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Graphs (1)Graphs (1)
Outline and Required Reading:Outline and Required Reading:
COSC 2011, Fall 2003, Section AInstructor: N. Vlajic
• The Graph ADT (§ 12.1)• Data Structures for Graphs (§ 12.2)
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Graph Graph – most general data structure - consists of set of vertices (V)and a set of edges (E) that connect vertices – G = {V, E}
Graphs
• vertices can also be referred to as nodes
• in Graph ADT, both vertices and edges are Positions, i.e theystore elements (e.g. v.element(), e.element())
Application Application – road maps, computer networks, WWW, activity charts(Are two points connected? What is the shortest path between them? etc. )
AB E
FDC
vertex =city
computerWeb-page
task
edge =roadcable
hyperlinktransition state
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Graphs: Types of Edges
Edge Edge – connects two vertices (u and v), can be undirected or directed(1) undirected edge {u,v} does not indicated direction – we can
can “travel” in either direction along the edge(2) directed edge (u,v) has a direction – we can “travel” only in
one direction along the edge
A B DC E
SelfSelf--LoopLoop – edge (undirected or directed) that originates and ends at thesame vertex
Parallel EdgesParallel Edges – two undirected edges with the same end vertices, ortwo directed edges with the same origin and destination
GFundirected edge directed edge self-edge parallel edges
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Graphs: Types of Graphs
Undirected GraphUndirected Graph – graph whose edges are all undirected
Directed GraphDirected Graph – graph whose edges are all directed
• undirected and mixed graphs can be converted into adirected graph by replacing each undirected edge {u,v}by the pair of directed edges (u,v) and (v,u)
Mixed GraphMixed Graph – graph that has both directed and undirected edges
A
BE
F
DC
A
BE
F
DC
How could vertex F be reached from vertex C in the two graphs?
Simple GraphSimple Graph – graph without parallel edges and self-loops
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Graphs: Types of Vertices
End Vertices (Endpoints) of an EdgeEnd Vertices (Endpoints) of an Edge – two vertices joined by the edge
Adjacent VerticesAdjacent Vertices – endpoints of the same edge
• in a directed edge, the first endpointis origin and the other is destination
A
BE
F
DC
mn
op
qr
s
Edge Incident on VertexEdge Incident on Vertex – vertex is one of the edge’s endpoints
Outgoing Edges of a VertexOutgoing Edges of a Vertex – directed edges that originate at the vertex
Incoming Edges of a VertexIncoming Edges of a Vertex – directed edges that terminated at the vertex
Degree of a VertexDegree of a Vertex – deg(v) = number of incoming and outgoing edges of v
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Graph: Basic Properties
Property 1:Property 1: If G is a graph with m edges then 2mdeg(v)Gv
=∑∈
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BE
F
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mn
op
qr
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ProofProof On the left-hand side, every undirected edge (u,v) is counted twice – once by its endpoint u and once by its endpoint v.
Property 2:Property 2: If G is a directed graph, then
ProofProof Each directed edge (u,v) contributes once to indeg sum of its origin, andonce to outdeg sum of its destination
moutdeg(v)indeg(v)GvGv
== ∑∑∈∈
A
BE
F
DC
mn
op
qr
s
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Graphs: Basic Properties (cont.)
Property 3:Property 3: In an undirected graph with n vertices and m edges (no self-loops and parallel edges) the following must hold
21)n(nm −≤
ProofProof Assume all n vertexes are mutually connected (case with max # of edges).
