© 2004 Goodrich, Tamassia Trees1 Lecture 5 Graphs Topics Graphs ( 圖 ) Depth First Search (...

Trees 1© 2004 Goodrich, Tamassia

Lecture 5 Graphs

Topics Graphs ( 圖 ) Depth First Search (深先搜尋 ) Breast First Search (寬先搜尋 )


GraphsA graph is a pair ( 對 ) (V, E), where

V is a set of nodes, called vertices (節點 ) E is a collection (集合 ) of pairs of vertices, called edges ( 邊 ) Vertices and edges are positions (位置 ) and store elements

Example: A vertex represents (代表 ) an airport (機場 ) and stores the

three-letter airport code ( 碼 ) An edge represents a flight route (飛航路線 ) between two

airports and stores the mileage (哩數 ) of the route

ORD PVD

MIADFW

SFO

LAX

LGA

HNL

849

802

13871743

1843

10991120

1233337

2555

142


GraphsEdge typesDirected edge (有向邊 )

ordered pair (有序對 ) of vertices (u,v)

first vertex u is the origin (起點 ) second vertex v is the destination( 目的地 )

e.g., a flightUndirected edge (有向邊 )

unordered pair of vertices (u,v) e.g., a flight route

Directed graph(有向圖 ) all the edges are directed e.g., route network

Undirected graph(有向圖 ) all the edges are undirected e.g., flight network

ORD PVDflight

AA 1206

ORD PVD849

miles


John

DavidPaul

brown.edu

cox.net

cs.brown.edu

att.netqwest.net

math.brown.edu

cslab1bcslab1a

GraphsApplications Electronic circuits (電路 )

Printed circuit board (電路板 )

Integrated circuit (IC)Transportation (交通 ) networks

Highway (高速公路 ) network

Flight (飛航 ) networkComputer networks

Local area network Internet Web

Databases Entity-relationship diagram ( 實體 - 關聯圖 )


GraphsTerminology (術語 ) End (終點 ) vertices (or endpoints) of an edge

U and V are the endpoints of a

Edges incident (接上 ) on a vertex

a, d, and b are incident on VAdjacent (接鄰 ) vertices

U and V are adjacentDegree ( 級 ) of a vertex

X has degree 5 Parallel (並行 ) edges

h and i are parallel edgesSelf-loop (自我迴圈 )

j is a self-loop

XU

V

W

Z

Y

a

c

b

e

d

f

g

h

i

j


P1

GraphsTerminology (cont.) Path (路徑 )

sequence (數列 ) of alternating (交替 ) vertices and edges

begins with a vertex ends with a vertex each edge is preceded (之前 )

and followed (之後 ) by its endpoints

Simple path (單純路徑 ) path such that all its vertices

and edges are distinct (不同 )Examples

P1=(V,b,X,h,Z) is a simple path

P2=(U,c,W,e,X,g,Y,f,W,d,V) is a path that is not simple

XU

V

W

Z

Y

a

c

b

e

d

f

g

hP2


GraphsTerminology (cont.) Cycle

circular sequence of alternating vertices and edges

each edge is preceded and followed by its endpoints

Simple cycle cycle such that all its vertices

and edges are distinct

Examples C1=(V,b,X,g,Y,f,W,c,U,a,) is

a simple cycle C2=(U,c,W,e,X,g,Y,f,W,d,V,a,

) is a cycle that is not simple

C1

XU

V

W

Z

Y

a

c

b

e

d

f

g

hC2


GraphsNotation

n number of vertices m number of edgesdeg(v) degree of vertex v

Property (性質 ) 1

v deg(v) 2m

Proof: each edge is counted (計算 ) twice

Property 2In an undirected graph

with no self-loops and no multiple (多數 ) edges

m n (n 1)2Proof: each vertex has

degree at most (n 1)

What is the bound (極限 ) for a directed graph?

