Data StructuresData Structures((数据结构数据结构 ))

Chapter 10:GraphsChapter 10:Graphs

VocabularyVocabulary Graph 图 Vertex 顶点 Edge 边 Arc 弧 Directed Graph 有向图 Undirected Graph 无向图 Adjacent Vertices 邻接点 Path 路径 Cycle 圈 Strongly connected 强连通 Weekly connected 弱连通 Disjoint 未连通 depth-first traversal 深度优先遍

历 Breadth-first traversal 广度优先

遍历

Adjacency matrix 邻接矩阵Adjacency list 邻接表Minimum Spanning tree 最小派生树

Introduction to GraphsIntroduction to Graphs

Linear ListTreeGraph

Recall that a list is a collection of components in which 1. each component (except one, the first)

has exactly 1 predecessor. 2. each component (except one, the last)

has exactly 1 successor. multiple successorsUnique predecessor

Figure 7-1 A tree

Each node may have multiple successors as well as multiple predecessors

Definition of graphDefinition of graph

A Graph is a collection of nodes, called

verticesvertices, and a collection of line segments,

called edges edges (or arc arc), that connecting pairs of

vertices.

In other words, a graph consists of two sets, a

set of verticesvertices and a set of lineslines.

TerminologyTerminology Graph may be either directeddirected or undirectedundirected:

Directed graph(Digraph) Each line has a directiondirection (arrow head)to its successor.

The lines in a directed graph are known as arcsarcs

G = (V, E),V:aggregate of Vertices , E:aggregate of edges 。 <vi, vj>

≠ <vj, vi>。

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TerminologyTerminology Graph may be either directeddirected or

undirectedundirected:Undirected graph

Each line has nono directiondirection.

The lines in a undirected graph are known as edgesedges

G = (V,E), V:aggreaget of Vertices , E:aggreage of edges .<vi,

vj> = <vj, vi>。

TerminologyTerminology PathPath:

A path is a sequence of verticesa sequence of vertices in which each vertex is

adjacent to the next one. In the following figure, {A, B, C, E} is one path and {A, B, E, F} is another.

Both directed and undirected graphs have paths.

TerminologyTerminology CycleCycle

A cycle is a pathpath consisting of at least three vertices

that starts and ends with the starts and ends with the same vertexsame vertex. In the following subfigure (b), B, C, D, E, B is a cycle.

In a digraph, a path can only follow the directionfollow the direction of the arcs

In an undirected graph, a path can move in either directionmove in either direction along the

edge

Cycle?

TerminologyTerminology LoopLoop

A loop is a special case of a cycle in which a a

single arcsingle arc beings and ends with the same beings and ends with the same

vertexvertex.

Connected Connected

Two vertices are said to be connected if there

is a path between them.

A graph is said to be connected if there is a A graph is said to be connected if there is a

path from any vertex to any other vertex.path from any vertex to any other vertex.

ConnectedConnected UnconnectedUnconnected

TerminologyTerminology Connected Connected (directed graph)

Strongly connected Strongly connected A directed graph is strongly connected if there is there is a patha path from each from each

vertex to every other vertexvertex to every other vertex in the digraph.

Weakly connectedWeakly connected A directed graph is weakly connected if at least two vertices are at least two vertices are not not

connectedconnected.

DisjointDisjoint A graph is disjoint if it is not connected.

TerminologyTerminology DegreeDegree

The degreedegree of a vertex is the number of lines

incident to it.

The degrees of the nodes A, C, D, F = 1

The degrees of the nodes B, E = 3

TerminologyTerminologyThe degreedegree of a vertex is the sum of the of the

indegree and outdegreeindegree and outdegree of lines incident to it. The outdegreeoutdegree of a vertex in a digraph is the number of arcs the number of arcs

leavingleaving the vertex the vertex.

The indegree indegree is the number of arcs the number of arcs enteringentering the vertex the vertex.

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OperationsOperations Add VertexAdd Vertex

OperationsOperations Delete VertexDelete Vertex

Add edgeAdd edge

OperationsOperations Delete edgeDelete edge

Find vertexFind vertex

Traverse GraphTraverse Graph Traverse graphTraverse graph

each vertex of the graphs be processed once each vertex of the graphs be processed once and only onceand only once

we must ensure that we process the data in each vertex only once. There are multiple paths to a vertex, we use a visited flag at each vertex to solve this problem.