A
DC
B
E
21)-n(n 1 ... 2)-(n 1)-(nm =+++=
)deg(v..)deg(v)deg(v)deg(vm },..,v,v{v-G inn
},v{v-G in3
}{v-G in2
G in1 1-n21211 ++++=
A
DC
B
E A
DC
B
E
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Graphs: Paths and Cycles
Path Path – sequence of alternating vertices and edges – starts at a vertex and ends at a vertex
Simple Path Simple Path – path such that all its vertices and edges are distinct
Cycle Cycle – path with identical start and end vertex
Simple Cycle Simple Cycle – cycle such that all its vertices and edges are distinct
A
BE
F
C D
mn
qo
r
ps
tA
BE
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C D
mn
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Path = (F, s, D, t, E, p, F, n, B) Simple Path = (C, q, A, o, E)
Cycle = (E, p, F, n, B, m, A, o, E, t, D, s, F, p, E) Simple Cycle = (E, t, D, r, C, q, A, o, E)
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Graphs: Subgraphs
Subgraph Subgraph – graph in G whose vertices and edges are subset of verticesand edges of G
BE
F
C D
m
qr
Subgraph ={A, B, C, D} U {m, q, r}
BE
F
C D
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ps
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Spanning Spanning Subgraph Subgraph – subgraph in G that contains all the vertices of G
Spanning Subgraph = = {A, B, C, D, E, F} U {m, o, p, s, t, r}
Connected Graph Connected Graph – for any two vertices there is a path between them
Connected Components of a Graph Connected Components of a Graph – maximal connected subgraphs
A A
BE
Fmn
o
r
pA
Graph with 2 Connected Components
C D
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Graphs: Forests and Trees
Forest Forest – undirected graph without cycles
Free Tree Free Tree – connected forest (free = has no root)
Spanning Tree Spanning Tree – subgraph of G that contains all of G’s vertices and enough of its edges to form a tree
• connected components of a forest are free trees
BE
C D
m
q
soAF
Forest
BE
m
q
soAF
p
Spanning Tree
• in a graph with cycles there is more than one spanning tree
C D
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Graphs: Forests and Trees
Example 1 [ spanning trees ]
BE
AF
C D
B
AF
C D
EB
AF
C D
E
B
AF
C D
EB F
D
EA
B F
D
E
C
A
C
Spanning Tree 1 Spanning Tree 2
Spanning Tree 3 Spanning Tree 4 Spanning Tree 5
Are there any other spanning threes besides these? Find them!
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Graphs: Forests and Trees (cont.)
Property 4:Property 4: Let G be an undirected graph with n vertices and m edges
(1) If G is a (spanning) tree
(2) If G is a forest
(3) If G is connected
1nm −=
1nm −≥
1nm −≤
BE
C DA
FE
C
AFB
D
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Graph ADT: Interface
General MethodsGeneral Methods public int numVertices();/* return the number of vertices in G */
public int numEdges();/* return the number of edges in G */
public Iterator vertices();/* return an iterator of the vertices of G */
public Iterator edges();/* return an iterator of the edges of G */
public Vertex aVertex();/* return an arbitrary vertex of G */
public int degree(Vertex v);/* return the degree of v */
public Iterator adjacentVertices(Vertex v);/* return an iterator of the vertices adjacent to v */
Usual MethodsUsual Methods size(), isEmpty(), elements(), positions(),replaceElement(p,o), swapElements(p,q)
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public Iterator incidentEdges(Vertex v);/* return an iterator of the edges incident upon v */public Vertex[ ] endVertices(Edge e);/* return a 2-D array storing the end vertices of e */
public Vertex opposite(Vertex v, Edge e);/* return the endpoint of edge e distinct from v */
public boolean areAdjacent(Vertex v, Vertex w);/* return whether vertices v and w are adjacent */
Graph ADT: Interface (cont.)
public Vertex insertVertex(Object o);/* insert and return a new (isolated) vertex storing the
object o at this position */
public Edge insertEdge(Vertex v, Vertex w, Object o);/* insert and return an undirected edge between vertices
v and w, storing the object o at this position */
public void removeVertex(Vertex v);/* remove vertex v and all its incident edges */
public void removeEdge(Edge e);/* remove edge e */
Update MethodsUpdate Methods
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Data Structures for Graphs
Graph Implementation TechniquesGraph Implementation Techniques
1. Edge List StructureEdge List Structure
2. Adjacency List StructureAdjacency List Structure
3. Adjacency MatrixAdjacency Matrix
• in all three techniques, all vertices are stored in a container
• in 1 and 2, only “existing” edges are stored in a container(space occupancy: O(n+m))
• in 3, a place holder for an edge between each pair of vertices is provided (space occupancy: O(n2)
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1. Edge List Structure
Vertex Object Vertex Object – contains references to• element• corresponding position in vertex sequence
Edge Object Edge Object – contains references to• element• corresponding position in edge sequence• origin vertex object• destination vertex object
A
CB
DA B C D
o p
q
r
o p q r
E
V
• counter for the number of (un)directed edges
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1. Edge List Structure (cont.)
