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Graphs Chapter 30 Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved X

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**Chapter Contents Some Examples and Terminology Traversals Road Maps**

Airline Routes Mazes Course Prerequisites Trees Traversals Breadth-First Traversal Dept-First Traversal

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**Chapter Contents Topological Order Paths**

Finding a Path Shortest Path in an Unweighted Graph Shortest Path in a Weighted Graph Java Interfaces for the ADT Graph

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**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

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

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

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**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

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**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

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Weights Fig A weighted graph.

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**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

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Connected Graphs Fig Undirected graphs

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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.

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**Airline Routes Note the graph with two subgraphs**

Each subgraph connected Entire graph disconnected Fig Airline routes

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

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

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**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, 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

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Trees Fig The visitation order of two traversals; (a) depth first

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

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**Breadth-First Traversal**

A breadth-first traversal visits a vertex and then each of the vertex's neighbors before advancing View algorithm for breadth-first traversal of nonempty graph beginning at a given vertex

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**Breadth-First Traversal**

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

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**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 View algorithm for depth-first traversal

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**Depth-First Traversal**

Fig A trace of a depth-first traversal beginning at vertex A of the directed graph

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**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

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Topological Order Fig. 30-8 Fig Three topological orders for the graph of Fig

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Topological Order Fig An impossible prerequisite structure for three courses as a directed graph with a cycle. Click to view algorithm for a topological sort

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

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**Shortest Path in an Unweighted Graph**

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

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**Shortest Path in an Unweighted Graph**

Click to view algorithm for finding shortest path Fig (a) The graph in 30-15a after the shortest-path algorithm has traversed from vertex A to vertex H; (b) the data in the vertex

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**Shortest Path in an Unweighted Graph**

Fig Finding the shortest path from vertex A to vertex H in the unweighted graph

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**Shortest Path in an Weighted Graph**

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

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**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

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**Shortest Path in an Weighted Graph**

Fig Finding the cheapest path from vertex A to vertex H in the weighted graph

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**Shortest Path in an Weighted Graph**

Click to view algorithm for finding cheapest path in a weighted graph Fig The graph in Fig a after finding the cheapest path from vertex A to vertex H.

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**Java Interfaces for the ADT Graph**

Methods in the BasicGraphInterface addVertex addEdge hasEdge isEmpty getNumberOfVertices getNumberOfEdges clear View interface for basic graph operations

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**Java Interfaces for the ADT Graph**

Fig A portion of the flight map in Fig

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**Java Interfaces for the ADT Graph**

Operations of the ADT Graph enable creation of a graph and Answer questions based on relationships among vertices View interface of operations on an existing graph

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