An Exact Parallel Algorithm for the Radio k -coloring Problem

. For a positive integer k , a radio k -coloring of a simple connected graph G = ( V ( G ) , E ( G )) is a mapping |


Introduction
Motivated by the channel assignment problem proposed by Hale [1] in wireless networks, radio labeling in graphs has been studied from various perspectives. Let ( , ) G V E  be a simple, connected and undirected graph. Let diam (G) = d and k d  be a positive integer. We use standard terms or notations as used in common texts such as [2,3]. For a positive integer k, a radio k-coloring f of a simple connected graph G is an assignment of nonnegative integers to the vertices of G such that for every two distinct vertices u and v of G,.| ( )-( )| 1-( , ) The span of a radio k-coloring f, rck( f ), is the maximum integer assigned to some vertex of G by f.
The radio k-chromatic number, rck( G ) of G is min{rck ( f )}, where the minimum is taken over all radio k-colorings f of G. If k is the diameter of G, then rck(G) is known as the radio number and is denoted by rn(G).
The radio k-coloring problem approaches to the radio coloring problem RCP when k = 2. RCP was first introduced by Griggs and Yeh [4] as the L(2, 1) labelling problem. The RCP in general has been proved to be NP-complete [4], even when restricted to planar graphs, split graphs, or the complements of bipartite graphs [5,6].
Exact results have been obtained for certain special class of graphs such as paths [7], cycles and trees [4].
Determining the radio number seems a difficult task, even for some basic graphs. For instance, the radio number for paths and cycles has been studied by Chartrand et al. [8,9], in which, the bounds of the radio numbers for paths and cycles were presented. Later on, the radio numbers for paths and cycles were completely settled in [10].
Badr and Moussa [11] proposed an improved upper bound of the radio kchromatic number for a given graph against another which is based on work by Saha and Panigrahi [12]. They introduced also a polynomial algorithm which (differs from the first algorithm and derived from Liu and Zhu [10]) which determines the radio number of the path graph n P . They [11] also proposed a new integer linear programming model [13][14][15][16] for the radio k-coloring problem.
In this work, we propose four algorithms (two serial algorithms and their parallel versions) which related to the radio k-coloring problem. One of them is an approximate algorithm that determines an upper bound of the radio number of a given graph. The other is an exact algorithm which finds the radio number of a graph G. The approximate algorithm is a polynomial time algorithm while the exact algorithm is an exponential time algorithm. The parallel algorithms are parallelized using the Message Passing Interface (MPI) standard. The experimental results, analysis prove the ability of the proposed algorithms to achieve a speedup 7 for 8 processors.

Main Contributions
In this section, we propose four algorithms (two serial algorithms and their parallel versions) related to the radio k-coloring problem. One of them is an approximate algorithm that determines an upper bound of the radio number of a given graph. The other is an exact algorithm which finds the radio number of a graph G. The approximate algorithm is a polynomial time algorithm while the exact algorithm is an exponential time algorithm.

2.
1. An upper bound of the radio number for a given graph G.
Badr and Moussa [11] proposed an upper bound algorithm of the radio number for a given graph as Algorithm 1. Here we analyze the complexity of Algorithm 1. Algorithm 1 consists of two steps, the first step is Floyed-Warshall's algorithm and the other is the remiander of Algorithm 1. We know the complexity of Floyed-Warshall's algorithm is O(n 3 ) because it has three inner for loops. The remainder of Algorithm 1 has three inner for loop only each of them has length n so the complexity of step 2 is O(n 3 ) . Thus the overall complexity for Algorithm 1 is 2.
2. An exact sequential algorithm for determining the radio number of a given graph G.
In this section, we propose an exact algorithm (Algorithm 2) which determines the radio number of a given graph with n vertices. Recall the radio k-coloring problem is NP-complete problem so Algorithm 2 is an exponential time algorithm. It based on the brute force technique. Algorithm 2 determines the upper bound k of the radio number of a graph G with order n then it generates all possible labeling sets of the graph G. It is clear that the number of all labeling sets for the graph G is . Not of all these labeling sets satisfy the following so the following step in Algorithm 2 checks whether or not a certain set whether satisfies the above inequality. Finally, Algorithm 2 determines the radio number of the graph G by taking {min(max( )) : where Li are the all possible labeling sets for G.
Algorithm 2: A serial algorithm for finding the radio number of a given graph Input: G be an n-vertex simple connected graph and k be a positive integer.
the adjacency matrix A[n][n] of G Output: A radio k-coloring of G.

Complexity of Algorithm 2:
In this section, we analyze the complexity of Algorithm 2. Algorithm 2 consists of more than one procedure: the first procedure is Floyed-Warshall's method.
We know the complexity of Floyed-Warshall's algorithm is O(n 3 ) because it has only three inner for loops. The second procedure of Algorithm 2 has three inner for loop only each of them has length n so the complexity of the second procedure is O(n 3 ).
The remain of Algorithm 1 has the checking subroutine which has complexity O(k n ) because the first vertex in G takes k possibilities, the second vertex takes (k-1) possibilities, and so on until the last vertex takes (k-n+1) possibilities. Thus the overall complexity for the Algorithm 1 is

Conclusion
In this work, we proposed four algorithms (two serial algorithms and their parallel versions) which related to the radio k-coloring problem. One of them is an approximate algorithm that determines an upper bound of the radio number for a given graph. The other is an exact algorithm which finds the radio number of a graph G. The approximate algorithm is a polynomial time algorithm while the exact algorithm is an exponential time algorithm. The parallel algorithms were parallelized using the Message Passing Interface (MPI) standard. The experimental results,