csc 213 – large scale programming. today’s goals review first two implementation for graph adt ...
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LECTURE 30:ADJACENCY-MATRIXBASED GRAPH
CSC 213 – Large Scale Programming
Today’s Goals
Review first two implementation for Graph ADT What fields & data used in edge-list based
approach Operations adjacency-list improves & how
it does this Consider when Graph used in real-life
problems For these cases, what operations are
important? How can we speed them up to make work
go faster? Could new implementation use arrays O(1)
time? Consider changes needed to enable using
matrices
edges
vertices
Edge-List Implementation
Base for all Graph implementations Sequences of
vertices & edges Each instance of Edge refers to end vertices
u w
u v w
a b
u
v
wa b
edges
a b
Adjacency-List Implementation Vertex maintains Sequence of Edges Only change
needed
Methods which use incident edges faster Costs some space Improves few
methods to O(1)
vertices
u v w
u wu
v
wa b
Graph ADT
Accessor methods vertices(): iterable for
vertices edges(): iterable for
edges endVertices(e): array
with endpoints of edge e
opposite(v,e): e’s endpoint that is not v
areAdjacent(v,w): check if v and w are adjacent
replace(v,x): make x new element at vertex v
replace(e,x): make x new element at edge e
Update methods insertVertex(x):
create vertex storing element x
insertEdge(v,w,x): add edge (v,w) with element x
removeVertex(v): remove v (& incident edges)
removeEdge(e):remove e
Retrieval methods incidentEdges(v): get
edges incident to v
Graph ADT
Accessor methods vertices(): iterable for
vertices edges(): iterable for
edges endVertices(e): array
with endpoints of edge e
opposite(v,e): e’s endpoint that is not v
areAdjacent(v,w): check if v and w are adjacent
replace(v,x): make x new element at vertex v
replace(e,x): make x new element at edge e
Update methods insertVertex(x):
create vertex storing element x
insertEdge(v,w,x): add edge (v,w) with element x
removeVertex(v): remove v (& incident edges)
removeEdge(e): remove e
Retrieval methods incidentEdges(v): get
edges incident to v
Can This Be Made Faster?
Testing for adjacency is very common Often check how or if vertices are
connected Checking in O(1) time speeds up Graph
algorithms
Can This Be Made Faster?
Testing for adjacency is very common Often check how or if vertices are connected Checking in O(1) time speeds up Graph
algorithms Can trade off lots of space to make
faster Unique integer ID assigned to each Vertex Matrix is created as doubly-subscripted array
of Edge matrix[sourceID][targetID] refers to Edge or null
edges
vertices
0 1 2
0
1
2
Adjacency Matrix Structure
Edge-List structurestill used as base
u v w
0 1 2
u
v
wa b
ba
edges
vertices
0 1 2
0
1
2
Adjacency Matrix Structure
Edge-List structurestill used as base
Vertex stores int Index found in
matrix u v w
0 1 2
u
v
wa b
ba
edges
vertices
0 1 2
0
1
2
Adjacency Matrix Structure
Edge-List structurestill used as base
Vertex stores int Index found in
matrix
Adjacency matrix in Graph class
u v w
0 1 2
u
v
wa b
ba
edges
vertices
0 1 2
0
1
2
Adjacency Matrix Structure
Edge-List structurestill used as base
Vertex stores int Index found in
matrix
Adjacency matrix in Graph class null if
not adjacent
u v w
0 1 2
u
v
wa b
ba
edges
vertices
0 1 2
0
1
2
Adjacency Matrix Structure
Edge-List structurestill used as base
Vertex stores int Index found in
matrix Adjacency matrix
in Graph class null if
not adjacent -or-
Edge incidentto both vertices
u v w
0 1 2
u
v
wa b
ba
edges
vertices
0 1 2
0
1
2
Adjacency Matrix Structure
Undirected edgesstored in both array locations
u v w
0 1 2
u
v
wa b
ba
edges
vertices
0 1 2
0
1
2
Adjacency Matrix Structure
Undirected edgesstored in both array locations
Directed edgesonly in array from source to target
u v w
0 1 2
u
v
wa b
ba
0 1 2
0
1
2
edges
vertices
Adjacency Matrix Structure
Undirected edgesstored in both array locations
Directed edgesonly in array from source to target
u v w
0 1 2
u
v
wa b
ba
Vertex in the Matrix
Another Vertex implementation Only change is a field for this Vertex
Make subclass of existing Vertex class Have 2 classes, which should we use? Does
it matter?
class AMVertex<V> extends Vertex<V>{-or-
class AMVertex<V,E> extends ALVertex<V,E> {private int rank;
// Also need to define getRank, but not setRank}
Inserting/Removing Vertex
Reallocates & copy adjacency matrix Insertion grows array creating locations for
vertex But we have choices when Vertex
removed Resize adjacency-matrix to prevent
“bubbles” Only good when vertices are constant
Inserting/Removing Vertex
Reallocates & copy adjacency matrix Insertion grows array creating locations for
vertex But we have choices when Vertex
removed Resize adjacency-matrix to prevent
“bubbles” Only good when vertices are constant
What else could we do & when is it useful?
n vertices & m edges no self-loops
Edge-List
Adjacency-List
Adjacency-Matrix
Space n + m n + m n2
incidentEdges(v) m deg(v) n + deg(v)
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
For Next Lecture
Finish up your coding of program #2; due today Can use virtual extension, if you still have it
Midterm #2 in class week from today Test will include all material through today Lab on graphs & implementations, so get
chance to use