dijkstra’s Algorithm) 1 e b 10 3 2 9 a 4 6 5 7 c d 2

25. Shortest Path Algorithms (Dijkstra’s Algorithm): Initialize a bcde 0 ∞ ∞ ∞ ∞ e b 8 1 8 10 3 2 9 0 a 4 6 5 7 8 8 2 d c

26. Shortest Path Algorithms (Dijkstra’s Algorithm): step 1 a bcde 0 10/a 5/a -- -- a bcde 0 10/a 5/a -- -- e b 1 8 10 a bcde 0 ∞ ∞ ∞ ∞ 10 3 2 9 0 a 4 6 5 7 5 8 2 d c

27. Shortest Path Algorithms (Dijkstra’s Algorithm): step 2 a bcde 0 10/a 5/a -- -- e b 0 8/c 5/a 7/c 14/c 0 8/c 5/a 7/c 14/c 1 14 8 10 3 2 9 0 a 4 6 5 7 5 7 2 d c

28. Shortest Path Algorithms (Dijkstra’s Algorithm): step 3 a bcde 0 10/a 5/a -- -- e b 0 8/c 5/a 7/c 14/c 1 11 8 0 8/c 5/a 7/c 11/d 08/c5/a 7/c 11/d 10 3 2 9 0 a 4 6 5 7 5 7 2 d c

29. Shortest Path Algorithms (Dijkstra’s Algorithm): step 4 a bcde 0 10/a 5/a -- -- e b 0 8/c 5/a 7/c 14/c 1 9 8 08/c5/a 7/c 11/d 08/c5/a 7/c9/d 10 3 2 9 0 a 4 6 5 7 5 7 2 d c

30. Shortest Path Algorithms (Dijkstra’s Algorithm): Done a bcde 08/c5/a 7/c9/b e b 1 9 8 10 3 2 9 0 a 4 6 5 7 5 7 2 Minimum distance from a to b : 8 Minimum distance from a to c: 5 Minimum distance from a to d: 7 Minimum distance from a to e: 9 d c

31. Example

32. Dijkstra’s Algorithm Dijkstra(G: graph with vertices v0…vn-1and weights w[u][v]) // computes shortest distance of vertex 0 to every other vertex create a set vertexSet that contains only vertex 0 d[0] = 0 for (v = 1 through n-1) d[v] = infinity for (step = 2 through n) find the smallestd[v] such that v is not in vertexSet add v to vertexSet for (all vertices u not in vertexSet) if (d[u] >d[v] + w[v][u]) d[u] =d[v] + w[v][u]

33. Shortest Path Algorithms Using a Priority Queue (Dijkstra’s Algorithm): step 1 e b 1 8 10 10 3 2 9 0 a 4 6 5 7 5 8 2 d c [5,c],[10,b]

34. Shortest Path Algorithms (Dijkstra’s Algorithm): step 2 e b 1 14 8 10 3 2 9 0 a 4 6 5 7 5 7 2 d c [7,d],[8,b],[14,e]

35. Shortest Path Algorithms (Dijkstra’s Algorithm): step 3 e b 1 11 8 10 3 2 9 0 a 4 6 5 7 5 7 2 d c [8,b],[11,e]

36. Shortest Path Algorithms (Dijkstra’s Algorithm): step 4 c b 1 9 8 10 3 2 9 0 a 4 6 5 7 5 7 2 e d [9,c]

37. Shortest Path Algorithms (Dijkstra’s Algorithm): Done c b 1 9 8 10 3 2 9 0 a 4 6 5 7 5 7 2 e d

38. Dijkstra’s Algorithm • How to obtain the shortest paths? • At each vertex maintain a pointer that tells you the vertex from which you arrived.

39. Outline • Introduction • Terminology • Implementing Graphs • Shortest Paths • Properties • Graph Traversals • Topological Sorting • Spanning Trees • Minimum Spanning Trees • Circuits

40. Graph ADT • Create • Empty? • Number of vertices? • Number of edges? • Edge exists between two vertices? • Add a vertex • Add an edge • Delete a vertex (and any edges adjacent to it) • Delete an edge • Retrieve a vertex

41. Classes for a Weighted UndirectedGraph • Vertex:??? • Edge: ??? • Graph: • organized collection of vertices and edges • Adjacency Matrix Implementation • Adjacency List Implementation

42. A B C D E Adjacency Matrix Implementation • Edges • square matrix of edges Vertices Adjacency Matrix: indicates whether an edge exists between two vertices mapping of vertex labels to array indices Possible Java implementation: int [][] A;

43. Adjacency Matrix Implementation • Directed graph: A[i][j] is 1 if there is an edge from vertex i to vertex j • Undirected graph: A[i][j] is 1 if there is an edge between vertex i and vertex j. What’s the relationship between A[i][j] and A[j][i]? • Weighted graph: A[i][j] gives the weight of the edge

44. d e f g Example • Adjacency matrix?

45. A B C D E Adjacency List Implementation Each vertex has a list of outgoing edges B D E A B C mapping of vertex labels to list of edges

46. Adjacency List Implementation • For undirected graphs each edge appears twice. Why?

47. Which Implementation? • Which implementation best supports common Graph Operations: • Is there an edge between vertex i and vertex j? • Find all vertices adjacent to vertex j • Which best uses space?

48. Implementation: Edge Class Class Edge { private Integer v,w; // vertices private int weight; public Edge(Integer first, Integer second, intedgeWeight){ v = first; w = second; weight = edgeWeight; } public intgetWeight() { return weight; } public Integer getV() { return v; } public Integer getW() { return w; }

49. Implementation: Graph Class class Graph { private intnumVertices; private intnumEdges; private Vector<TreeMap<Integer, Integer>> adjList; public Graph(intn) { numVertices = n; numEdges = 0; adjList = new Vector<TreeMap<Integer, Integer>>(); for (inti=0; i<numVertices; i++) { adjList.add(newTreeMap<Integer, Integer>()); } }

50. Implementation: Graph Class public void addEdge(Integerv, Integer, w, int weight){ // precondition: the vertices v and w must exist //postcondition: the edge (v,w) is part of the graph adjList.get(v).put(w, weight); adjList.get(w).put(v, weight); numEdges++; }