1. Mobile and Wireless Computing Institute for Computer Science, University of Freiburg Western Australian Interactive Virtual Environments Centre (IVEC) Alternative Formulation of Full Reversal  Associate with each node i, a pair, where i is the unique node ID and is an integer.  Set of pairs are ordered lexicographically. Direction of any link goes from to iff  At each iteration, a node with no outgoing links increases its so as to reverse all its incoming links.

2. Mobile and Wireless Computing Institute for Computer Science, University of Freiburg Western Australian Interactive Virtual Environments Centre (IVEC) Partial Reversal Method  Every node i, other than the destination, keeps a list of its neighbouring nodes j that have reversed the direction of the corresponding links (i,j).  At each iteration, each node i that has no outgoing links reverses the directions of links (i,j) for all j that do not appear in its list. Then it empties the list.  If there is no such j (all its neighbours appear in the list), i reverses all incoming links and empties the list.

3. Mobile and Wireless Computing Institute for Computer Science, University of Freiburg Western Australian Interactive Virtual Environments Centre (IVEC) Partial Reversal Example 1 23 4 56 dest

4. Mobile and Wireless Computing Institute for Computer Science, University of Freiburg Western Australian Interactive Virtual Environments Centre (IVEC) Partial Reversal Example Nodes that reverse 1 23 4 56 dest Link failure

5. Mobile and Wireless Computing Institute for Computer Science, University of Freiburg Western Australian Interactive Virtual Environments Centre (IVEC) Partial Reversal Example Nodes that reverse 1 23 4 56 dest Link failure

6. Mobile and Wireless Computing Institute for Computer Science, University of Freiburg Western Australian Interactive Virtual Environments Centre (IVEC) Partial Reversal Example Nodes that reverse 1 23 4 56 dest Link failure

7. Mobile and Wireless Computing Institute for Computer Science, University of Freiburg Western Australian Interactive Virtual Environments Centre (IVEC) Partial Reversal Example 1 23 4 56 dest Nodes that reverse Link failure

8. Mobile and Wireless Computing Institute for Computer Science, University of Freiburg Western Australian Interactive Virtual Environments Centre (IVEC) Partial Reversal Example 1 23 4 56 dest Nodes that reverse Link failure

9. Mobile and Wireless Computing Institute for Computer Science, University of Freiburg Western Australian Interactive Virtual Environments Centre (IVEC) Height-Based Formulation of Partial Reversal  We associate with every node a triple where and are integers.  The set of triples are ordered lexicographically. Let N be the set of nodes in the network.  The initial set of triples satisfies for all, and for any link we have if and only if link in the initial DAG is directed from to

10. Mobile and Wireless Computing Institute for Computer Science, University of Freiburg Western Australian Interactive Virtual Environments Centre (IVEC) Height-Based Formulation of Partial Reversal  The set of triples forms a total order and the directed graph is loop-free regardless of the values for and.  The triple associated with each node can be viewed as a height and links are directed from higher to lower height.  The parameter is a reference level and and differentiate the relative node heights with a common reference level.

11. Mobile and Wireless Computing Institute for Computer Science, University of Freiburg Western Australian Interactive Virtual Environments Centre (IVEC) Changing Heights  A new iteration of the distributed algorithm is triggered when one or more of the nodes do not have any downstream links.  Suppose is the set of one-hop neighbours of node. The k-th iteration is implemented in the following way.

