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. 29-1 A portion of a road map.

NodesNodes

EdgesEdges

5

Road Maps

Fig. 29-2 A directed graph representing a portion of a city's street map.

6

Paths

A sequence of edges that connect two vertices in a graphIn 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. 29-3 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. 29-4 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. 29-5 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. 29-6 Airline routes

13

Mazes

Fig. 29-7 (a) A maze; (b) its representation as a graph

14

Course Prerequisites

Fig. 29-8 The prerequisite structure for a selection of courses as a directed graph without cycles.

15

TreesAll trees are graphs• But not all graphs are trees

A tree is a connected graph without cyclesTraversals• 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. 29-9 The visitation order of two traversals; (a) depth first; (b) breadth first.

17

Breadth-First TraversalAlgorithm for breadth-first traversal of nonempty graph beginning at a given vertex

Algorithm getBreadthFirstTraversal(originVertex)vertexQueue = a new queue to hold neighborstraversalOrder = a new queue for the resulting traversal orderMark originVertex as visitedtraversalOrder.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 visitedtraversalOrder.enqueue(nextNeighbor)vertexQueue.enqueue(nextNeighbor)

}}return traversalOrder

A breadth-first traversal visits a vertex and then each of the

vertex's neighbors before advancing

A breadth-first traversal visits a vertex and then each of the

vertex's neighbors before advancing

18

Breadth-First TraversalFig. 29-10 (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. 29-11 A trace of a depth-

first traversal beginning at

vertex A of the directed graph in Fig. 29-10a.

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. 29-12 Three topological orders for the graph of Fig. 29-8.

23

Topological Order

Fig. 29-13 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 visitedn = number of vertices in the graphfor (counter = 1 to n){ nextVertex = an unvisited vertex whose neighbors, if any, are all visited

Mark nextVertex as visitedstack.push(nextVertex)

}return stack

25

Topological Order

Fig. 29-14 Finding a topological order

for the graph in Fig. 29-8.

26

Shortest Path in an Unweighted Graph

Fig. 29-15 (a) an unweighted graph and (b) the possible paths from vertex A to vertex H.

27

Shortest Path in an Unweighted Graph

Fig. 29-16 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. 29-17 Finding the shortest path from vertex A to vertex H in the unweighted graph in Fig. 29-15a.

29

Shortest Path in an Weighted Graph

Fig. 29-18 (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. 29-19 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. 29-20 The graph in Fig. 29-18a 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. 29-21 A portion of the flight map in Fig. 29-6.

Operations of the ADT graph enable creation of a graph and

answer questions based on relationships among vertices

Operations of the ADT graph enable creation of a graph and

answer questions based on relationships among vertices