Example n 4 m 6 deg(v) 3


GraphsOperations (操作 )Vertices and edges

are positions store elements

Accessor (擷取 ) methods endVertices(e): an array

of the two endvertices of e

opposite(v, e): the vertex opposite (對方 ) of v on e

areAdjacent(v, w): true iff v and w are adjacent

replace(v, x): replace (取代 ) element at vertex v with x

replace(e, x): replace element at edge e with x

Update (更新 ) methods insertVertex(o): insert a

vertex storing element o insertEdge(v, w, o):

insert (插入 ) an edge (v,w) storing element o

removeVertex(v): remove (刪除 ) vertex v (and its incident edges)

removeEdge(e): remove edge e

Iterator (重複子 ) methods

incidentEdges(v): edges incident to v

vertices(): all vertices in the graph

edges(): all edges in the graph


GraphsRepresented (表現 ) by edge List (邊串列 )Vertex list structureVertex object

Element (元素 ) Reference (參考 ) to

position (位置 ) in vertex sequence (數列 )

Edge object element origin vertex object destination vertex object reference to position in

edge sequenceVertex sequence

sequence of vertex objects

Edge sequence sequence of edge objects

v

u

w

a c

b

a

zd

u v w z

b c d


GraphsRepresented by adjacency List (接鄰串列 )Edge list structureIncidence sequence for each vertex

sequence of references to edge objects of incident edges

Augmented (擴增式 ) edge objects

references to associated (有關 ) positions in incidence sequences of end vertices

u

v

w

a b

a

u v w

b


GraphsRepresented by adjacency matrix (接鄰矩陣 )Augmented vertex objects

Integer key (index, 索引 ) associated (關聯 ) with vertex

2D-array adjacency array

Reference to edge object for adjacent vertices

Null for nonadjacent vertices

The “old fashioned (樣式 )” version just has 0 for no edge and 1 for edge

u

v

w

a b

0 1 2

0

1

2 a

u v w0 1 2

b


Graphs

n vertices, m edges no parallel edges no self-loops Bounds are “big-

Oh”

Edge

List

AdjacencyList

Adjacency Matrix

Space n m n m n2

incidentEdges(v) m deg(v) n

areAdjacent (v, w)

m min(deg(v), deg(w)) 1

insertVertex(o) 1 1 n2

insertEdge(v, w, o)

1 1 1

removeVertex(v)

m deg(v) n2

removeEdge(e) 1 1 1

Asymptotic Performance (漸進式效能 )


GraphsA subgraph (子圖 ) S of a graph G is a graph such that

The vertices of S are a subset (子集 ) of the vertices of G

The edges of S are a subset of the edges of G

A spanning subgraph ( 展距子圖 ) of G is a subgraph that contains all the vertices of G

Subgraph

Spanning subgraph


Graphs

A graph is connected ( 相連 ) if there is a path between every pair of verticesA connected component ( 部分 ) of a graph G is a maximal connected subgraph ( 最大相連子圖 ) of G

Connected graph

Non connected graph with two connected components


GraphsA (free) tree is an undirected graph T such that

T is connected T has no cyclesThis definition of tree is

different from the one of a rooted tree

A forest (森林 ) is an undirected graph without cyclesThe connected components of a forest are trees

Tree

Forest


GraphsA spanning tree of a connected graph is a spanning subgraph that is a treeA spanning tree is not unique unless the graph is a treeSpanning trees have applications to the design of communication networksA spanning forest of a graph is a spanning subgraph that is a forest

Graph

Spanning tree


Depth-First SearchDepth-first search (DFS) is a general (通用 ) technique for traversing ( 走 ) a graphA DFS traversal of a graph G

Visits (拜訪 ) all the vertices and edges of G

Determines (決定 ) whether G is connected

Computes (計算 ) the connected components of G

Computes a spanning forest of G

DFS on a graph with n vertices and m edges takes O(nm ) timeDFS can be further extended to solve other graph problems

Find and report a path between two given vertices

Find a cycle in the graph

Depth-first search is to graphs what Euler tour is to binary trees


Depth-First SearchThe algorithm uses a mechanism (機制 ) for setting ( 設定 ) and getting “labels” ( 標籤 ) of vertices and edges

