1. Part A - Terminology and Traversals
Chapter 15
Graphs
Modified
Part A - Terminology and Traversals

2. Chapter Scope Directed and undirected graphs
Weighted graphs (networks)
Common graph algorithms
Java Software Structures, 4th Edition, Lewis/Chase

3. Terminology and Traversals
Java Software Structures, 4th Edition, Lewis/Chase

4. Graphs
Like trees, graphs are made up of nodes and the connections between those nodes.
In graph terminology, we refer to the nodes as vertices and refer to the connections among them as edges
Vertices are typically referenced by a name or label
Edges are referenced by a pairing of the vertices (A, B) that they connect
An undirected graph is a graph where the pairings representing the edges are unordered (order doesn’t matter)
Java Software Structures, 4th Edition, Lewis/Chase

5. Undirected Graphs An undirected graph:
Java Software Structures, 4th Edition, Lewis/Chase

6. Undirected Graphs
An edge in an undirected graph can be traversed in either direction
Two vertices are said to be adjacent if there is an edge connecting them
Adjacent vertices are sometimes referred to as neighbors
An edge of a graph that connects a vertex to itself is called a self-loop or sling
An undirected graph is considered complete if it has the maximum possible number of edges connecting vertices. How many edges?
Java Software Structures, 4th Edition, Lewis/Chase

7. Undirected Graphs
A path is a sequence of edges that connects two vertices in a graph
The length of a path in is the number of edges in the path (or the number of vertices minus 1)
An undirected graph is considered connected if for any two vertices in the graph there is a path between them
Java Software Structures, 4th Edition, Lewis/Chase

8. Undirected Graphs
An example of an undirected graph that is not connected:
Java Software Structures, 4th Edition, Lewis/Chase

9. Cycles
A cycle is a path in which the first and last vertices are the same and none of the edges are repeated
A graph that has no cycles is called acyclic
Java Software Structures, 4th Edition, Lewis/Chase

10. Directed Graphs
A directed graph, sometimes referred to as a digraph, is a graph where the edges are ordered pairs of vertices (edges have arrows)
This means that the edges (A, B) and (B, A) are separate, directional edges in a directed graph
Java Software Structures, 4th Edition, Lewis/Chase

11. Directed Graphs A directed graph with vertices A, B, C, D
edges (A, B), (A, D), (B, C), (B, D) and (D, C)
Java Software Structures, 4th Edition, Lewis/Chase

12. Directed Graphs
Previous definitions change slightly for directed graphs
a path in a direct graph is a sequence of directed edges that connects two vertices in a graph
a directed graph is connected if for any two vertices in the graph there is a path between them
if a directed graph has no cycles, it is possible to arrange the vertices in a sequence such that vertex A precedes vertex B in the sequence, if an edge exists from A to B
Java Software Structures, 4th Edition, Lewis/Chase

13. Directed Graphs
A connected directed graph and an unconnected directed graph:
Java Software Structures, 4th Edition, Lewis/Chase

14. A tree as a graph A tree is an undirected acyclic graph.
A rooted tree has one vertex designated as the root.
A directed tree is a directed graph with edges directed away from the root, and which would be a tree if the directions on the edges were ignored

15. Weighted Graphs
A weighted graph, sometimes called a network, is a graph with weights (or costs) associated with each edge
The weight of a path in a weighted graph is the sum of the weights of the edges in the path
Weighted graphs may be either undirected or directed
For weighted graphs, we represent each edge with a triple including the starting vertex, ending vertex, and the weight (Boston, New York, 120)
Java Software Structures, 4th Edition, Lewis/Chase

16. Weighted Graphs
We could use an undirected network to represent flights between cities
The weights are the cost of flight
Java Software Structures, 4th Edition, Lewis/Chase

17. Weighted Graphs
A directed version of the graph could show different costs depending on the direction
Java Software Structures, 4th Edition, Lewis/Chase

18. Common Graph Algorithms
There are a number of a common algorithms that may apply to undirected, directed, and/or weighted graphs
These include
various traversal algorithms
algorithms for finding the shortest path length (edge count)
algorithms for finding the least costly path in a network
algorithms for answering simple questions (such as connectivity)
Java Software Structures, 4th Edition, Lewis/Chase

