The document discusses the graph coloring problem. It defines a graph as a pair of vertices and edges, and defines graph coloring as assigning colors to vertices such that no adjacent vertices have the same color. It explains that the graph coloring problem can be solved using an adjacency matrix representation of the graph and parallel processing elements (PEs). Each of the C^n possible colorings is checked by a PE in O(n^2) time to determine if it is valid. The overall complexity of this parallel algorithm is O(n^2).

1. By
Dr. Heman Pathak
Associate Professor
Dept.of Comp. Sc.
KGC - Dehradun

2.  A graph G is a pair(V, E) of a set of vertices V
and a set of edges E.
 The edges are unordered pairs of the form(i, j)
where i, j∈V.
 Two vertices i and j are said to be adjacent if
and only if (i, j)∈E and non-adjacent otherwise.
 Assume that a graph has n vertices.
 A graph can be represented by an Adjacency
Matrix n X n.
2
GRAPH COLORING

3. 3
GRAPH COLORING

4.  The graph coloring problem is an
assignment of colors to the vertices
such that no two adjacent vertices are
assigned the same color.
 A k-coloring of a graph G is a coloring
of G using k colors.
4
THE GRAPH COLORING PROBLEM

5. 5
THE GRAPH COLORING PROBLEM
Colours (C) Vertices (n) Combinations
1 1 1
1 2 1
1 3 1
2 1 2
2 2 4
2 3 8
3 1 3
3 2 9
3 3 27
In general Cn Combinations for C color and n Vertices

6.  No. of Vertices : n
 No of Colors : C
 Graph : A[n][n]
 Possible Solutions : Cn
 No. of PEs required : Cn
 P(i0,i1,i2,………….in-1) : corresponds to a coloring of
vertex 0 with i0, a coloring of vertex 1 with i1,…………..and a
coloring of vertex n-1 with in-1.
 Each PE initially sets its value in the n dimensional candidate
array to 1.
THE GRAPH COLORING PROBLEM
6

7.  It then spend (n2)time determining whether, for the
particular assignment of color to vertices it represents,
two adjacent vertices have been given the same color.
 If A[j,k] = 1 and ij = ik, then the coloring is not valid.
 If PE detects invalid coloring, it sets it candidate to 0.
 If after n2 comparisons its candidate array is still 1, then
coloring is valid.
 By summing over Cn elements in the candidate array, it
can be determined whether there exists a valid coloring
or not.
THE GRAPH COLORING PROBLEM
7

8. THE GRAPH COLORING PROBLEM
8
V0 V1 V2
V0 0 1 0
V1 1 0 1
V2 0 1 0

9. THE GRAPH COLORING PROBLEM
9

10. THE GRAPH COLORING PROBLEM
 It takes (log 𝐶 𝑛) steps to spawn 𝐶 𝑛 PEs
 Every processor executes the doubly-nested for
loops in (n2).
 Summing the 𝐶 𝑛 elements of answer requires
(log 𝐶 𝑛) using 𝐶 𝑛 PEs
 Overall complexity of the algorithm is
(n2 + 𝒍𝒐𝒈 𝑪 𝒏) = (n2 + 𝒏 𝒍𝒐𝒈 𝑪)
 Because c<n then complexity reduces to (n2). 10