1. 1 Graphs ORD DFW SFO LAX 802 1743 1843 1233 337 Many slides taken from Goodrich, Tamassia 2004

2. © 2004 Goodrich, Tamassia 2 Graphs A graph is a pair (V, E), where V is a set of nodes, called vertices (node=vertex) E is a collection of pairs of vertices, called edges Vertices and edges are positions and store elements Example: A vertex represents an airport and stores the three-letter airport code An edge represents a flight route between two airports and stores the mileage of the route ORD PVD MIA DFW SFO LAX LGA HNL 849 802 1387 1743 1843 1099 1120 1233 337 2555 142

3. © 2004 Goodrich, Tamassia 3 Edge Types Directed edge ordered pair of vertices (u,v) first vertex u is the origin second vertex v is the destination e.g., a flight Undirected edge unordered pair of vertices (u,v) e.g., a flight route Directed graph all the edges are directed e.g., route network Undirected graph all the edges are undirected e.g., flight network ORDPVD flight AA 1206 ORDPVD 849 miles

4. © 2004 Goodrich, Tamassia 4 Applications Electronic circuits Printed circuit board Integrated circuit Transportation networks Highway network Flight network Computer networks Local area network Internet Web Databases Entity-relationship diagram

5. © 2004 Goodrich, Tamassia 5 Terminology End vertices (or endpoints) of an edge U and V are the endpoints of a Edges incident on a vertex a, d, and b are incident on V Adjacent vertices U and V are adjacent Degree of a vertex X has degree 5 Parallel edges h and i are parallel edges Self-loop j is a self-loop XU V W Z Y a c b e d f g h i j

6. © 2004 Goodrich, Tamassia 6 P1P1 Terminology (cont.) Path sequence of alternating vertices and edges begins with a vertex ends with a vertex each edge is preceded and followed by its endpoints Simple path path such that all its vertices and edges are distinct Examples P 1 =(V,b,X,h,Z) is a simple path P 2 =(U,c,W,e,X,g,Y,f,W,d,V) is a path that is not simple XU V W Z Y a c b e d f g hP2P2

7. © 2004 Goodrich, Tamassia 7 Terminology (cont.) Cycle circular sequence of alternating vertices and edges each edge is preceded and followed by its endpoints Simple cycle cycle such that all its vertices and edges are distinct Examples C 1 =(V,b,X,g,Y,f,W,c,U,a,  ) is a simple cycle C 2 =(U,c,W,e,X,g,Y,f,W,d,V,a,  ) is a cycle that is not simple C1C1 XU V W Z Y a c b e d f g hC2C2

8. © 2004 Goodrich, Tamassia 8 Properties Notation n number of vertices m number of edges deg(v) degree of vertex v Property 1  v deg(v)  2m Proof: each edge is counted twice Property 2 In an undirected graph with no self-loops and no multiple edges m  n (n  1)  2 Proof: each vertex has degree at most (n  1) What is the bound for a directed graph? Example n  4 m  6 deg(v)  3

9. © 2004 Goodrich, Tamassia 9 Main Methods of the Graph ADT Vertices and edges are positions store elements Accessor methods endVertices(e): an array of the two endvertices of e opposite(v, e): the vertex opposite of v on e areAdjacent(v, w): true iff v and w are adjacent replace(v, x): replace element at vertex v with x replace(e, x): replace element at edge e with x Update methods insertVertex(o): insert a vertex storing element o insertEdge(v, w, o): insert an edge (v,w) storing element o removeVertex(v): remove vertex v (and its incident edges) removeEdge(e): remove edge e Iterator methods incidentEdges(v): edges incident to v vertices(): all vertices in the graph edges(): all edges in the graph

10. © 2004 Goodrich, Tamassia 10 Implementation of Graphs Vertex adjacency lists Adjacency matrix

11. © 2004 Goodrich, Tamassia 11 Example problem Store the airport flight graph Given an airport, output all airports that have direct flights Given an airport, output all airports that have flights with at most 1 connection Given two airports, find minimum number of hops needed to connect them

12. 12 Shortest Paths C B A E D F 0 328 58 4 8 71 25 2 39

13. © 2004 Goodrich, Tamassia 13 Weighted Graphs In a weighted graph, each edge has an associated numerical value, called the weight of the edge Edge weights may represent, distances, costs, etc. Example: In a flight route graph, the weight of an edge represents the distance in miles between the endpoint airports ORD PVD MIA DFW SFO LAX LGA HNL 849 802 1387 1743 1843 1099 1120 1233 337 2555 142 1205