19. Graph Traversals
There are two main types of graph traversal algorithms
breadth-first: behaves much like a level-order traversal of a tree
depth-first: behaves much like the preorder traversal of a tree
One difference from trees: there is no root node present in a graph
Graph traversals may start at any vertex in the graph
Java Software Structures, 4th Edition, Lewis/Chase

20. Graph Traversals
Java Software Structures, 4th Edition, Lewis/Chase

21. BFT starting from vertex v
create a queue Q visit v, mark v as visited and put v into Q while Q is non-empty remove the head element u of Q visit, mark & enqueue all (unvisited) neighbors of u
Java Software Structures, 4th Edition, Lewis/Chase

22. DFT starting from vertex v
create a stack S visit, mark v as visited and push v onto S while S is non-empty peek at the top u of S if u has an (unvisited) neighbor w, visit w, mark w and push it onto S else pop S
Java Software Structures, 4th Edition, Lewis/Chase

23. Part B – Data Structures for Graphs
Chapter 15
Graphs
Modified
Part B – Data Structures for Graphs

24. Basic Data Structures for Graphs
Adjacency lists
Adjacency matrices
Java Software Structures, 4th Edition, Lewis/Chase

25. Java Software Structures, 4th Edition, Lewis/Chase

26. For weighted graph - network
Can add weight field to nodes in adjacency lists.

27. Part C – Graph Algorithms 1
Chapter 15
Graphs
Modified
Part C – Graph Algorithms 1

28. Basic Graph Algorithms
Un-weighted graphs/digraphs
Connectivity
Cycle detection
Shortest path
Spanning tree
Topological sort
Java Software Structures, 4th Edition, Lewis/Chase

29. Basic Graph Algorithms
Weighted graphs/digraphs
Minimum spanning tree
Cheapest path
Traveling salesman problem
Java Software Structures, 4th Edition, Lewis/Chase

30. Dynamic graphs Adding and removing vertices Adding and removing edges
Java Software Structures, 4th Edition, Lewis/Chase

31. Java Software Structures, 4th Edition, Lewis/Chase

32. Connectivity
For a graph (undirected) how can we determine if it is connected?
Answer: Do a depth first traversal from any vertex and see if all of the vertices are reached.
For a digraph, how do can we determine if it is connected?
Answer: Do a depth first traversal from every vertex and see if all other vertices are reached.
Java Software Structures, 4th Edition, Lewis/Chase

33. Java Software Structures, 4th Edition, Lewis/Chase

34. Cycle detection
For a graph, how can we detect if it contains any cycles?
Do a depth first traversal and look for “back edges”, edges from current node to a node already visited.
For a digraph –
Do the same, for every vertex as starting vertex.
Java Software Structures, 4th Edition, Lewis/Chase

35. Finding shortest path (un-weighted)
For a graph, how do we find the shortest path (fewest edges) from vertex A to vertex B?
Do a breadth first traversal starting at A and going until B is reached, recording the path length and predecessor at each node visited.
Need more info stored at each node visited.
For a digraph –
Same
Java Software Structures, 4th Edition, Lewis/Chase

36. Shortest path form A to B
create a queue Q visit A, mark A as visited, set path length=0, set pred=null, and put A into Q repeat until B is visited remove the head element u of Q for each w: (unvisited) neighbors of u visit w, mark w, set path length of w, set pred of w=u, and enqueue w Follow the preds backwards from B to A
Java Software Structures, 4th Edition, Lewis/Chase

37. Java Software Structures, 4th Edition, Lewis/Chase

38. Part D – Graph Algorithms 2
Chapter 15
Graphs
Modified
Part D – Graph Algorithms 2

39. Basic Graph Algorithms
Un-weighted graphs/digraphs
Spanning tree
Topological sort
Java Software Structures, 4th Edition, Lewis/Chase

40. Java Software Structures, 4th Edition, Lewis/Chase

41. Constructing a spanning tree
Assume an undirected graph
A spanning tree is a subgraph of the original graph which includes all of the vertices but only some of the edges.
If the graph has N vertices, the spanning tree will have N-1 edges.
It will have no cycles; it is a tree
It uses the minimum number of edges to connect all of the vertices together -- Unique?
Lots of practical applications
If a network contains any cycles, an edge can be removed and the graph is still connected.
Java Software Structures, 4th Edition, Lewis/Chase

42. Finding a spanning tree
Do a depth first traversal and record all edges traversed to get to a “new” edge.
Java Software Structures, 4th Edition, Lewis/Chase

