1. GRAPHS

2. Definition The Graph Data Structure Drawing of a Graph
set V of vertices
collection E of edges (pairs of vertices in V)
Drawing of a Graph
vertex <-> circle/oval
edge <-> line connecting the vertex pair

3. Sample Uses Airports and flights
Persons and acquaintance relationships
Intersections and streets
Computers and connections; the Internet

4. Graph Example Graph G=(V,E): V={a,b,c,d}, E={(a,b),(b,c),(b,d),(a,d)}

5. Adjacency Vertices connected by an edge are adjacent
An edge e is incident on a vertex v (and vice-versa) if v is an endpoint of e
Degree of a vertex v, deg(v)
number of edges incident on v
if directed graph, indeg(v) and outdeg(v)

6. Graph Properties Graph G with n vertices and m edges
v in G deg(v) = 2m
v in G indeg(v) = v in G outdeg(v) = m
m is O(n2)
m <= n(n-1)/2 if G is undirected
m <= n(n-1) if G is directed

7. Subgraphs Graph H a subgraph of graph G Spanning subgraph
vertex set of H is a subset of vertex set of G
edge set of H is a subset of edge set of G
Spanning subgraph
vertex set of H identical to vertex set of G

8. Paths and Cycles (simple) Path (simple) Cycle
sequence of distinct vertices such that consecutive vertices are connected through edges
(simple) Cycle
same as path except that first and last vertices are identical
Directed paths and cycles
edge directions matter

9. Connectivity
Connected graph
there is a path between any two vertices

10. Graph Methods Container methods General or Global methods
numVertices(),numEdges(),vertices(),edges()
Directed Edge methods
indegree(v), inadjacentVertices(v), inincidentEdges(v)
Update methods
adding/removing vertices or edges

11. General Methods numvertices() -returns the number of vertices in G
numedges() - returns the number of edges in G
vertices() - return an enumeration of the vertex positions in G
edges() - returns an enumeration of the edge positions in G

12. General Methods Degree (v) - returns the degree of v
adjacentvertices(v) - returns an enumeration of the vertices adjacent to v
incidentedges(v) - returns an enumeration of the edges incident upon v
endvertices(e) - return an array of size 2 storing the end vertices of e

13. Directed Edges
Directededges() - return an enumeration of all directed edges
indegree(v) - return the indegree of v
outdegree(v) - return the outdegree of v
origin (v) - return the origin of directed edge e
other functions - inadjacentvertices(), outadjacentvertices(), destination()

14. Updating Graphs
Insertedge(v,w,o) - insert and return an undirected edge between vertices v and w storing the object o at this position
Insertvertex(o) - insert and return a new vertex storing the object o at this position
removevertex (v) - remove the vertex v and all incident edges
removeedge(e) - remove edge e

15. Implementation of a graph data structure

16. Graph Implementations
Edge List
vertices and edges are nodes
edge nodes refer to incident vertex nodes
Adjacency List
same as edge list but vertex nodes also refer to incident edge nodes
Adjacency Matrix
2D matrix of vertices; entries are edges

17. Implementation Examples
Illustrate the following graph using the different implementations 4 a b 5 1 3 c d 2

18. Edge List

19. Adjacency List

20. Adjacency Matrix

21. Implementation Examples- Query
Illustrate the following graph using the different implementations 4 a b 8 5 e 1 3 6 c 7 d 2

22. Comparing Implementations
Edge List
simplest but most inefficient
Adjacency List
many vertex operations are O(d) time where d is the degree of the vertex
Adjacency Matrix
adjacency test is O(1)
addition/removal of a vertex: resize 2D array

23. Graph Traversals
Traversal - A systematic procedure for exploring a graph by examining all of its vertices and edges
Consistency - rules to determine the order of visits (top-down, alphabetical,etc.) - make the path consistent with the actual graph (don’t change conventions)

24. Graph Traversals Depth First Search (DFS) Breadth First Search (BFS)
visit a starting vertex s
recursively visit unvisited adjacent vertices
Breadth First Search (BFS)
visit starting vertex s
visit unvisited vertices adjacent to s, then visit unvisited vertices adjacent to them, and so on … (visit by levels)

25. DFS Traversal
Depth First Search - recursively visit (traverse) unvisited vertices
Dead End - reaching a vertex where all the incident edges go to a previously visited vertex
Back Track - If a dead-end vertex is reached, go to the previous vertex

26. Traversal Examples DFS: a,b,c,d,h,e,f,g,l,k,j,i c b d a e g f h i j l

27. DFS Traversal
Path of the DFS traversal - path=a,b,c,d,h - h is a dead end vertex - back track to d,c,b,a and go to another adjacent (unvisited) vertex -> e - path=a,e,f,g,k,j,i - i is a dead end vertex - back track to all the way back to a (also a dead end vertex)
finished

28. BFS Traversal Visit adjacent vertices by levels
discovery edges - edges that lead to an unvisited vertex
cross edge - edges that lead to a previously visited vertex

29. Traversal Examples BFS: a,b,e,i,c,f,j,d,g,k,h,l c b d a e g f h i j l

30. DFS Algorithm Algorithm DFS(v) Input: Vertex v
Output: Depends on the application
mark v as visited; perform processing
for each vertex w adjacent to v do
if w is unmarked then
DFS(w)

31. BFS Algorithm Algorithm BFS(s) Input: Starting vertex s
Output: Depends on the application
// traversal algorithm uses a queue Q
// loop terminates when Q is empty

32. BFS Algorithm continued
Q  new Queue()
mark s as visited and Q.enqueue(s)
while not Q.isEmpty()
v  Q.dequeue()
perform processing on v
for each vertex w adjacent to v do
if w is unmarked then
mark w as visited and Q.enqueue(w)

33. Query
Let G be a graph whose vertices are the integers 1 through 8 and let the adjacent vertices of each vertex be given by the table below : Vertex Adjacent Vertices 1 (2,3,4) 2 (1,3,4) 3 (1,2,4) 4 (1,2,3,6) 5 (6,7,8) 6 (4,5,7) 7 (5,6,8) 8 (5,7)

34. Query
Assume that in the traversal of G, the adjacent vertices of a given vertex are returned in the same order as they are listed in the above table:
a) Draw G
b) Give the sequence of vertices of G visited using a DFS traversal order starting at vertex 1
c) Give the sequence of vertices visited using a BFS traversal starting at vertex 1

35. Weighted Graphs Overview

36. Weighted Graphs
Graph G = (V,E) such that there are weights/costs associated with each edge
w((a,b)): cost of edge (a,b)
representation: matrix entries <-> costs a 20 35 e b 12 32 c 65 d

37. Problems on Weighted Graphs
Minimum-cost Spanning Tree
Given a weighted graph G, determine a spanning tree with minimum total edge cost
Single-source shortest paths
Given a weighted graph G and a source vertex v in G, determine the shortest paths from v to all other vertices in G
Path length: sum of all edges in path

38. Weighted Graphs-Concerns
Minimum Cost
Shortest Path

39. Weighted Graphs Shortest Path - Dijkstra’s Algorithm
Minimum Spanning Trees - Kruskal’s Algorithm