Graphs

1. Graphs

2. Graph Definition • A graph G=(V,E) consists a set of vertices, V, and edges, E. • Each edge is a pair (v,w) where v,w are elements of V • If the edge pair is ordered, then the graph is directed (digraph) and we use the notation <v,w> • Sometimes edges have a third component called cost

3. Simple Graph • a pair (V,E) • V • finite set of vertices • points • E • irreflexive, symmetric relation on V • edges between vertices

4. Multigraph • A graph than contains • more than one edge • from a vertex to another vertex

5. More Definitions -Adjacency • If the edge (u,v) is in E then u and v are adjacent and the edge (u,v) is incident on u and v • If the edge <u,v> is in E then u is adjacent to v and v is adjacent from u • The number of edges incident on a vertex is the degree of the vertex • Digraph nodes had in-degree and out-degree

6. Adjacent, Incident • Vertices u and v in undirected graph G are called adjacent if {u,v} is an edge of G • If edge e = {u,v}, e is called incident with vertices u and v • Edge econnectsu and v

7. Degree of a vertex • the number of edges incident to (touching) the vertex deg(A) = 5

8. Directed Edges • A directed edge (u,v) • shows a path from point u to point v • includes an arrow • u is called the initial vertex of (u,v) • v is called the terminal vertex of (u,v)

9. In-Degree, Out-Degree • deg-(v) • the in-degree of vertex v • the number of edges with v as terminal vertex • deg+(v) • the out-degree of vertex v • the number of edges with v as initial vertex

10. More Definitions -Paths • A path is a sequence of vertices, v1, v2,..., vn such that (vi, vi+1) is an element of E for 1<=i<=n • The length of the path is the number of edges in the path (n-1) • A simple path is a path with all distinct vertices (except perhaps the first and last)

11. Number of Edges in a Graph • The maximum number of edges in a graph is n(n-1)/2 (where n is number of vertices) • The maximum number of edges in a digraph is n(n-1) • A graph with the maximum number of edges is a complete graph • A graph with few edges is sparse • A nearly complete graph is dense

12. Complete Graph • A complete graph on n vertices: Kn • a simple graph • containing exactly one edge • between each pair of distinct vertices

13. Directed graph • a pair (V,E) • V • finite set of vertices (points) • E is an relation on V • For any pair u,v in V, • E may contain one edge (u,v) from u to v, • and (v,u) another edge from v to u.

14. Example Digraph A B D E C F G

15. Example Digraph • V = {A, B, C, D, E, F, G} • E = {<A,F>,<A,B>,<C,A>,<F,C>,<B,F>, <G,B>,<B,D>,<D,E>,<E,G>} • (A,B,D,E,G) is a path of length 4 • (A,F,C,A) is a simple cycle • (A,B,D,E,G,B,F,C,A) is a cycle (not simple)

16. More Definitions -Cycles • A cycle is a path of length at least one where the starting and ending vertices are the same • A cycle is a simple cycle if the path is a simple path • An acyclic graph is a graph with no cycles • A directed acyclic graph is sometimes called a DAG

17. Cycles Cn • A cycle n vertices (n>=3) • {v1, v2, v3, … vn} • edges between {v1,v2}, {v2,v3}, …, {vn-1, vn}, {vn, v1}

18. Bipartite Graph • A simple graph is bipartite • vertex set V can be partitioned into disjoint sets V1 and V2 • every edge in the graph connects a vertex in V1 and a vertex in V2

19. Subgraph • A subgraph of graph G = (V,E) • is a graph H = (W,F) • where WÍV and FÍE

20. Graph Union • The union of two simple graphs • G1 = (V1, E1) and G2 = (V2, E2) • G1È G2 • is a simple graph • with vertex set: V1È V2 • and edge set: E1È E2

21. Graphs Representations

22. Adjacency Matrix AG • A graph representation • A n´n zero-one matrix • A = [aij]

23. Graph Representations - Adjacency Matrix • If G=(V,E) is a graph with n vertices, then G can be represented by a nxn, say A, where A[i,j]=1 iff (i,j) is in E (<i,j> for a digraph), and A[i,j]=0 if there is no such edge in E • Space requirements are n2 • O(1) to determine if (i,j) is in E • O(n2) to initialize, determine number of edges, etc.

