dschap11

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Graphs

1

Chapter 11



Chapter Objectives

Data Structures Using Java2

y Learn about graphs

y Become familiar with the basic terminology of graph theory

y Discover how to represent graphs in computer memory

y Explore graphs as ADTsy Examine and implement various graph traversal algorithms



Chapter Objectives


y Learn how to implement the shortest path algorithm

y Examine and implement the minimal spanning tree

algorithm

yExplore the topological sort



Königsberg Bridge Problem


In 1736, the following problem was posed:

y River Pregel (Pregolya) flows around the island Kneiphof

y Divides into twoy River has four land areas ( A, B,C , D)

y Bridges are labeled a, b, c, d , e, f , g



Graphs




Königsberg Bridge Problem


y The Königsberg bridge problem

y Starting at one land area, is it possible to walk across all the bridges exactly

once and return to the starting land area?

y In 1736, Euler represented Königsberg bridge problem as graph;

Answered the question in the negative.

y This marked (as recorded) the birth of graph theory.



Graphs




Graph Definitions and Notations


y A graph G is a pair, g = (V , E), where V is a finite nonempty

set, called the set of vertices of G, and E V x V

y Elements of E are the pair of elements of V . E is called the set

of e

dge

s



Graph Definitions and Notations


y Let V (G) denote the set of vertices, and E(G) denote the set

of edges of a graph G. If the elements of E(G) are ordered

pairs, g is called a directed graph or digraph; Otherwise, g is

called an undirected graph

y In an undirected graph, the pairs (u, v ) and (v , u) represent

the same edge



Various Undirected Graphs




Various Directed Graphs




Graph Representation: Adjacency

Matrix

y Let G be a graph with n vertices, where n > 0

y Let V (G) = {v 1, v 2, ..., v n}

y The adjacency matrix AG is a two-dimensional n × n matrixsuch that the (i, j)th entry of AG is 1 if there is an edge from

v i to v j; otherwise, the (i, j)th entry is zero





Matrix




Graph Representation:

Adjacen

cy Lists


y In adjacency list representation, corresponding to eachvertex, v , is a linked list such that each node of the linked listcontains the vertex u, such that (v , u) E(G)

y Array, A, of size n, such that A[i] is a ref erence variable pointing to address of first node of linked list containing the vertices to which v i is adjacent

y Each node has two components, (vertex and link)

y Component vertex contains index of vertex adjacent tovertex i



Graph Representation:

Adjacency List





Matrix




Operations on Graphs


y Create the graph: store in memory using a particular graph

representation

y Clear the graph: make the graph empty

y

Determine whether the graph is emptyy Traverse the graph

y Print the graph



class LinkedListGraph


public class LinkedListGraph extends UnorderedLinkedList

{

//default constructor

public LinkedListGraph()

{

super();

}

//copy constructor public LinkedListGraph(LinkedListGraph otherGraph)

{

super(otherGraph);

}

//Method to retrieve the vertices adjacent to a given

//vertex.

//Postcondition: The vertices adjacent to a given

// vertex from linked list are retrieved // in the array adjacencyList.

// The number of vertices are returned.



class LinkedListGraph (continued)


public int getAdjacentVertices(DataElement[] adjacencyList)

{

LinkedListNode current;

int length = 0;current = first;

11

while(current != null)

{

adjacencyList[len++] = current.info.getCopy();

current = current.link;

}

return length;}

}



Graph Traversals


y Depth first traversal

y Mark node v as visited

y Visit the node

y For each vertex u adjacent to v

y If u is not visited

y Start the depth first traversal at u



Depth First Traversal




Breadth First Traversal


The general algorithm is:a. for each vertex v in the graph

if v is not visited

add v to the queue //start the breadth// first search at v

b. Mark v as visited c. while the queue is not empty

c.1. Remove vertex u from the queue

c.2. Retrieve the vertices adjacent to u

c.3. for each vertex w that is adjacent to u

if w is not visited

c.3.1. Add w to the queue

c.3.2. Mark w as visited



Shortest Path Algorithm


y Weight of the edge: edges connecting two vertices can be assigned a nonnegative real number

y Weight of the path P: sum of the weights of all the edges on

the path P ;Weight of v from u via P

y Shortest path: path with smallest weight

y Shortest path algorithm: greedy algorithm developed byDijkstra



Shortest Path Algorithm


Let G be a graph with n vertices, where n > 0.