12. Mobile and Wireless Computing Institute for Computer Science, University of Freiburg Western Australian Interactive Virtual Environments Centre (IVEC) Changing Heights  A node, other than the destination, for which, increases to and sets : If there exists a neighbour with ; otherwise,  For all other nodes : and

13. Mobile and Wireless Computing Institute for Computer Science, University of Freiburg Western Australian Interactive Virtual Environments Centre (IVEC) Example of Height Modification in Partial Reversal Link failure (0,4,1)(0,3,2)(0,2,3)(0,5,4) (0,2,5) (0,1,6) (0,0,0) Dest Nodes that modify heights

14. Mobile and Wireless Computing Institute for Computer Science, University of Freiburg Western Australian Interactive Virtual Environments Centre (IVEC) Example of Height Modification in Partial Reversal Link failure (0,4,1)(0,3,2)(0,2,3)(0,5,4) (0,2,5) (1,1,6) (0,0,0) Dest Nodes that modify heights

15. Mobile and Wireless Computing Institute for Computer Science, University of Freiburg Western Australian Interactive Virtual Environments Centre (IVEC) Example of Height Modification in Partial Reversal Link failure (0,4,1)(0,3,2)(1,0,3)(0,5,4) (1,0,5) (1,1,6) (0,0,0) Dest Nodes that modify heights

16. Mobile and Wireless Computing Institute for Computer Science, University of Freiburg Western Australian Interactive Virtual Environments Centre (IVEC) Example of Height Modification in Partial Reversal (0,4,1)(1,-1,2)(1,0,3)(0,5,4) (1,0,5) (1,1,6) (0,0,0) Dest Link failure Nodes that modify heights

17. Mobile and Wireless Computing Institute for Computer Science, University of Freiburg Western Australian Interactive Virtual Environments Centre (IVEC) Example of Height Modification in Partial Reversal Link failure (1,-2,1)(1,-1,2)(1,0,3)(0,5,4) (1,0,5) (1,1,6) (0,0,0) Dest Nodes that modify heights

18. Mobile and Wireless Computing Institute for Computer Science, University of Freiburg Western Australian Interactive Virtual Environments Centre (IVEC) Properties of Partial Reversal  The DAG is transformed from destination- disoriented to destination-oriented at the end of the partial reversal.  No node with a valid route to the destination ever participates in the reversal process.  A valid route is a sequence of directed links from a node to the destination.

19. Mobile and Wireless Computing Institute for Computer Science, University of Freiburg Western Australian Interactive Virtual Environments Centre (IVEC) Partial Reversal is Loop Free Proof by contradiction: Assume there is a loop, then 1 2 3 4 5 k This is impossible since the heights have a total order.

20. Mobile and Wireless Computing Institute for Computer Science, University of Freiburg Western Australian Interactive Virtual Environments Centre (IVEC) Routing Using the GF Algorithm  The GF algorithm is completely distributed and each node executes the algorithm independently.  Suppose a node i has a packet for a destination node D. Either i wants to send the packet to D, or some other node j has sent the packet to i.  In either case, i forwards the packet to one of its downstream neighbours.  Since the network is a DAG, i will never get back this packet again and the packet will reach its destination D.

21. Mobile and Wireless Computing Institute for Computer Science, University of Freiburg Western Australian Interactive Virtual Environments Centre (IVEC) Getting Information from Neighbours  Each node exchanges information with its neighbours to keep track of the neighbours´ height.  Recall that each node has a height for each destination. Hence, each node needs to keep a list of heights for all the destinations in the network.  Each node exchanges this list with its neighbours periodically.

22. Mobile and Wireless Computing Institute for Computer Science, University of Freiburg Western Australian Interactive Virtual Environments Centre (IVEC) DAG Maintenance  A node forwards its packet for a destination D either if it does not need to reverse its links in the DAG for D, or when it has completed reversing its links.  Note that there are potentially N DAGs for a network with N nodes because each node may be a destination.  The link reversals are done independently for each DAG depending on a node´s situation with respect to that DAG.

23. Mobile and Wireless Computing Institute for Computer Science, University of Freiburg Western Australian Interactive Virtual Environments Centre (IVEC) Properties of Gafni-Bertsekas Algorithm  The GF algorithm has many good properties, e.g., maintaining multiple routes, only local propagation of network modification, loop freedom etc.  However, the GF algorithm works only for connected networks and cannot be used in general for mobile ad hoc networks.