Depth-first TraversalDepth-first Traversal

Breadth-first TraversalBreadth-first Traversal

Depth-first TraversalDepth-first Traversal We process all of a vertex’s descendents before

we move to an adjacent vertex.

Depth-first traversal of a tree

Depth-first TraversalDepth-first Traversal depth-first traversal of a graph

processing the first vertex

Select any vertex adjacent to the first vertex and process it

Select an adjacent vertex until we reach a vertex with no adjacent

entries, back out of the structure.(stack)

The order in which the adjacent vertices are processed depends on

how the graph is physically stored.

In the depth –first traversal all of a node’s

descendents are processed before moving to an

adjacent

Depth-first TraversalDepth-first Traversal

Depth-first TraversalDepth-first Traversal We Begin by pushing the first vertex A into

the stack We then loop, pop the stack, and , after

processing the vertex. Push all of the adjacent vertices into the stack. Such as process Vertex X at step 2, we pop x from the stack process it, and then push the adjacent vertices G and H into the stack.

When the stack is empty, the traversal is completes.

Breadth-first TraversalBreadth-first Traversal We processing all adjacent vertices of a

vertex before going to the next level

Breadth-first traversal of a tree

Breadth-first TraversalBreadth-first Traversal Breadth-first traversal of a graph

processing the first vertex

Processing all of the first adjacent vertices

Pick the first adjacent vertex and processing all of its adjacent

vertices, then the second adjacent vertex and so forth until we

finished.(Queue)

In the Breadth –first traversal all adjacent vertices

are processed before processing the descendents

of a vertex.

Breadth-first TraversalBreadth-first Traversal

Breadth-first TraversalBreadth-first Traversal We begin by enqueuing vertex A in the

queue We the loop, dequeuing the queue and

processing the vertex from the front of the queue. After processing the vertex, we place all of its adjacent vertices into the queue.

When the queue is empty, the traversal is complete

Graph Storage StructureGraph Storage Structure Represent a graph we need to store two

setsThe vertices of the graphThe edges or arcs of the graph

Two most common structures Arrays Linked list

Adjacency MatrixAdjacency Matrix One-dimensional array to store the vertices Two-dimensional array to store the edges

or arcs

Adjacency MatrixAdjacency Matrix

Adjacency ListAdjacency List Two-dimensional linked list to store the

edges or arcs

Graph AlgorithmsGraph Algorithms Graph data Structure

Graph AlgorithmsGraph Algorithms Create Graph Insert Vertex Delete Vertex Insert Arc Delete Arc Retrieve Vertex First Arc Traverse

Depth-first Traversal AlgorithmDepth-first Traversal AlgorithmAlgorithm depthfirst (val graph<metadata>

Processing the keys of the graph is depth-first order.

Pre graph is a pointer to a graph head structure

Post vertices “processed”

1 If (empty graph)

1 Return

Set processed flags to not processed

2 walkPtr=graph.first

3 Loop (walkPtr)

1 walkPtr->processed = 0

2 walkPtr =walkPtr->nextVertex

4 End loop

Process each vertex in list

5 createStack(stack)

6 walkPtr=graph.first

7 loop(walkPtr not null)

1 if (walkPtr->Processed <2)

1 if (walkPtr->processed <1)

Push and set flag to stack

1 puchStack(stack,walker)

2 walkPtr->processed =1

2 end if Process vertex at stack top 3 loop (not emptyStack(stack))

1 popStack(stack,vertexPtr)2 process(vertex->dataPtr)3 vertexPtr->processed =2

Push all Vertices from adjacency list 4 arcwalkPtr=vertexPr->arc 5 loop( arcwalkPtr not null) 1 vertToPtr=arcwalkPtr->destination

2 if (vertToPtr->processed is 0) 1 puchStack(sack,VertToPtr) 2 vertToPtr->Processed =1 3 end if 4 arcwalkPtr=arcwalkPtr->nextArc

6 end loop 4 end loop 2 end if 3 walkPtr-walkPtr->nextVertex 8 end loop 9 destroyStack(stack)10 ReturnEnd depthfirst

Breadth-first Traversal AlgorithmBreadth-first Traversal AlgorithmAlgorithm Breadthfirst (val graph<metadata>

Processing the keys of the graph is Breadth-first order.