Let V (G) = {v 1, v 2, ..., v n}. Let W be a

two-dimensional n X n matrix such that:



Shortest Path


The

ge

ne

ral algorithm is:

1. Initialize the array smallestWeight so that

smallestWeight[u] = weights[vertex, u]

2. Set smallestWeight[vertex] = 0

3. Find the vertex, v, that is closest to vertex for which

the shortest path has not been determined 4. Mark v as the (next) vertex for which the smallest weight

is found

5. For each vertex w in G, such that the shortest path from vertex to w has not been determined and an edge (v, w)exists, if the weight of the path to w via v is smallerthan its current weight, update the weight of w to the

weight of v + the weight of the edge (v, w)

Because there are n vertices, repeat steps 3 through 5 n ² 1times



Shortest Path




Minimal Spanning Tree




Min

imalS

pann

in

g Tree


y A company needs to shut down the maximum number of

connections and still be able to fly from one city to another

(may not be directly).

ySee Figure 11-14





y (Free) tree T : simple graph such that if u and v are twovertices in T , then there is a unique path from u to v

y

Roote

d tree

: tree

in whic

h a partic

ular ve

rte

x is de

signate

das a root

y Weighted tree: tree in which weight is assigned to the edgesin T

y If T is a weighted tree, the weight of T , denoted by W (T ), is

the sum of the weights of all the edges in T





y A tree T is called a spanning tree of graph G if T is a subgraph

of G such that V (T ) = V (G),

yAll the vertices of G are in T .





y Theorem: A graph G has a spanning tree if and only if G is

connected.

yIn order to determine a spanning tree of a graph, the graphmust be connected.

y Let G be a weighted graph.A minimal spanning tree of G is a

spanning tree with the minimum weight.



Prim·s Algorithm


y Builds tree iteratively by adding edges until minimal spanning

tree obtained

yStart with a source vertex

y At each iteration, new edge that does not complete a cycle is

added to tree



Prim·s Algorithm


General form of Prim·s algorithm (let n = number of vertices in G):

1. Set V(T) = {source}

2. Set E(T) = empty

3. for i = 1 to n3.1 minWeight = infinity;

3.2 for j = 1 to n

if vj is in V(T)

for k = 1 to n

if vk is not in T and weight[vj][vk] < minWeight

{endVertex = vk;

edge = (vj, vk);

minWeight = weight[vj][vk];

}

3.3 V(T) = V(T) {endVertex};

3.4 E(T) = E(T) {edge};



Prim·s Algorithm




Spanning Tree as an ADT


Class: MSTree

public MSTree extends Graph

Instance Variables:

protected int source;

protected double[][] weights;

protected int[] edges;

protected double[] edgeWeights;



Spanning Tree as an ADT


Constructors and Instance Methods:

public MSTree() //default constructor

public MSTree(int size)

//constructor with a parameter

public void createSpanningGraph()

throws IOException, FileNotFoundException

//Method to create the graph and the weight matrix.

public void minimalSpanning(int sVertex)

//Method to create the edges of the minimal

//spanning tree. The weight of the edges is also

//saved in the array edgeWeights.

public void printTreeAndWeight()

//Method to output the edges of the minimal

//spanning tree and the weight of the minimal

//spanning tree.



Topological Order


y Let G be a directed graph and V (G) = {v 1, v 2, ..., v n}, where

n > 0.

y A topological ordering of V (G) is a linear ordering v i1, v i2, ...,

v

in

of the vertices such that if v ij

is a predecessor of v ik

, j k, 1

<= j <= n, and 1 <= k <= n, then v ij precedes v ik, that is, j

< k in this linear ordering.



Topological Order


y Because the graph has no cycles:

y There exists a vertex u in G such that u has no predecessor.

y There exists a vertex v in G such that v has no successor.



Topological Order


Class: TopologicalOrder

class TopologicalOrder extends Graph

Constructors and Instance Methods:

public void bfTopOrder()

//output the vertices in breadth first topological order

public TopologicalOrder(int size)

//constructor with a parameter

public TopologicalOrder()//default constructor



Breadth First Topological Order


1. Create the array predCount and initialize it so that

predCount[i] is the number of predecessors of the vertex

vi

2. Initialize the queue, say queue, to all those vertices vk so

that predCount[k] is zero. (Clearly, queue is not empty

because the graph has no cycles.)





3. while the queue is not empty

A. Remove the front element, u, of the queue

B. Put u in the next available position, say

topologicalOrder[topIndex], and increment topIndex

C. For all the immediate successors w of u

1. Decrement the predecessor count of w by 1

2. if the predecessor count of w is zero, add w to queue



Chapter Summary


y Graphs

y Graphs as ADTs

y Traversal algorithms

y

Shortest path algorithmsy Minimal spanning trees

y Topological sort

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