Algorithm DFS(G, v)Input graph G and a start vertex v of G Output labeling of the edges of G

in the connected component of v as discovery edges and back edges

setLabel(v, VISITED)for all e G.incidentEdges(v)

if getLabel(e) UNEXPLORED

w opposite(v,e)if getLabel(w)

UNEXPLOREDsetLabel(e, DISCOVERY)DFS(G, w)

elsesetLabel(e, BACK)

Algorithm DFS(G)Input graph GOutput labeling of the edges of G

as discovery ( 發現 ) edges and

back (回頭 ) edgesfor all u G.vertices()

setLabel(u, UNEXPLORED)for all e G.edges()

setLabel(e, UNEXPLORED)for all v G.vertices()

if getLabel(v) UNEXPLORED

DFS(G, v)


Depth-First Search - Example

DB

A

C

E

DB

A

C

E

DB

A

C

E

discovery edgeback edge

A visited vertex

A Unexplored (未展開 ) vertex

unexplored edge


Depth-First Search - Example (cont.)

DB

A

C

E

DB

A

C

E

DB

A

C

E

DB

A

C

E


Depth-First Search and Maze Traversal The DFS algorithm is similar to a classic (經典 ) strategy (策略 ) for exploring (探索 ) a maze ( 迷宮 )

We mark each intersection (交口 ), corner (角落 ) and dead end (vertex) visited

We mark each corridor ( 走廊 ) (edge ) traversed

We keep track of the path back to the entrance (入口 ) (start vertex) by means of a rope (繩索 ) (recursion stack)


Depth-First Search - Properties

Property 1DFS(G, v) visits all the vertices and edges in the connected component of v

Property 2The discovery edges labeled by DFS(G, v) form a spanning tree of the connected component of v

DB

A

C

E


Depth-First Search - Analysis

Setting/getting a vertex/edge label takes O(1) timeEach vertex is labeled twice

once as UNEXPLORED once as VISITED

Each edge is labeled twice once as UNEXPLORED once as DISCOVERY or BACK

Method incidentEdges is called once for each vertexDFS runs in O(n m) time provided (設若 ) the graph is represented by the adjacency list structure

Recall that v deg(v) 2m


Depth-First Search - Path Finding

We can specialize (特例化 ) the DFS algorithm to find a path between two given vertices u and z using the template (樣板 ) method patternWe call DFS(G, u) with u as the start vertexWe use a stack S to keep track of the path between the start vertex and the current vertexAs soon as destination vertex z is encountered ( 碰到 ), we return the path as the contents (內容 ) of the stack

Algorithm pathDFS(G, v, z)setLabel(v, VISITED)S.push(v)if v z

return S.elements()for all e G.incidentEdges(v)

if getLabel(e) UNEXPLORED


UNEXPLORED setLabel(e, DISCOVERY)S.push(e)pathDFS(G, w, z)S.pop(e)

else setLabel(e, BACK)

S.pop(v)


Depth-First Search - Cycle Finding

We can specialize the DFS algorithm to find a simple cycle using the template method patternWe use a stack S to keep track of the path between the start vertex and the current (現在 ) vertexAs soon as a back edge (v, w) is encountered, we return the cycle as the portion (部分 ) of the stack from the top to vertex w

Algorithm cycleDFS(G, v, z)setLabel(v, VISITED)S.push(v)for all e G.incidentEdges(v)

if getLabel(e) UNEXPLOREDw opposite(v,e)S.push(e)if getLabel(w) UNEXPLORED

setLabel(e, DISCOVERY)pathDFS(G, w, z)S.pop(e)

elseT new empty stackrepeat

o S.pop()T.push(o)

until o wreturn T.elements()

S.pop(v)


Breadth-First SearchBreadth-first search (BFS) is a general technique for traversing a graphA BFS traversal of a graph G