Pre graph is a pointer to a graph head structure

Post vertices “processed

1 If (empty graph)

1 return

2 End if

Fist se all processed flags to not processed

Falg:0– not processed, 1– enqueued, 2– processed

3 createqueue(queue)

4 walkPtr=graph.first

5 Loop (walkPtr not null)

1 walkPtr->processed =0

2 walkPtr=walkPtr->nextVertex

6 End loop

Process each vertex in vertex list

7 walkPtr =graph.first

8 Loop (walkPtr not null)

1 if (walkPtr->Processed <2)

1 if (walkPtr->Processed <1)

Enqueue and set processed flag to 1 1 enqueue(queue,walkPtr) 2 walkPtr->Processed =1 2 end if

How process descendents of vertex at queue first 3 loop (not emptyQueue(queue))

1 dequeue(queue,vertexPtr)Process Vertex and flag as

processed2 process(vertexPtr)3 vertxPtr->processed =2Enqueue all vertices from

adjacency list 4 arcPtr=vertexPtr->arc5 loop (arcPtr not null) 1 toPtr =arcPtr->destination 2 if (toPtr -> processed =1) 1 enqueue(queue,toPtr) 2 toPtr->processed =1 3 end if 4 arcPtr=arcPtr->nextArc6 end loop

4 end loop 2 end if 3 walkPtr=walPtr->nextVertex

9 end loop 10 destroyQueue(queue)11 returnEnd breadthfirst

NetworksNetworks A network is a graph whose lines are

weighted. It is also known as a weighted graph.

City airline Network

Minimum Spanning TreeMinimum Spanning Tree A spanning tree is a tree that contains all of the vertices in the graph

A minimum spanning tree of a network such that the sum of its weights are guaranteed to be minimal.

if there are duplicate weights, then these may be one or more minimum spanning tree.

Minimum Spanning TreeMinimum Spanning Tree

City airline Network

Minimum Spanning TreeMinimum Spanning Tree

Minimum Spanning TreeMinimum Spanning Tree From all the vertices in the tree. Select

the edge with minimal value to a vertex not currently in the tree and insert it into the tree.

Shortest pathShortest path We find the shortest path between to vertices in

network The Dijkstra algorithm is used to find the shortest path

between any two nodes in a graph

Example : we need to find the shortest path from vertex A to any other vertex in the graph.

Shortest pathShortest path

Shortest pathShortest path

Shortest pathShortest path Insert the first vertex into the tree From every vertex already in the tree , examine the total

path length to all adjacent vertices not in the tree. Select the edge with the minimum total path weight and insert it into the tree

Repeat step 2 until all vertices are in the tree

SummarySummary A graph is a collection of nodes, called vertices, and a collection of line segments connection pairs of nodes, called edges or arcs.

Graphs may be directed or undirected. A directed graph, or digraph is a graph is which each line has s direction. An undirected graph is a graph in which there is no direction on the lines. A line in a directed graph is called an arc.

In a graph, two vertices are said to be adjacent if an edge directly connects them

A path is a sequence of vertices in which each vertex is adjacent to the next one

A cycle is a path of at least three vertices that starts and ends with the same vertex

A loop is a special case of a cycle is which a single arc begins with the same vertex

SummarySummary A graph is said to be connected if ,for any two vertices, there is a

path from one to the other. A graph is disjointed if it is not connected.

The degree of a vertex is the number of the vertices adjacent to it. The outdegree of a vertex is the number of arcs leaving the node; the indegree of a vertex is the number of arcs entering the node.

Six operations have been defined for a graph:add a vertex, delete a vertex, add an edge, delete an edge, find a node, and traverse the graph.

There are two standard graph traversals: depth-first and breadth first. In the depth-first traversal, all of the node’s descendents are

processed before moving to an adjacent node In the breadth-first traversal, all of the adjacent vertices are

processed before processing the descendents of a vertex

SummarySummary To represent a graph in a computer, we need to store

two sets of information: the first sets represents the vertices and the second sets represents the edges.

The most common methods used to store a graph are the adjacency matrix method and the adjacency list methods

A network is a graph whose lines are weighted. A spanning tree is a graph whose lines are weighted A minimum spanning tree is a spanning tree in which

the total weight of the edges is the minimum. Another common algorithm in a graph is to find the

shotest pathe between two vertices.