Wireless Ad Hoc and Sensor Networks

374 Wireless Ad Hoc and Sensor Networksa 1a ks n 1 n 2 n k n k +1 n k +2 dFIGURE 8.8Case II scenario.For example, n 1 is s’s one-hop neighbor and n k+2 is n k ’s two-hop neighbor.Consider the following two situations:n 11. Suppose that on the optimal route, the first intermediate nodeis not selected as MPR by source s . However, n 2 is the two-hopneighbor of s . Based on the basic idea of MPR selection that allthe two-hop neighbors of s must be covered by its MPR set, smust have another neighbor a 1 , which is selected as its MPR, andis connected to n 2 . According to the MPR selection criteria ofOEDR, s selects a 1 instead of n 1 as its MPR because the cost toreach n 2 using a 1 is less than or equal to the cost to reach n 2using n 1 . Because route s −> n 1 −> n 2 −> … −> d is an optimalpath, => s −> a 1 −> n 2 −> … −> d is also an optimal path.This implies that the source’s MPR are on the optimal path.2. Assume that on the optimal route s −> n 1 −> n 2 −> … −>n k −> n k+1 −> … −> d, all the nodes on segment n 1 −> …−> n k are selected as MPR by their previous node, we now provethat the next hop node of n k on the optimal route is n k ’s MPR.Suppose that n k+1 is not n k ’s MPR. Same as in I, n k+2 is the twohopneighbor of n k , so n k must have another neighbor a k ,which is the MPR of n k and has a connection to n k+2 . Again,n k selects a k instead of n k+1 as its MPR because the cost toreach n k+2 , using a k , is less than or equal to the cost toreach n k+2 , using n k+1 .Because, route s −> … −> n k −> n k+1 −> n k+2 −> … −> d isan optimal path,=> s −> … −> n k −> a k −>n k+2 −> … −> d is also an optimalpath.This implies that in an optimal route, the ( k + 1)th intermediatenode is the MPR of the ( k)th intermediate node.Based on I and II all the intermediate nodes of an optimal path areMPRs of the previous node.LEMMA 8.4.2A node can correctly compute the optimal path for the whole network topology.