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### Graphs

Chapter 29

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

Chapter ContentsSome 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

Road Maps

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

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

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

Weights

Fig. 29-3 A weighted graph.

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

Connected Graphs

Fig. 29-4 Undirected graphs

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.

Airline Routes

- Note the graph with two subgraphs
- Each subgraph connected
- Entire graph disconnected

Fig. 29-6 Airline routes

Mazes

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

Course Prerequisites

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

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

Trees

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

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 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

Breadth-First Traversal

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

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

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.

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

Topological Order

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

Topological Order

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

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 visitedMark 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

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.

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.

Java Interfaces for the ADT Graph

- Methods in the BasicGraphInterface
- addVertex
- addEdge
- hasEdge
- isEmpty
- getNumberOfVertices
- getNumberOfEdges
- clear

Java Interfaces for the ADT Graph

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.