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Data Structures
(数据结构)
Chapter 10:Graphs
Graph图
Vocabulary
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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
Adjacency matrix 邻接矩阵
Adjacency list 邻接表
Minimum Spanning tree 最小派生树
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 successors
Unique predecessor
Introduction to Graphs
Linear
Tree
Graph
Figure 7-1 A tree
List
Each
node
may
have
multiple successors as well
as multiple predecessors
Definition of graph
 A Graph
is a collection of nodes, called
vertices, and a collection of line segments,
called edges (or arc), that connecting pairs of
vertices.
 In
other words, a graph consists of two sets, a
set of vertices and a set of lines.
Terminology
Graph may be either directed or undirected:
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 Directed
graph(Digraph)
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Each line has a direction (arrow head)to its successor.
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The lines in a directed graph are known as arcs
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G = (V, E),V:aggregate of Vertices , E:aggregate of edges。
<vi, vj> ≠ <vj, vi>。
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Terminology
Graph may be either directed or undirected:
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 Undirected
graph
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Each line has no direction.
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The lines in a undirected graph are known as edges
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G = (V,E), V:aggreaget of Vertices , E:aggreage of edges .<vi,
vj> = <vj, vi>。
Terminology
Path:
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 A path
is a sequence of vertices in which
each vertex is adjacent to the next one.
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In the following figure, {A, B, C, E} is one path and {A, B, E, F}
is another.
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Terminology
Cycle
 A cycle
is a path consisting of at least three
vertices that starts and ends with the same
vertex.
Cycle?
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In the following subfigure (b), B, C, D, E, B is a cycle.
In a digraph, a path can only follow the direction of the arcs
In an undirected graph, a path can move in either direction along
Terminology
Loop
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 A loop
is a special case of a cycle in which a
single arc beings and ends with the same
vertex.
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Connected
 Two
vertices are
said to be connected
if there
Connected
Unconnected
is a path between them.
 A graph
is said to be connected if there is a
path from any vertex to any other vertex.
Terminology
 Connected (directed graph)
 Strongly
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A directed graph is strongly connected if there is a path from each
vertex to every other vertex in the digraph.
 Weakly
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connected
connected
A directed graph is weakly connected if at least two vertices are not
connected.
 Disjoint
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A graph is disjoint if it is not connected.
Terminology
Degree
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
The degree of a vertex is the number of lines
incident to it.
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The degrees of the nodes A, C, D, F = 1
Terminology
 The degree of a vertex is the sum of the
indegree and outdegree of lines incident to it.
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The outdegree of a vertex in a digraph is the number of arcs
leaving the vertex.
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The indegree is the number of arcs entering the vertex.
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Operations
Add Vertex
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Operations
Delete Vertex
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Add edge
Operations
Delete edge
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Find vertex
Traverse
Graph
Traverse graph
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each vertex of the graphs be processed once
and 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
Traversal
 Breadth-first
Traversal
Depth-first
Traversal
We process all of
a vertex’s descendents before
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we move to an adjacent vertex.
Depth-first traversal of a tree
Depth-first
Traversal
depth-first traversal
of a graph
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 processing
 Select
the first vertex
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
In the depth –first traversal all of a node’s descendents
arestructure.(stack)
processed before moving to an adjacent
The order in which the adjacent vertices are
processed depends on how the graph is
Depth-first Traversal
Depth-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
Traversal
We processing all
adjacent vertices of a

vertex before going to the next level
Breadth-first traversal of a tree
Breadth-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
In the Breadth –first traversal all adjacent vertices are
adjacentbefore
vertex
and so forth
until we of a
processed
processing
the descendents
finished.(Queue)
vertex.
Breadth-first Traversal
Breadth-first
Traversal
We begin by enqueuing
vertex A in the
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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 Structure
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Represent a graph we need to store two
sets
 The
vertices of the graph
 The edges or arcs of the graph
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Two most common structures
 Arrays
 Linked
list
Adjacency
Matrix
One-dimensional
array to store the vertices
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Two-dimensional array to store the edges
or arcs
Adjacency Matrix
Adjacency
Listlinked list to store the
Two-dimensional
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edges or arcs
Graph
Algorithms
Graph data
Structure
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Graph
Algorithms
Create Graph
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Insert Vertex
 Delete Vertex
 Insert Arc
 Delete Arc
 Retrieve Vertex
 First Arc
 Traverse
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Algorithm 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)
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 Return
End depthfirst
Depth-first Traversal Algorithm
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 processed
2 process(vertexPtr)
3 vertxPtr->processed =2
Enqueue all vertices from adjacency list
4 arcPtr=vertexPtr->arc
5 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->nextArc
6 end loop
4 end loop
2 end if
3 walkPtr=walPtr->nextVertex
9 end loop
10 destroyQueue(queue)
11 return
End breadthfirst
Breadth-first Traversal Algorithm
Algorithm 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
Networks
A network is a graph whose lines are
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weighted. It is also known as a weighted
graph.
City airline Network
Minimum
Spanning
Tree
A spanning tree
is a tree that
contains all of
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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 Tree
City airline Network
Minimum Spanning Tree
Minimum
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
We find thepath
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 path
Shortest path
Shortest
path
Insert the first
vertex into the tree
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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
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Summary
A graph is a collection of nodes, called vertices,
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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
Summary
A graph is said to be connected if ,for any two
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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: depthfirst and breadth first.
Summary
To represent a graph in a computer, we need to
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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