24. Adjacency Matrix Example A B C D E F G

25. Adjacency List • A graph representation • which specifies all vertices that are adjacent • to each vertex of the graph

26. Graph Representations -Adjacency Lists • If |E| << n2 then some operations can be more efficient if we use adjacency lists • For each vertex we keep a list of adjacent vertices • O(|E|) +O(|V|) space requirement • O(|E|) creation, scanning all edges, etc.

27. B F D F A E G F C G B Adjacency List Example A B C D E

28. Incidence Matrix AG • A graph representation • A n´m zero-one matrix • n vertices • m edges • M = [mij]

29. Incidence Matrix AG

30. Connectivity

31. Path of length n from u to v(undirected graph) • a sequence of edges e1, e2, … en • such that • f (e1) = {x0, x1} • f (e2) = {x1, x2} • f (en) = {xn-1, xn} • where x0 = u & xn = v • The path is denoted by its vertex sequence • {x1, x2, …, xn}

32. Path Examples • a,d,c,f,e • a simple path of length 4 • d,e,c,a • not a path • b,c,f,e,b • a circuit of length 4 • a,b,e,d,a,b • path, but not simple

33. Circuit (cycle) • a path in a graph in which • first and last vertices • are the same vertex • Simple Circuit • no edges are repeated

34. Connected graph • A graph in which there is • a path between every pair of distinct vertices

35. Connected Components • A graph not connected • is the union of two or more connected subgraphs • called connected components

36. Cut Vertices / Cut Edges • Cut Vertices • A vertex which when removed (with edges) • produces a subgraph with more connected components than in the original graph

37. Strongly Connected DiGraph • a directed graph in which • there exists a path from a to b • and a path from b to a • whenever a and b are vertices in the graph

38. Weakly Connected Digraph • a directed graph which contains a path • between any two vertices • in the underlying undirected graph

39. Shortest Path Algorithms • Algorithms exist to find the shortest path between two vertices (Dijkstra)

40. Planar graph • A graph drawn without any edges crossing • crossing of edges • lines intersect at some point • other than the common endpoint

41. Breadth-First Graph Processing • Put the starting node on a queue of nodes to be processed • Remove the front node from the queues and process each of its adjacent nodes, putting them on the queue if they have not been visited before • Continue processing until no nodes remain on the queue • O(|E|)

42. Depth-First Graph Processing • Start with a starting node • Visit each of the nodes adjacent to the current node, calling the depth-first search on each that has not been visited • O(|E|)

43. Shortest Unweighted Path • Performs a breadth-first processing of the nodes, assigning each of them a cost which is the length of the path from the starting node to the node • Uses a Queue of vertices to be examined • Picking a vertex is O(|V|) for whole algorithm • Each edge is examined once during the algorithm so O(|E|) - dominate factor

44. 0 A B D E C F G 0 1 2 A B D E C F G 1 Shortest Unweighted Path Example 0 1 A B D E C F G 1 0 1 2 A B D E C 2 F G 1

45. Shortest Unweighted Path Example (cont) 0 1 0 1 2 2 A A B D B D E C E C 3 3 2 2 F G F G 1 1 0 1 2 0 1 A 2 B D A B D E C 3 E C 3 2 2 F G F G 1 4 1 4

46. Weighted Shortest Path • In unweighted shortest path, each node has its distance set only once • If edges have positive weights, then may need to reset distances more than once

47. Example of Cost Resetting 3 v 12 w 7 0 Start 8 w’s cost was set to 12 when node u visited. When node v is visited needs to be reset u 4

48. Weighted Shortest Path Algorithm (Dijkstra’s) • Label the starting vertex with a cost of zero and pick as the current vertex • Label each of the vertex adjacent to current vertex with cost of current vertex plus cost of edge to the vertex • Pick the vertex with the smallest cost as the new current vertex • Continue 1 & 2 until no vertices remainhttp://carnap.ss.uci.edu/java/dijkstra/DijkstraApplet.html

49. Dijkstra’s Algorithm Requirements • At each step we want vertex with smallest cost as next current vertex • If we scan table each time, this requires O(|V|2) operations • Use a priority queue which has nlogn complexity • Actual implementation permits duplicate vertices in priority queue

50. 0 3 3 1 A B D 1 2 8 1 E 3 C 5 4 F G 3 8 0 3 4 0 3 4 5 6 6 Example of Dijkstra’s Algorithm 0 3 1 A B D 1 2 8 1 E 3 C 5 4 F G 3 3 1 A 3 1 B D A B D 1 1 2 8 2 8 1 E 3 1 C E 3 C 5 4 5 4 F G 3 F G 3 1