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2.3.3 Virtual Coordinates Another approach due to Ratnasamy et al. [13] involves assigning virtual coordinates to each node and then applying standard geographic routing over these coordinates. Virtual coordinates can be constructed using only local connectivity information (which is available since nodes always know their neighbors). Assuming that each node knows its own routing table (consisting of all nodes in its distance two neighborhood), the following routing algorithm can be executed. First, the packet is forwarded to the node in the routing table closest (in virtual coordinates) to the destination, if (and only if) that node is closer to the destination than the current node. Second, if the node is closer to the destination than any other node in its routing table, then the packet is considered to have arrived to its destination. Third, if a packet is neither able to make greedy progress nor has reached a stopping point, then it performs an expanding ring search until a closer node is found (or a maximum TTL has been exceeded). If such geometric coordinates are not known, then Ratnasamy et al. [13] base their algorithm on the so-called perimeter nodes (these are nodes lying on the outer boundary of the system). They distinguish and analyze three scenarios on the perimeter nodes: (a) Perimeter nodes know their location, (b) perimeter nodes know that they are perimeter nodes, but do not know their location, and (c) nodes know neither their location nor whether they are on the perimeter. In the sequel, we provide details of their construction for each of the three cases.

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step (n)

rotate() moves the elements marked by the range [first,middle) to the end of the container The element addressed by middle becomes the first element of the container For example, given the word "hissboo", a rotation around the element 'b' would turn the word into "boohiss"

rotate_copy() template< class ForwardIterator, class OutputIterator > OutputIterator rotate_copy( ForwardIterator first, ForwardIterator middle, ForwardIterator last, OutputIterator result );

rotate_copy() behaves the same as rotate(), except that the rotated sequence is copied to the container marked off by result The returned OutputIterator points one past the last element copied The original sequence is unchanged

Perimeter Nodes Know Their Location. In the rst case an iterative relaxation procedure is being used in order to build the coordinates of nonperimeter nodes using an analogy borrowed from the theory of graph embeddings (see Linial et al. [14]). Links are represented by forces pulling adjacent neighbors together. It is assumed that the force in the x-direction is proportional to the difference in the x-coordinates (similarly for the y-coordinates). Assuming its neighbors are held xed, then a node s equilibrium position (the one where the sum of the forces is zero) is where its x-coordinate is the average of its neighbors x-coordinates (similarly, for y-coordinates). Each nonperimeter node u = (xu , yu ) periodically updates its virtual

#include <algorithm> #include <vector> #include <iostreamh> /* * generates: original element sequence: 1 3 5 7 9 0 2 4 6 8 10 rotate on middle element(0) :: 0 2 4 6 8 10 1 3 5 7 9 rotate on next to last element(8) :: 8 10 1 3 5 7 9 0 2 4 6 rotate_copy on middle element :: 7 9 0 2 4 6 8 10 1 3 5 */ int main() { int ia[] = { 1, 3, 5, 7, 9, 0, 2, 4, 6, 8, 10 }; vector< int, allocator > vec( ia, ia+11 ); ostream_iterator< int > ofile( cout, " " ); cout "original element sequence:\n"; copy( vecbegin(), vecend(), ofile ); cout '\n'; rotate( &ia[0], &ia[5], &ia[11] ); cout "rotate on middle element(0) ::\n"; copy( ia, ia+11, ofile ); cout '\n'; rotate( vecbegin(), vecend()-2, vecend() ); cout "rotate on next to last element(8) ::\n"; copy( vecbegin(), vecend(), ofile ); cout '\n'; vector< int, allocator > vec_res( vecsize() ); rotate_copy( vecbegin(), vecbegin()+vecsize()/2, vecend(), vec_resbegin() ); cout "rotate_copy on middle element ::\n"; copy( vec_resbegin(), vec_resend(), ofile ); cout '\n'; } search() template< class ForwardIterator1, class ForwardIterator2 > ForwardIterator search( ForwardIterator1 first1, ForwardIterator1 last1,

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ForwardIterator2 first2, ForwardIterator2 last2 ); template< class ForwardIterator1, class ForwardIterator2, class BinaryPredicate > ForwardIterator search( ForwardIterator1 first1, ForwardIterator1 last1, ForwardIterator2 first2, ForwardIterator2 last2, BinaryPredicate pred );

Given two ranges, search() returns an iterator addressing the first position in the range marked by [first1,last1), in which the second range occurs as a subsequence If the subsequence does not occur, last1 is returned For example, within Mississippi, the subsequence iss occurs twice, and search() returns an iterator to the start of the first instance By default, the equality operator is used for comparison of the elements; the second version allows the user to supply a comparison operation to be used

1. G. Holland and N. Vaidya. Analysis of TCP performance over mobile ad hoc networks. In Proceedings of ACM/IEEE MobiCom, 1999, pp. 219 230. 2. S. Xu and T. Saadawi. Performance evaluation of TCP algorithm in multi-hop wireless packet networks. Wireless Communications and Mobile Computing, 2:85 100, 2002.