Edge List Structure Edge List Structure – the simples implementation of a graph• each vertex / edge is represented by an object
• all vertices are stored in container V, all edges are stored in container E
• direct access from edges to the vertices they areincident upon is provided – endVertices, origin,destination, opposite run in O(1) time
• access to edges incident upon a given vertex isinefficient – all edge objects must be inspected
A
CB
DA B C D
o p
q
r
o p q r
E
V
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1. Edge List Structure (cont.)
Graph Method
size, isEmpty, replaceElement, swapElementnumVertices, numEdges, aVertex
degree, inDegree, outDegreeelements, positions
endVertices, origin, destination, oppositeincidentEdges, adjacentVertices, areAdjacent
removeVertex…
Time
O(1)O(1)O(1)
O(n+m)O(1)O(m)O(m)
Running Times
each vertexstores a
counter forthese values
Space Requirements – O(m+n), where m = # of edges, n = # of vertices
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2. Adjacency List Structure
Vertex Object Vertex Object – as in Edge List Structure, plus
• reference to incidence container I(v) that stores referencesto all edges incident on v
Edge Object Edge Object – as in Edge List Structure, plus
• reference to the corresponding positions of the edge in theincident containers I(u) and I(v)
Adjacency List Structure Adjacency List Structure – extends Edge List Structure to support direct access to the incident edges (andadjacent vertices)• vertex-based methods run in O(1) time
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2. Adjacency List Structure (cont.)
A
CB
B A C
o p
o p
E
V
I(B) I(A) I(C)
What is the cost and what steps are involved in executing adjacentVertices(A)?
What is the cost and what steps are involved in executing incidentEdges(A)?
What is the cost and what steps are involved in executing areAdjacent(A,B)?
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Graph Method
size, isEmpty, replaceElement, swapElementnumVertices, numEdges, aVertex
degree, inDegree, outDegreeelements, positions
endVertices, origin, destination, oppositeincidentEdges, adjacentVertices,
areAdjacentremoveVertex
…
Time
O(1)O(1)O(1)
O(n+m)O(1)
O(deg(v))O(min(deg(v1),deg(v2)))
O(deg(v))
Running Times
2. Adjacency List Structure (cont.)
Space Requirements – O(m+n) + ∑∑∑∑deg(v) = O(m+n)
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3. Adjacency Matrix Structure
Vertex Object Vertex Object – as in Edge List Structure, plus• distinct integer in the range 0, 1, .., n-1, called “index of v”
Edge Object Edge Object – as in Edge List Structure
• A(i,j) = reference to edge (i,j), if such edge exists;• otherwise A(i,j) = null
Adjacency Matrix Structure Adjacency Matrix Structure – extends Edge List Structure and enables that adjacency between a pair of verticesbe determined in constant time• cost: O(n2) space usage
Adjacency Matrix Adjacency Matrix – an n x n array (A)
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3. Adjacency Matrix Structure (cont.)
A
CBo p
B A C
V
o p
E
0 1 2
0 1 2012
∅∅
∅0
1
2 ∅
∅
What is the cost and what steps are involved in executing areAdjacent(A,B)?
What is the cost and what steps are involved in executing incidentEdges(A)?
What is the cost and what steps are involved in executing adjacentVertices(A)?
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3. Adjacency Matrix Structure (cont.)
Graph Method
size, isEmpty, replaceElement, swapElementnumVertices, numEdges, aVertex
degree, inDegree, outDegreeelements, positions
endVertices, origin, destination, oppositeincidentEdges, adjacentVertices,
areAdjacentinsertVertex, removeVertex
…
Time
O(1)O(1)O(1)
O(n+m)O(1)O(n)O(1)O(n2)
Running Times
Space Requirements – O(m+n) + O(n2) = O(n2)
a whole newarray of
smaller/largersize must be
created
