Download 31GraphsDigraphsADT

FUNDAMENTALS: DATA STRUCTURES & ALGORITHMS
SPECIFICATION OF THE ADT – GRAPH/DIGRAPH
Additional definitions
Subgraph
Connected
Adjacency
Spanning Tree
a graph containing a subset of nodes and edges
contains a path between every pair of nodes
if there is an edge between the two nodes
a graph with sufficient edges removed to eliminate cycles
Name
Elements
Structure
- Graph or Digraph
- set of instances from a GraphNode ADT
- each instance of GraphNode connected to zero or more other GraphNode objects determined by
the set of edges.
Domain Sets
G – set of all ordered pairs <n, e> (see earlier definition for a graph)
N – set of all GraphNode objects
E – set of ordered pairs of GraphNode references from the set N
K – set of instances of KeyNode ADT
Operations
create:
g
insert: g, n
g (uses ‘traverse’ to locate position in the tree)
Also delete, update and retrieve operations
traverse: g, k
n ( several methods of maintaining the graph structure; depth first,
breadth first, priority first, spanning tree)
path: g
e (possibilities: between two nodes, all paths, shortest paths etc.)
______________________________
Representation and Implementation
a)Matrix Representation
Matrices are of considerable use in representing graphs/digraphs, particularly static graphs.
Assuming digraph G = <N, E> where N = set of nodes { n1, n2, n3, . . . nm } and E the set ordered pairs from N then:
ADJACENCY MATRIX (A)
PRECEDENCE MATRIX (P)
aij = 1 if < ni, nj >  E
pij = 1 if ni precedes nj
aij = 0 if < ni, nj >  E
pij = 0 if ni does not precede nj
PATH MATRIX (L)
lij = 1 if a path exists between ni and nj
lij = 0 if no path exists between ni and nj
Example
1
2
O6
O
O
5
O
O
SUCCEDENCE MATRIX (S)
sij = 1 if ni succedes nj
sij = 0 if ni does not succede nj
A =
O
010000
001100
000100
110000
001000
000000
L =
1 11100
111100
111100
111100
111110
000000
Relationship between the Adjacency matrix and the Path matrix for a digraph
Suppose A is the adjacency matrix. Let matrix B = Ak
Then bij is the total number of distinct sequences < n1, . .> . . . . . . <. . , nj > that:
i) have length k
ii) correspond to paths in the digraph
Proof: For k = 1 then B = A and the theorem is true (through the definition of the Adjacency matrix A)
Assume true for some k = h and let matrix C = Ah For a given i, j and k we have:
cik = sequences of length h
< n1, . . .> . . . . . . . . . . . . . . . . . <. . . , nk >
= sequences of length h+1 < n1, . . .> . . . . . . . . . . . . . . . . . <. . . , nk >, < nk, nj >
i.e. cik . akj = cik if akj is an arc (sequence of one or more edges), otherwise it is 0
Hence total sequences of length h+1
=
< n1, . . .> . . . . . . . . . . . . . . . . . <. . . , n j >
n
=  cik . akj
i.e. the (i,j)th element of the matrix Ah+1
Corollaries
i)
If Ak = 0 for some k <= n then the digraph contains no cycles i.e. is acyclic.
ii)
If L is the path matrix of a digraph and Q = A + A 2 + A3 + A4 . . . . . . . . . . . + An then:
lij = 1 if and only if qij is not zero. If boolean matrix operations are used (see later) i.e.
Q = A  A2  A3  A4  . . . . . . . . . .  An where Ak = Ak-1 ^ A then Q = L
OPERATIONS ON MATRIX REPRESENTATIONS FOR DIGRAPHS
The Adjacency to Path matrix for a Digraph
If A = Adjacency matrix of size n. If B = Ak (A  A . . .) then bij = 0 reflects a distinct
path in the digraph of length k.
a) If Ak = 0 for some k <= n then the digraph is acyclic.
b) Boolean operations used:
C=AA
cij =
(a
ik
 akj )
k=1
D=AA
c)
dij = aij  aij
The path matrix:
P = A  A2  A3  A4 . . . . . . . . . . . . . An
Example
o1
A
o2
=
0
1
1
1
0
0
0
0
0
1
0
0
0
0
0
1
1
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
1
1
0
0
0
0
0
1
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
o3
A2
=
o5
4o
o6
A3
=
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
A4
=
0
0
0
0
0
0
Since the Path matrix P is equivalent to the Adjacency matrix A, then P2 contains the
second precedents to each node, P3 the third precedents etc.etc.
Representation and Implementation of Graphs
Matrix Representation
Matrices are of considerable use in representing graphs/digraphs,
particularly static graphs.
Assuming digraph G = <N, E> where N = set of nodes { n1, n2, n3, . . . nm }
and E the set ordered pairs from N then:
ADJACENCY MATRIX (A)
aij = 1 if < ni, nj >  E
aij = 0 if < ni, nj >  E
nj
PRECEDENCE MATRIX (P)
pij = 1 if ni precedes nj
pij = 0 if ni does not precede
PATH MATRIX (L)
SUCCEDENCE MATRIX (S)
lij = 1 if a path exists between ni and nj
sij = 1 if ni succedes nj
lij = 0 if no path exists between ni and nj
sij = 0 if ni does not
succede nj
Example
1
O
2
O
O6
5
O
O
4
O
3
A =
010000
001100
000100
110000
001010
000000
L =
111100
111100
111100
111100
111110
000000