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# Graphs - PowerPoint PPT Presentation

Graphs. Chapter 29. 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

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Presentation Transcript

### Graphs

Chapter 29

Airline Routes

Mazes

Course Prerequisites

Trees

Traversals

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

Chapter Contents

• 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

Nodes

Edges

Fig. 29-1 A portion of a road map.

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

• 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

• 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

Fig. 29-3 A weighted graph.

• 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

Fig. 29-4 Undirected graphs

• 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.

• Note the graph with two subgraphs

• Each subgraph connected

• Entire graph disconnected

Fig. 29-6 Airline routes

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

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

• 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

Fig. 29-9 The visitation order of two traversals; (a) depth first; (b) breadth first.

• Algorithm 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 frontVertexMark 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

Fig. 29-10 (ctd.) A trace of a breadth-first traversal for a directed graph, beginning at vertex A.

• 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

Fig. 29-11 A trace of a depth-first traversal beginning at vertex A of the directed graph in Fig. 29-10a.

• Given a directed graph without cycles

• In a topological order

• Vertex a precedes vertex b whenever

• A directed edge exists from a to b

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

Fig. 29-13 An impossible prerequisite structure for three courses as a directed graph with a cycle.

• 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

Topological Order

Fig. 29-14 Finding a topological order for the graph in Fig. 29-8.

Shortest Path in an Unweighted Graph

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

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

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.

Shortest Path in an Weighted Graph

Fig. 29-18 (a) A weighted graph and (b) the possible paths from vertex A to vertex H.

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

Fig. 29-19 Finding the cheapest path from vertex A to vertex H in the weighted graph in Fig 29-18a.

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.

• Methods in the BasicGraphInterface

• hasEdge

• isEmpty

• getNumberOfVertices

• getNumberOfEdges

• clear

Operations of the ADT graph enable creation of a graph and answer questions based on relationships among vertices

Fig. 29-21 A portion of the flight map in Fig. 29-6.