1. 1/14 Ad Hoc Networking, 3.2.2004 Eli M. Gafni and Dimitri P. Bertsekas Distributed Algorithm for Generating Loop-free Routes in Networks With Frequently Changing Topology IEEE Transactions on Communications, vol. Com-29, no. 1, Jan 1981 Ah Hoc Networking 3.2.2004 Lasse Leppänen

2. 2/14 Ad Hoc Networking, 3.2.2004 Contents 1.Introduction 2.A Problem on Acyclic Directed Graphs 3.Two Algorithms for Solving Problem 4.A general Class of Algorithms 5.Conclusions

3. 3/14 Ad Hoc Networking, 3.2.2004 1. Introduction (1/2) The idea of ad hoc networking is not so new. Already in the beginning of 80’s researchers were talking about mobile packet radio (PR) networks. In such networks, it is necessary to use intermediate nodes as repeaters to transfer a message from source to destination. => Routing problem. Most of the routing schemes considered for PR networks involve the use of central station (CS). Collecting information of network connectivity. Set up routes. One possibility is to use shortest path algorithms. In this paper, a shortest path algorithm is assumed to use to determine primary routes. It is assumed that each node has a primary route to CS. Central station have to be informed of the topological changes. Each node send information to CS immediately. => Great deal of information if topological changes are frequent. Node can take a time before sending information. => Reduction of messages if topological changes are temporary.

4. 4/14 Ad Hoc Networking, 3.2.2004 1. Introduction (2/2) Primary routes can be affected by topology changes. => Finding alternate route. With wired NW flooding can be used in failure situation. With PR network flooding is quite unsuitable, because of collisions. So it is necessary to use a contingency routing algorithm to cope effectively with topological changes. Desirable properties for contingency algorithm: 1. It should provide some redundancy in the form of additional routes to reduce the frequency with any node will lose all its routes to CS. 2. It should not rely on instructions from the CS in establishing new routes when all existing routes of any node fail. 3. It should not employ flooding or create serious problems due to collisions. 4. It must ensure that each route is loop-free at all times. 5. It must be capable of incorporating awakened links into existing routes with little communication overhead.

5. 5/14 Ad Hoc Networking, 3.2.2004 2. A Problem on Acyclic Directed Graphs (1/2) Terminology: Directed graph is acyclic if it contains no directed cycle. ADG is destination oriented if there exists a directed path from every node to the destination node. Otherwise, ADG is destination disoriented. Connected ADG is destination disoriented if there exists a node other than the destination that have no outgoing link. In this paper the following problem is considered: Transform a destination disoriented ADG to destination oriented ADG by reversing the direction of some links. Problem is closely related to the contingency routing problem if destination is associated with the central station.

6. 6/14 Ad Hoc Networking, 3.2.2004 2. A Problem on Acyclic Directed Graphs (2/2) Any communication link has a direction and messages can be send in this direction only. If resulting directed graph has no direction cycles and it is destination oriented, then every link is the part of a route leading to the destination. Each node may have several downstream links. If due to topological changes some node will be left without downstream neighbor, ADG has become destination disoriented. => Direction of some links has to be reversed. There is large number of algorithms that solve the problem, e.g. using assign positive weights on all links and use a shortest path algorithms. In this paper problem is not only solved but the algorithm has also the desirable properties discussed earlier.

7. 7/14 Ad Hoc Networking, 3.2.2004 3. Two Algorithms for Solving Problem (1/4) In this paper, two algorithms are introduced. I Full Reversal Method At each iteration each node other than the destination that has no outgoing link reverses the direction of all its incoming links. II Partial Reversal Method Every node than destination keeps a list of its neighboring nodes that have reversed the direction of the corresponding links. At each iteration each node that has no outgoing link reverses the directions of the links which do not appear on the list.

8. 8/14 Ad Hoc Networking, 3.2.2004 3. Two Algorithms for Solving Problem (2/4) Example1: Full Reversal Method D R 1 st iteration D R 2 nd iteration D R 3 rd iteration R D R 4 th iteration D 5 th iteration R R

9. 9/14 Ad Hoc Networking, 3.2.2004 3. Two Algorithms for Solving Problem (3/4) Example2: Partial Reversal Method D R 1 st iteration D R 2 nd iteration D R 3 rd iteration R D R 4 th iteration D 5 th iteration

10. 10/14 Ad Hoc Networking, 3.2.2004 3. Two Algorithms for Solving Problem (4/4) Both examples are extreme because they require a large number of reversals. In typical situation in most networks (particularly with relatively high connectivity) the reversal process not require a long chain of iterations and process is done infrequently. Both algorithms: 1. If the graph is connected, the reversal process will terminate after a finite number of iterations. 2. The directed graph generated at each iteration is acyclic. 3. The direction of any link that have a direct path to the destination in the initial ADG will never be reversed. Algorithms allows also the addition of new directed links in an ADG without forming a cycle.

11. 11/14 Ad Hoc Networking, 3.2.2004 4. A General Class of Algorithms The two algorithms presented are representative of a general class of algorithms for given problem, which are based on a generalized numbering system. In this chapter two propositions are given: 1. Regardless of the timing and order of reversals the same final destination oriented ADG will be obtained within a finite number of iterations. 2. A node that lies on a directed path to the destination in the initial ADG essentially will not participate in the algorithm. Both propositions are proven in the appendix of the paper.

12. 12/14 Ad Hoc Networking, 3.2.2004 5. Summary (1/2) Proposed algorithms form the basis for developing contingency algorithms for mobile PR networks with central station. Each node has a generalized number and the directions of links are always from higher to lower number. Numbers preclude the formation of loops and provide reliable secondary routes to the destination (central station). Some practical issues have to be resolved before these algorithms can be implemented in real networks, e.g. Error detection to ensure that each node operates on the basis of correct numbers for all its neighbors. Some numbers can become too large during the algorithm process. It is interesting to compare the algorithms with distributed shortest path algorithms based on minimum number of hops. Algorithms in this paper do not guarantee the generation of shortest paths.

13. 13/14 Ad Hoc Networking, 3.2.2004 5. Summary (2/2) They offer two substantial advantages: Because of the multiplicity of available routes the contingency algorithm will be activated rarely when a node loses all its routes. Nodes that have not lost all its available routes do not participate in the reversal process or communication exchange (proposition2). In order to awake a new link no direction reversals or communication will be necessary. Algorithms guarantee loop freedom of generated routes at all time. This is not the case for some of the distributed shortest path algorithms, e.g. the original ARPANET algorithm.

14. 14/14 Ad Hoc Networking, 3.2.2004 Thank you for your attention!