Wireless Ad Hoc and Sensor Networks

Optimized Energy and Delay-Based Routing 373CASE II When the node in N2 () s has more than one neighbor in Ns (),the MPR nodes are selected, based on the OEDR MPR selection criteria.Consider a node s whose one-hop neighbors are given by Ns (), and aparticular node d in N 2 () s , with multiple nodes n1, n2, …, nk( k > 1)belongingto Ns () as its neighbors. Let the cost to reach these one-hop neighborsfrom s be C sni , , and the cost to reach d from n i be given by C n . Accordingi , dto the MPR selection criteria of OEDR, the MPR node to cover d from s isselected as the node n i , with cost of Min{( Csn , + Cn , d),( Csn , + Cn , d), …,1 1 2 2( Csn , + C )}. Hence, the MPR selection criteria of OEDR will result in ank nk,doptimal route from s to its two-hop neighbors in N 2 () s .LEMMA 8.4.1The intermediate nodes on the optimal path are selected as MPRs by the previousnodes on the path.PROOF The proof follows on similar lines to that of Ying et al. (2003).A node in the route may not be selected as the MPR by the previous node,if the node does not provide a connection to that node’s two-hop neighborsor the node does not meet the MPR selection criteria.CASE I The node in Ns () of the previous node s, does not provide a connectionto any node in N2 () s . Consider the graph shown in Figure 8.7.Node n 2 only connects to node s ’s one-hop neighbor n 1 . The two possiblepaths from s to d are s −> n 1 −>d and s −> n 2 −> n 1 −> d. Accordingto Assumption 1, n 2 is not on the optimal path from s to d .CASE II There is an optimal path from source to destination such thatall the intermediate nodes on the path are selected as MPRs by theirprevious nodes on the same path. Without loss of generality, we supposethat in an optimal path, s −> n 1 −> n 2 −> … −> n k −> n k+1 −> … −>d, there are nodes in the route which are not selected as MPRs by theirprevious nodes. Also, based on the result of Case I, we can assume that foreach node on the path, its next node on the path is its one-hop neighbor,and the node two hops away from it is its two-hop neighbor (see Figure 8.8).sn 2n 1dFIGURE 8.7Case I scenario.