43. Find Spanning tree – DFT starting from vertex v
create a stack S Initialize spanning tree edge list to empty mark v as visited and push v onto S while S is non-empty peek at the top u of S if u has an (unvisited) neighbor w, add edge from u to w to spanning tree, mark w and push it onto S else pop S
Java Software Structures, 4th Edition, Lewis/Chase

44. Java Software Structures, 4th Edition, Lewis/Chase

45. Topological sort
A diagraph is acyclic if it has no cycles. Such a graph is often referred to as a directed acyclic graph, or DAG, for short. DAGs are used in many applications to indicate precedence among events. A directed graph G is acyclic if and only if a depth-first search of G yields no back edges. Topological sort of a DAG: a linear ordering (sequence) of vertices such that if (u, v) is an edge of the DAG, then u appears somewhere before v in that ordering.
Java Software Structures, 4th Edition, Lewis/Chase

46. Topological sort
Do a DFT of the DAG and put vertices onto the front of a linked list as they are finished.
Java Software Structures, 4th Edition, Lewis/Chase

47. Alternate algorithm for top. sort of DAG
Compute the indegrees of all vertices
Find a vertex U with indegree 0 and append it to the ordering
If there is no such vertex then there is a cycle and the vertices cannot be ordered. Stop.
Remove U and all its edges (U,V) from the graph.
Update the indegrees of the remaining vertices.
Repeat steps 2 through 4 while there are vertices to be processed.
Java Software Structures, 4th Edition, Lewis/Chase

48. Part E – Graph Algorithms 3
Chapter 15
Graphs
Modified
Part E – Graph Algorithms 3

49. Basic Graph Algorithms
Weighted graphs/digraphs
Minimum spanning tree
Cheapest path
Traveling salesman problem
Java Software Structures, 4th Edition, Lewis/Chase

50. Java Software Structures, 4th Edition, Lewis/Chase

51. MST for weighted graph
We can also assign a weight to each edge, which is a number representing how unfavorable it is, and use this to assign a weight to a spanning tree by computing the sum of the weights of the edges in that spanning tree. A minimum spanning tree (MST) or minimum weight spanning tree is a spanning tree with weight less than or equal to the weight of every other spanning tree.
We will assume positive weights only.
Java Software Structures, 4th Edition, Lewis/Chase

52. Greedy algorithms
A greedy algorithm is an algorithm that follows the problem solving heuristic of making the locally optimal choice at each stage with the hope of finding a global optimum.
Java Software Structures, 4th Edition, Lewis/Chase

53. Prim’s algorithm
Initialize a tree with a single vertex, chosen arbitrarily from the graph.
Grow the tree by one edge: Of the edges that connect the tree to vertices not yet in the tree, find the minimum-weight edge, and transfer it to the tree.
Repeat step 2 (until all vertices are in the tree).
Java Software Structures, 4th Edition, Lewis/Chase

54. Kruskal's algorithm
create a forest F (a set of trees), where each vertex in the graph is a separate tree
create a set S containing all the edges in the graph
while S is nonempty and F is not yet spanning
remove an edge with minimum weight from S
if that edge connects two different trees, then add it to the forest, combining two trees into a single tree
At the termination of the algorithm, the forest forms a minimum spanning forest of the graph. If the graph is connected, the forest has a single component and forms a minimum spanning tree.
Java Software Structures, 4th Edition, Lewis/Chase

55. Shortest (cheapest) path in a weighted graph
Dijkstra’s shortest path algorithm computes the shortest paths to all vertices from a starting vertex.
It is a greedy algorithm
It works by growing a tree, adding the “closest” node to the current tree to get the next version of the tree.
The node added is the one that is adjacent to a node already in the tree and closest to the start node using only edges already included in the tree and one frontier edge connecting it that tree.
After adding a node, the distances of the nodes already in the tree must be updated because the new node may provide a shorter path to some of those nodes.
Java Software Structures, 4th Edition, Lewis/Chase

