Graphs Chapter 29

2 Chapter Contents Some Examples and Terminology Road Maps Airline Routes Mazes Course Prerequisites Trees Traversals Breadth-First Traversal Dept-First Traversal Topological Order Paths Finding a Path Shortest Path in an Unweighted Graph Shortest Pat in a Weighted Graph Java Interfaces for the ADT Graph

3 Some Examples and Terminology Vertices or nodes are connected by edges A graph is a collection of distinct vertices and distinct edges Edges can be directed or undirected When it has directed edges it is called a digraph A subgraph is a portion of a graph that itself is a graph

4 Road Maps Fig A portion of a road map. Nodes Edges

5 Road Maps Fig A directed graph representing a portion of a city's street map.

6 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 Simple path does not pass through any vertex more than once A graph with no cycles is acyclic

7 Weights A weighted graph has values on its edges Weights or costs 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

8 Weights Fig A weighted graph.

9 Connected Graphs A connected 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

10 Connected Graphs Fig Undirected graphs

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

12 Airline Routes Note the graph with two subgraphs Each subgraph connected Entire graph disconnected Fig Airline routes

13 Mazes Fig (a) A maze; (b) its representation as a graph

14 Course Prerequisites Fig The prerequisite structure for a selection of courses as a directed graph without cycles.

15 Trees All trees are graphs But not all graphs are trees A tree is a connected graph without cycles Traversals Preorder, inorder, postorder traversals are examples of depth-first traversal Level-order traversal of a tree is an example of breadth-first traversal Visit a node For a tree: process the node's data For a graph: mark the node as visited

16 Trees Fig The visitation order of two traversals; (a) depth first; (b) breadth first.

17 Breadth-First Traversal Algorithm for breadth-first traversal of nonempty graph beginning at a given vertex Algorithm getBreadthFirstTraversal(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

18 Breadth-First Traversal Fig (ctd.) A trace of a breadth-first traversal for a directed graph, beginning at vertex A.

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

20 Depth-First Traversal Fig A trace of a depth- first traversal beginning at vertex A of the directed graph in Fig a.

21 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

22 Topological Order Fig Three topological orders for the graph of Fig

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

24 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 whose neighbors, if any, are all visited Mark nextVertex as visited stack.push(nextVertex) } return stack

25 Topological Order Fig Finding a topological order for the graph in Fig

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

27 Shortest Path in an Unweighted Graph Fig The graph in 29-15a after the shortest-path algorithm has traversed from vertex A to vertex H

28 Shortest Path in an Unweighted Graph Fig Finding the shortest path from vertex A to vertex H in the unweighted graph in Fig a.

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

30 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

31 Shortest Path in an Weighted Graph Fig Finding the cheapest path from vertex A to vertex H in the weighted graph in Fig 29-18a.

32 Shortest Path in an Weighted Graph Fig The graph in Fig a after finding the cheapest path from vertex A to vertex H.

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

34 Java Interfaces for the ADT Graph Fig A portion of the flight map in Fig Operations of the ADT graph enable creation of a graph and answer questions based on relationships among vertices