© 2004 goodrich, tamassia trees1 lecture 5 graphs topics graphs ( 圖 ) depth first search (...
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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 )
Trees 2© 2004 Goodrich, Tamassia
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
Trees 3© 2004 Goodrich, Tamassia
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
Trees 4© 2004 Goodrich, Tamassia
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 )
Trees 5© 2004 Goodrich, Tamassia
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
Trees 6© 2004 Goodrich, Tamassia
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
Trees 7© 2004 Goodrich, Tamassia
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
Trees 8© 2004 Goodrich, Tamassia
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
Trees 9© 2004 Goodrich, Tamassia
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
Trees 10© 2004 Goodrich, Tamassia
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
Trees 11© 2004 Goodrich, Tamassia
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
Trees 12© 2004 Goodrich, Tamassia
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
Trees 13© 2004 Goodrich, Tamassia
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 )
Trees 14© 2004 Goodrich, Tamassia
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
Trees 15© 2004 Goodrich, Tamassia
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
Trees 16© 2004 Goodrich, Tamassia
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
Trees 17© 2004 Goodrich, Tamassia
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
Trees 18© 2004 Goodrich, Tamassia
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
Trees 19© 2004 Goodrich, Tamassia
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)
Trees 20© 2004 Goodrich, Tamassia
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
Trees 21© 2004 Goodrich, Tamassia
Depth-First Search - Example (cont.)
DB
A
C
E
DB
A
C
E
DB
A
C
E
DB
A
C
E
Trees 22© 2004 Goodrich, Tamassia
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)
Trees 23© 2004 Goodrich, Tamassia
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
Trees 24© 2004 Goodrich, Tamassia
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
Trees 25© 2004 Goodrich, Tamassia
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
w opposite(v,e)if getLabel(w)
UNEXPLORED setLabel(e, DISCOVERY)S.push(e)pathDFS(G, w, z)S.pop(e)
else setLabel(e, BACK)
S.pop(v)
Trees 26© 2004 Goodrich, Tamassia
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)
Trees 27© 2004 Goodrich, Tamassia
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
Trees 28© 2004 Goodrich, Tamassia
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
w opposite(v,e)if getLabel(w)
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)
Trees 29© 2004 Goodrich, Tamassia
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
Trees 30© 2004 Goodrich, Tamassia
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
Trees 31© 2004 Goodrich, Tamassia
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
Trees 32© 2004 Goodrich, Tamassia
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
Trees 33© 2004 Goodrich, Tamassia
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
Trees 34© 2004 Goodrich, Tamassia
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 )
Trees 35© 2004 Goodrich, Tamassia
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
Trees 36© 2004 Goodrich, Tamassia
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