14. © 2004 Goodrich, Tamassia 14 Shortest Paths Given a weighted graph and two vertices u and v, we want to find a path of minimum total weight between u and v. Length of a path is the sum of the weights of its edges. Example: Shortest path between Providence and Honolulu Applications Internet packet routing Flight reservations Driving directions ORD PVD MIA DFW SFO LAX LGA HNL 849 802 1387 1743 1843 1099 1120 1233 337 2555 142 1205

15. © 2004 Goodrich, Tamassia 15 Shortest Path Properties Property 1: A subpath of a shortest path is itself a shortest path Property 2: There is a tree of shortest paths from a start vertex to all the other vertices Example: Tree of shortest paths from Providence ORD PVD MIA DFW SFO LAX LGA HNL 849 802 1387 1743 1843 1099 1120 1233 337 2555 142 1205

16. © 2004 Goodrich, Tamassia 16 Dijkstra’s Algorithm The distance of a vertex v from a vertex s is the length of a shortest path between s and v Dijkstra’s algorithm computes the distances of all the vertices from a given start vertex s Assumptions: the graph is connected the edges are undirected the edge weights are nonnegative We grow a “cloud” of vertices, beginning with s and eventually covering all the vertices We store with each vertex v a label d(v) representing the distance of v from s in the subgraph consisting of the cloud and its adjacent vertices At each step We add to the cloud the vertex u outside the cloud with the smallest distance label, d(u) We update the labels of the vertices adjacent to u

17. © 2004 Goodrich, Tamassia 17 Edge Relaxation Consider an edge e  (u,z) such that u is the vertex most recently added to the cloud z is not in the cloud The relaxation of edge e updates distance d(z) as follows: d(z)  min{d(z),d(u)  weight(e)} d(z)  75 d(u)  50 10 z s u d(z)  60 d(u)  50 10 z s u e e

18. © 2004 Goodrich, Tamassia 18 Example CB A E D F 0 428  4 8 71 25 2 39 C B A E D F 0 328 511 4 8 71 25 2 39 C B A E D F 0 328 58 4 8 71 25 2 39 C B A E D F 0 327 58 4 8 71 25 2 39

19. © 2004 Goodrich, Tamassia 19 Example (cont.) CB A E D F 0 327 58 4 8 71 25 2 39 CB A E D F 0 327 58 4 8 71 25 2 39

20. © 2004 Goodrich, Tamassia 20 Dijkstra’s Algorithm A priority queue stores the vertices outside the cloud: Key: distance, Element: vertex n nlogn (priority heap queue) n nlogn (priority heap queue) Sum n (deg(n))=2m 2m 2mlogn (priority heap queue) Algorithm DijkstraDistances(G, s) Q  new heap-based priority queue for all v  G.vertices() if v  s setDistance(v, 0) else setDistance(v,  ) Q.insert(getDistance(v), v) while  Q.isEmpty() u  Q.removeMin() for all e= (z,u)  G.incidentEdges(u) /* relax edge e */ r  getDistance(u)  weight(e) if r  getDistance(z) setDistance(z,r) Recompute position of z in Q O(mlogn)

21. © 2004 Goodrich, Tamassia 21 Shortest Paths Tree Using the template method pattern, we can extend Dijkstra’s algorithm to return a tree of shortest paths from the start vertex to all other vertices We store with each vertex a third label: parent edge in the shortest path tree In the edge relaxation step, we update the parent label Algorithm DijkstraShortestPathsTree(G, s) … for all v  G.vertices() … setParent(v,  ) … for all e  G.incidentEdges(u) { relax edge e } z  G.opposite(u,e) r  getDistance(u)  weight(e) if r  getDistance(z) setDistance(z,r) setParent(z,e) Q.replaceKey(getLocator(z),r)

22. © 2004 Goodrich, Tamassia 22 Why It Doesn’t Work for Negative-Weight Edges If a node with a negative incident edge were to be added late to the cloud, it could mess up distances for vertices already in the cloud. CB A E D F 0 457 59 4 8 71 25 6 0-8 Dijkstra’s algorithm is based on the greedy method. It adds vertices by increasing distance. C’s true distance is 1, but it is already in the cloud with d(C)=5!

23. © 2004 Goodrich, Tamassia 23 Other Graph issues Traversal depth-first Breadth-first Very hard (NP-complete) problems on graphs Coloring  Given a graph, can we assign one of three colors (say red, blue, or green) to each vertex, such that no adjacent vertices have the same color? Traveling salesman (Hamiltonian cycle)  path that travels through every vertex once, and winds up where it started