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A new authentication model for ad hoc networks International Journal of Information Security 11 5 : Han B. Zone-based virtual backbone formation in wireless ad hoc networks Ad Hoc Networks 7 1 : Pan Y. Approximation algorithms for load-balanced virtual backbone construction in wireless sensor networks Theoretical Computer Science Kim D. Zhang Z. Zou F.

Constructing minimum connected dominating sets with bounded diameters in wireless networks IEEE Transactions on Parallel and Distributed Systems 20 2 : Wan P. Tighter approximation bounds for minimum cds in unit disk graphs Algorithmica 61 4 : Zhu Q.

Liu L. Minimum total communication power connected dominating set in wireless networks in X. Wang R. Zheng T.

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Jing and X. Kai Eds. On the construction of k-connected m-dominating sets in wireless networks Journal of combinatorial optimization 23 1 : Liao Q. Portfolio optimization of computer and mobile botnets International Journal of Information Security 13 1 : Lin Z. Xu Wang D. A coloring based backbone construction algorithm in wireless ad hoc network in Y. Chung and J. Moreira Eds. Liu Q. Hong Y. Chinchuluun P.

Pardalos R. Enkhbat and E. Pistikopoulos Eds. Misra R. Minimum connected dominating set using a collaborative cover heuristic for ad hoc sensor networks IEEE Transactions on Parallel and Distributed Systems 21 3 : Each black node will join the tree when it receives the invitation for the first time together with the gray node which relays the invitation to itself.

This process should be repeated until all black nodes are in the tree. Theorem 7 guarantees that whenever there is any black node outside the current dominating tree, at least one black node would join the dominating tree. Thus eventually all black nodes will join the dominating tree.

A reporting process described as follows, if necessary, can be performed along the spanning tree T to notify the root of the completion. So the construction of the dominating tree requires O n messages and n time. The same is true for the optional reporting process. Therefore, the total message complexity and time complexity of our algorithm are O n log n and O n respectively.

Finally, bound with the size of the dominating tree. Since each gray node appearing in the dominating tree is the parent of at least one black node, the total number of gray nodes in the dominating tree is at most one less than the number of black nodes. From Lemma 8, the total number of nodes in the dominating tree is at most. This set contains Dominators which are the elements of dominating set. D is a dominating set in G.

D induces a connected subgraph of G. In Fig. Similarly in Fig.

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## Constant-Factor Approximation for Minimum-Weight (Connected) Dominating Sets in Unit Disk Graphs

As in Figs. For example in Fig. Connected graph: A graph G is said to be connected if there is a path between every pair of distinct vertices of a graph G. A graph which is not connected is called disconnected graph. Edge theory of graphs was introduced by Stephen Hedetniemi and Renu Laskar [7].

It has been mentioned that many of the concepts in graph theory has equivalent formulations as concepts for edge graphs. One such formulation is the Y -dominating set of a edge graph. The proposed algorithm, first phase total edge dominating set is constructed and to make the total dominating set as connected one, in the second phase connected edges are found with the help of connector nodes neighborhood based selection criteria.

In the final phase exhaustive local search procedure in applied to reduce the number of edges connected nodes in the CDS, make it as an optimal minimum connected edge dominating set. In the first phase of the algorithm, for a given graph G the corresponding edge graph E G is constructed and its Y- dominating set is found. Procedure is described in Algorithm 2.

From the above if S1,S2, This process is repeated until there is no common element connecting at least two of the subsets. In the third phase, drop the redundant elements in the set S to get a MCEDS set using the exhaustive local search procedure in Algorithm 3.

## Approximation Algorithmic Performance for CEDS in Wireless Network

In which the set of vertices in the graph below with larger radius represents the total edge dominating set. Based on the TED algorithm procedure, now the vertex 16 is added to the total edge dominating set. Similarly the vertex 20 is added to the total edge dominating set, which is shown in the Figure 3. Theorem 2 : The proposed TED algorithm returns a connected edge dominating set, and has a time complexity of O nm.

Let G V,E be an undirected connected graph. Every subset is connected to all the other sets through some other sets or directly by adding connector nodes between them using the neighborhood search procedure in the second phase. The third phase of the algorithm removes some more vertices 6. Therefore it is enough to prove that updated S is still a CDS. The computational complexity of the algorithm is as follows: In Algorithm 1 while loop is executed at most n times. To remove redundant nodes in the CDS, obtained by Algorithm 1 and Algorithm 2, Algorithm 3 will perform its procedures at most d times.

## A better constant-factor approximation for weighted dominating set in unit disk graph

Thus, the computational complexity of the TD algorithm is given by O mn. In the simulation graphs, the following condition is constantly implemented to generate random network instances i. Each node has been assigned to a fixed transmission radio range m. For each fixed number of nodes and the transmission radio range, network instances are created.

- Hydraulic engineering IV: Proceedings of the 4th International Technical Conference on Hydraulic Engineering (CHE 2016, Hong Kong, 16-17 July 2016).
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- A better constant-factor approximation for weighted dominating set in unit disk graph.
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- Constant-Factor Approximation for Minimum-Weight (Connected) Dominating Sets in Unit Disk Graphs.

Before start of the simulation, all the networks are checked to make sure of that their connectivity. All the four algorithms were ran on the network instances, Average size of CDS is taken as the size of the CDS produced by each algorithm and the obtained results are shown in the Figure 5. From the results shown it is clear that, for all the four algorithms the size of the CDS increases when the number of nodes in the network increases. Moreover from the obtained results it is observed that the proposed TED algorithm find better average results than other compared algorithms.

Figure 5: Comparison of average size for different transmission ranges.

In the simulation, graph compares the size of CDS of all the four algorithms when the transmission radio range varies. Each node has been assigned to a transmission range starting from 50m, each node further has been assigned transmission ranges up to m. For each n and r, network instances were created and simulations are carried out on all these instances. The process is repeated for another set of nodes, randomly deployed in the same area. The average size of CDS constructed by each algorithm for two different set of nodes of different transmission ranges shown in the Figure 5.

Figure 6: Comparison of average size of CDS in different area densities, here graph shows the effectiveness of the TED algorithm when the area of the network density is varied. Advertisement Hide. A better constant-factor approximation for weighted dominating set in unit disk graph. Article First Online: 01 March This is a preview of subscription content, log in to check access. LNCS, vol Springer, Berlin, pp 3—14 Google Scholar. Bar-Yehuda R, Moran S On approximation problems related to the independent set and vertex cover problem.

Dai WF, Gao M, Stojmenovic I On calculating power-aware connected dominating sets for efficient routing in ad hoc wireless networks. J Commun Netw 4 1 —70 Google Scholar.