Visits all the vertices and edges of G

Determines whether G is connected

Computes the connected components of G

Computes a spanning forest of G

BFS on a graph with n vertices and m edges takes O(nm ) timeBFS can be further ( 更進一步 ) extended ( 擴充 ) to solve other graph problems

Find and report a path with the minimum (最小 ) number of edges between two given vertices

Find a simple cycle, if there is one


Breadth-First SearchThe algorithm uses a mechanism for setting and getting “labels” of vertices and edges

Algorithm BFS(G, s)L0 new empty sequenceL0.insertLast(s)setLabel(s, VISITED)i 0 while Li.isEmpty()

Li 1 new empty sequence for all v Li.elements()

for all e G.incidentEdges(v) if getLabel(e) UNEXPLORED


UNEXPLOREDsetLabel(e, DISCOVERY)setLabel(w, VISITED)Li 1.insertLast(w)

elsesetLabel(e, CROSS)

i i 1

Algorithm BFS(G)Input graph GOutput labeling of the edges

and partition of the vertices of G

for all u G.vertices()setLabel(u, UNEXPLORED)

for all e G.edges()setLabel(e, UNEXPLORED)

for all v G.vertices()if getLabel(v)

UNEXPLORED

BFS(G, v)


Breadth-First Search - Example

CB

A

E

D

discovery edgecross ( 反 ) edge

A visited vertex

A unexplored vertex

unexplored edge

L0

L1

F

CB

A

E

D

L0

L1

F

CB

A

E

D

L0

L1

F


Breadth-First Search - Example (cont.)

CB

A

E

D

L0

L1

F

CB

A

E

D

L0

L1

FL2

CB

A

E

D

L0

L1

FL2

CB

A

E

D

L0

L1

FL2


Breadth-First Search - Example (cont.)

CB

A

E

D

L0

L1

FL2

CB

A

E

D

L0

L1

FL2

CB

A

E

D

L0

L1

FL2


Breadth-First Search - Properties

Notation (符號 )Gs: connected component of s

Property 1BFS(G, s) visits all the vertices and edges of Gs

Property 2The discovery edges labeled by BFS(G, s) form a spanning tree Ts of Gs

Property 3For each vertex v in Li

The path of Ts from s to v has i edges

Every path from s to v in Gs has at least i edges

CB

A

E

D

L0

L1

FL2

CB

A

E

D

F


Breadth-First Search - Analysis

Setting/getting a vertex/edge label takes O(1) timeEach vertex is labeled twice

once as UNEXPLORED once as VISITED

Each edge is labeled twice once as UNEXPLORED once as DISCOVERY or CROSS

Each vertex is inserted once into a sequence Li Method incidentEdges is called once for each vertexBFS runs in O(n m) time provided the graph is represented by the adjacency list structure

Recall that v deg(v) 2m


Breadth-First Search - Applications

Using the template method pattern, we can specialize the BFS traversal of a graph G to solve (解決 ) the following problems in O(n m) time Compute the connected components of G Compute a spanning forest of G Find a simple cycle in G, or report that G is a

forest Given two vertices of G, find a path in G

between them with the minimum number of edges, or report that no such path exists (存在 )


DFS vs. BFS

CB

A

E

D

L0

L1

FL2

CB

A

E

D

F

DFS BFS

Applications DFS BFSSpanning forest, connected components, paths, cycles

Shortest paths

Biconnected components


DFS vs. BFS (cont.)Back edge (v,w)

w is an ancestor ( 祖先 ) of v in the tree of discovery edges

Cross edge (v,w) w is in the same level

as v or in the next level in the tree of discovery edges

CB

A

E

D

L0

L1

FL2

CB

A

E

D

F

DFS BFS

© 2004 Goodrich, Tamassia Trees1 Lecture 5 Graphs Topics Graphs ( 圖 ) Depth First Search (...

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Transcript of © 2004 Goodrich, Tamassia Trees1 Lecture 5 Graphs Topics Graphs ( 圖 ) Depth First Search (...