1. Graphs

2. Some Examples and Terminology
A graph is a set of vertices (nodes) and a set of edges (arcs) such that each edge is associated with exactly two vertices
Edges can be undirected or directed (digraph)
A subgraph is a portion of a graph that itself is a graph

3. A portion of a road map treated as a graph
Nodes
Edges
A portion of a road map treated as a graph

4. A directed graph representing part of a city street map

5. (a) A maze; (b) its representation as a graph

6. The prerequisite structure for a selection of courses as a directed graph without cycles.

7. Paths A sequence of edges that connect two vertices in a graph
In a directed graph the direction of the edges must be considered
Called a directed path
A cycle is a path that begins and ends at same vertex and does not pass through any vertex more than once
A graph with no cycles is acyclic

8. Weights A weighted graph has values on its edges
Weights, costs, etc.
A path in a weighted graph also has weight or cost
The sum of the edge weights
Examples of weights
Miles between nodes on a map
Driving time between nodes
Taxi cost between node locations

9. A weighted graph

10. Connected Graphs A connected graph A complete graph
Has a path between every pair of distinct vertices
A complete graph
Has an edge between every pair of distinct vertices
A disconnected graph
Not connected

11. Connected Graphs
Undirected graphs

12. Vertex A is adjacent to B, but B is not adjacent to A.
Adjacent Vertices
Two vertices are adjacent in an undirected graph if they are joined by an edge
Sometimes adjacent vertices are called neighbors
Vertex A is adjacent to B, but B is not adjacent to A.

13. Note the graph with two subgraphs
Each subgraph connected
Entire graph disconnected
Airline routes

14. Trees All trees are graphs A tree is a connected graph without cycles
But not all graphs are trees
A tree is a connected graph without cycles
Traversals
Preorder (and, technically, inorder and postorder) traversals are examples of depth-first traversal
Level-order traversal of a tree is an example of breadth-first traversal
Visit a node
Process the node’s data and/or mark the node as visited

15. Trees
Visitation order of two traversals; (a) depth first; (b) breadth first.

16. Depth-First Traversal
Visits a vertex, then
A neighbor of the vertex,
A neighbor of the neighbor,
Etc.
Advance as far as possible from the original vertex
Then back up by one vertex
Considers the next neighbor

17. Algorithm depthFirstTraversal (originVertex)
traversalOrder = a new queue for the resulting traversal order
vertexStack = a new stack to hold vertices as they are visited
Mark originVertex as visited
traversalOrder.enqueue (originVertex)
vertexStack.push (originVertex)
while (!vertexStack.isEmpty ()) {
topVertex = vertexStack.peek ()
if (topVertex has an unvisited neighbor) {
nextNeighbor = next unvisited neighbor of topVertex
Mark nextNeighbor as visited
traversalOrder.enqueue (nextNeighbor)
vertexStack.push (nextNeighbor) }
else // all neighbors are visited
vertexStack.pop ()
return traversalOrder

18. Depth-First Traversal
Trace of a depth-first traversal beginning at vertex A.
Assumes that children are placed on the stack in reverse alphabetic (or numeric order).

19. Breadth-First Traversal
Algorithm for breadth-first traversal of nonempty graph beginning at a given vertex
Algorithm breadthFirstTraversal(originVertex) vertexQueue = a new queue to hold neighbors traversalOrder = a new queue for the resulting traversal order Mark originVertex as visited traversalOrder.enqueue(originVertex) vertexQueue.enqueue(originVertex) while (!vertexQueue.isEmpty()) { frontVertex = vertexQueue.dequeue() while (frontVertex has an unvisited neighbor) { nextNeighbor = next unvisited neighbor of frontVertex Mark nextNeighbor as visited traversalOrder.enqueue(nextNeighbor) vertexQueue.enqueue(nextNeighbor) } } return traversalOrder
A breadth-first traversal visits a vertex and then each of the vertex's neighbors before advancing

20. Breadth-First Traversal
A trace of a breadth-first traversal for a directed graph, beginning at vertex A.
Assumes that children are placed in the queue in alphabetic (or numeric order).

21. Implementations of the ADT GraphB C D 1
A directed graph and implementations using adjacency lists and an adjacency matrix.

23. Topological Order Given a directed graph without cycles
In a topological order
Vertex a precedes vertex b whenever
a directed edge exists from a to b

24. Topological Order
Three topological orders for the indicated graph.

25. Topological Order
An impossible prerequisite structure for three courses as a directed graph with a cycle.

26. Topological Order Algorithm for a topological sort
Algorithm getTopologicalSort() vertexStack = a new stack to hold vertices as they are visited n = number of vertices in the graph for (counter = 1 to n) {
nextVertex = an unvisited vertex having no unvisited successors Mark nextVertex as visited stack.push(nextVertex) } return stack

27. Topological Sorting
Algorithm getTopologicalSort() vertexStack = a new stack to hold vertices as they are visited n = number of vertices in the graph for (counter = 1 to n) {
nextVertex = an unvisited vertex having no unvisited successors Mark nextVertex as visited stack.push(nextVertex) } return stack

28. Finding a topological order

29. Shortest Path in an Unweighted Graph
(a) an unweighted graph and (b) the possible paths from vertex A to vertex H.

30. Shortest Path in an Unweighted Graph
The previous graph after the shortest-path algorithm has traversed from vertex A to vertex H

31. Shortest Path in an Unweighted Graph
Finding the shortest path from vertex A to vertex H.

32. Shortest Path in an Weighted Graph
(a) A weighted graph and (b) the possible paths from vertex A to vertex H.

33. Shortest Path in an Weighted Graph
Shortest path between two given vertices
Smallest edge-weight sum
Algorithm based on breadth-first traversal
Several paths in a weighted graph might have same minimum edge-weight sum
Algorithm given by text finds only one of these paths

34. Shortest Path in an Weighted Graph
Finding the shortest path from vertex A to vertex H

35. Shortest Path in an Weighted Graph
The previous graph after finding the shortest path from vertex A to vertex H.

36. Java Interfaces for the ADT Graph
Methods in the BasicGraphInterface
addVertex
addEdge
hasEdge
isEmpty
getNumberOfVertices
getNumberOfEdges
clear