graphs chapter 19. 2 chapter contents some examples and terminology road maps airline routes mazes...

Graphs

Chapter 19

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

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 Vertices or nodes are connected by edges A subgraph is a portion of a graph that

itself is a graph

4

Road Maps

A portion of a road map. Undirected edges

NodesNodes

EdgesEdges

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Street Maps

A directed graph representing a city's street map. Directed edges

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Path

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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Weight

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

A weighted graph.

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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

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Connected Graphs

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

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

Airline routes

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Mazes

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

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Course Prerequisites

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

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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

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Trees

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

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Breadth-First Traversal(ctd.) A trace of a breadth-first traversal for a directed graph,

beginning at vertex A.

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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 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

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Depth-First Traversal

Visits a vertex, then A neighbor of the vertex, A neighbor of the neighbor, Etc. Visit them, and push them into stack

Advance as much as possible from the original vertex

When no unvisited neighbor exists, back up by one level by popping up current vertex Considers the next neighbor branch

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Depth-First TraversalA trace of a depth-first traversal

beginning at vertex A of the directed graph

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Depth-First TraversalAlgorithm getDepthFirstTraversal(originVertex)

vertexStack = a new stack to hold vertices as they are visitedtraversalOrder = a new queue for the resulting traversal orderMark originVertex as visitedtraversalOrder.enqueue(originVertex)vertexStack.push(originVertex)while (!vertexStack.isEmpty()){

topVertex = vertexStack.peek()if (topVertex has an unvisited neighbor){ nextNeighbor = next unvisited neighbor of topVertex

Mark nextNeighbor as visitedtraversalOrder.enqueue(nextNeighbor)vertexStack.push(nextNeighbor)

}else // all neighbors are visited

vertexStack.pop()

}return traversalOrder

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Graph Traversal Exercises

Breadth-First and Depth-First Traversal starting from aBreadth-First and Depth-First Traversal starting from a

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Some of the possible Answers

Breadth-first a f h e g i d j k c l n b m o

Depth-first a f e d c b g h i j k l m n o

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Topological Order

Given a directed graph without cycles (DAG) An ordering of vertices in a directed acyclic

graph. In a topological order

Vertex a precedes vertex b whenever A directed edge exists from a to b

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Topological Order

First find any vertex with no incoming edges. Print this vertex, and remove it, along with its

edges from the graph. Then we apply the same strategy to the rest

of graph. Indegree of node: Incoming number of edges

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Topological Order

Three topological orders for the graph

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Topological Order

An impossible prerequisite structure for three courses as a directed graph with a cycle.

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Topological Orderundershorts

pants

belt

socks

shoes

shirt

tie

jacket

watch

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Topological Order

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Topological Order

The graph shown has many valid topological sorts, including:

7,5,3,11,8,2,9,10 7,5,11,2,3,10,8,9 3,7,8,5,11,10,9,2

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Shortest Path in an Unweighted Graph

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

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The graph after the shortest-path algorithm has traversed from vertex A to vertex H

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

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Shortest Path in an Weighted Graph

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

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Shortest path between two given vertices Smallest edge-weight sum

Algorithm based on breadth-first traversal (Dijkstra algorithm)

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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Finding the cheapest path from vertex A to vertex H

in the weighted graph

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The graph after finding the cheapest path from vertex A to vertex H.

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Shortest Path from A to E?

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Minimum Spanning Tree

The minimum spanning tree (MST) of a graph defines the cheapest subset of edges that keeps the graph in one connected component.

Telephone companies are particularly interested in minimum spanning trees, because the minimum spanning tree of a set of sites defines the wiring scheme that connects the sites using as little wire as possible.

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Greedy Algorithm

Greedy algorithms work in phases. In each phase, a decision is made that appears to be good, without regard for future consequences.

A greedy algorithm makes the locally optimum choice at each phase with the hope of finding the global optimum.

41

Kruskal's Algorithm

Let G be a connected graph with n vertices Initialize n components, each one containing

one vertex of G. Now sort the edges in increasing order by

weight and set T = the empty set. Now examine each edge in turn. If an edge

joins two components, add it to T and merge the two components into one. If not, discard the edge.

Stop when only one component remains.

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Example

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Kruskal's Algorithm Ordered edges: [1,4], [6,7], [3,4], [1,2], [4,2], [3,1], [4,7], [3,6], [7,5], [4,5], [6,4], [2,5]

Components T (edges)1,2,3,4,5,6,7 empty[1,4], 2, 3, 5,6,7 [1,4][1,4], 2,3,5,[6,7] [1,4], [6,7][1,4,3], 2, 5,[6,7] [1,4], [6,7], [3,4][1,4,3,2], 5, [6,7] [1,4], [6,7], [3,4], [1,2][1,4,3,2,6,7], 5 [1,4], [6,7], [3,4], [1,2], [4,7][1,4,3,2,6,7,5] [1,4], [6,7], [3,4], [1,2], [4,7], [7,5]

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MST

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Find a MST

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Methods in the BasicGraphInterface addVertex addEdge hasEdge isEmpty getNumberOfVertices getNumberOfEdges clear

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A portion of the flight map

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

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The Adjacency Matrix

(a) A directed graph and (b) its adjacency matrix.

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The Adjacency Matrix

Adjacency matrix uses fixed amount of space Depends on number of vertices Does not depend on number of edges

Typically the matrix will be sparse Presence of an edge between two vertices

can be known immediately All neighbors of a vertex found by scanning

entire row for that vertex

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The Adjacency List

Adjacency lists for the directed

graph

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The Adjacency List

Represents only edges that originate from the vertex

Space not reserved for edges that do not exist Uses less memory than corresponding adjacency

matrix Thus more often used than adjacency matrix

graphs chapter 19. 2 chapter contents some examples and terminology road maps airline routes mazes...

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