56. Let the node at which we are starting be called the initial node
Let the node at which we are starting be called the initial node. Let the distance of node Y be its distance from the initial node. Dijkstra's algorithm will assign some initial distance values and will try to improve them step by step.
Assign to every node a tentative distance value: set it to zero for our initial node and to infinity for all other nodes.
Mark all nodes unvisited. Set the initial node as current. Create a set of the unvisited nodes called the unvisited set consisting of all the nodes.
For the current node, consider all of its unvisited neighbors and calculate their tentative distances. Compare the newly calculated tentative distance to the current assigned value and assign the smaller one. For example, if the current node A is marked with a distance of 6, and the edge connecting it with a neighbor B has length 2, then the distance to B (through A) will be = 8. If B was previously marked with a distance greater than 8 then change it to 8. Otherwise, keep the current value.
Java Software Structures, 4th Edition, Lewis/Chase

57. When we are done considering all of the neighbors of the current node, mark the current node as visited and remove it from the unvisited set. A visited node will never be checked again.
If all nodes have been marked visited or if the smallest tentative distance among the nodes in the unvisited set is infinity (occurs when there is no connection between the initial node and remaining unvisited nodes), then stop. The algorithm has finished.
Select the unvisited node with the smallest tentative distance, and set it as the new "current node" then go back to step 3.
Java Software Structures, 4th Edition, Lewis/Chase

58. First 5 steps in computing the shortest paths from A.
CN5E by Tanenbaum & Wetherall, © Pearson Education-Prentice Hall and D. Wetherall, 2011

59. Dijkstra's algorithm picks the unvisited vertex with the lowest-distance. It then calculates the distance through it to each unvisited neighbor, and updates the neighbor's distance if smaller. Marked visited (set to red) when done with neighbors.
Java Software Structures, 4th Edition, Lewis/Chase

60. Animated demos
Java Software Structures, 4th Edition, Lewis/Chase

61. Traveling salesman problem
Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city?
Java Software Structures, 4th Edition, Lewis/Chase

62. Brute Force Solutions of TSP
The most direct solution would be to try all permutations (ordered combinations) and see which one is cheapest (using brute force search). The running time for this is O(N!) where N is the number of cities, so this solution becomes impractical somewhere around 20 cities.
Java Software Structures, 4th Edition, Lewis/Chase

63. An exact solution for 15,112 German towns was found in 2001 using the cutting-plane method proposed by George Dantzig, Ray Fulkerson, and Selmer M. Johnson in 1954, based on linear programming.
The computations were performed on a network of 110 processors located at Rice University and Princeton University. The total computation time was equivalent to 22.6 years on a single 500 MHz Alpha processor.
Java Software Structures, 4th Edition, Lewis/Chase

64. In March 2005, the travelling salesman problem of visiting all 33,810 points in a circuit board was solved using Concorde TSP Solver: a tour of length 66,048,945 units was found and it was proven that no shorter tour exists. The computation took approximately 15.7 CPU-years.
In April 2006 an instance with 85,900 points was solved using Concorde TSP Solver, taking over 136 CPU-years.
Java Software Structures, 4th Edition, Lewis/Chase

65. TSP is an NP-hard problem
Comp 314/Comp 332
Java Software Structures, 4th Edition, Lewis/Chase

66. Part F – Graph Algorithms 4
Chapter 15
Graphs
Modified
Part F – Graph Algorithms 4

67. Dynamic graphs Adding and removing vertices Adding and removing edges
Java Software Structures, 4th Edition, Lewis/Chase

68. Java Software Structures, 4th Edition, Lewis/Chase

69. Dynamic structures of graphs
So far we have worked with static data structures for graphs.
We didn’t add vertices or edges, and we didn’t delete vertices or edges.
Now let’s consider how to handle those cases
Java Software Structures, 4th Edition, Lewis/Chase

70. With adjacency matrices
Adding a vertex means adding a row and column to a matrix
Adding an edge (assuming the vertices are already there) means filling in an entry (or two) in the matrix
Deleting a vertex means removing a row and a column from the matrix, which deletes the edges incident with that vertex
Deleting an edge means zeroing out an entry in the matrix.
Arrays are not the most flexible structures for adding and deleting.
Java Software Structures, 4th Edition, Lewis/Chase

71. With adjacency lists
Adding a vertex means adding a new list (with any additional vertex info you are keeping).
Adding an edge means adding node(s) to all list(s) for vertices that edge is incident with.
Deleting a vertex means removing a list; this will delete the edges from that vertex, and other lists may have to be updated as well
Deleting an edge means removing the node(s) for it in the appropriate list(s).
This structure is more flexible, especially if you use an ArrayList, or any list for the vertex info and links.
Java Software Structures, 4th Edition, Lewis/Chase