1 Self-stabilizing Algorithms and Frequency Assignment Problems

2 Ad-hoc networks Future network: –Each handset functions as a base-station. –No (or few) stationary base-stations. –Communication must be through other hand- sets.

3 Ad-hoc network

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5 A B

6 A B

7 Features Dynamic topology –Each node can only see its neighbors –No total knowledge of the system –Can change due to movements, faults, someone turning on/off Distributed algorithms –Each node must make its decisions based on its own and its neighbors states –The system must converge between each dynamic change

8 Self-stabilizing algorithms A move from ad-hoc methods to provable correct distributed algorithms. Each node i performs a set of rules on the form: If p(i) then m(i) where p(i) is based on its own and its neighbors variables, while m(i) only sets its own variables. Major challenge: Can neither predict the order in which the rules are performed among the nodes nor what the initial configuration is.

9 Self-stabilizing algorithms We desire guaranteed and as fast convergence as possible (time = # of rule applications) The solution is represented as the union of the states of all the nodes Examples of problems: –Minimum spanning tree –Leader election –Frequency assignment (graph coloring)

10 Models To show correctness: –Every stable configuration is a legal solution –Any configuration will lead to a stable configuration Different models: –Fair daemon –Adversarial daemon –Rules execute sequentially –Two adjacent nodes do not execute a rule simultaneously

11 Complexity issues Even solutions to simple problems can have exponential worst case running time Solutions: –Show fast convergence through simulations –Restrict algorithms to special graph classes

12 Example: Graph coloring color 1 color 2 color 3 color 4 Rule 1: If (my_color == color of any neighbor) then my_color = new legal color Rule 2: If (my_color != lowest legal color) then my_color = lowest legal color

13 Results: Leader election in a tree –One node should be marked as the leader –All other should be marked that they are not the leader With J.Blair: Solves problem in O(n 2 ) moves Also solves a number of bottom-up algorithms on trees using the same number of moves Can construct general feedback systems that stabilizes in O(n 3 ) moves

14 Results (cont): Optimal k-packing on trees Find maximum set of nodes that are at least k edges apart from each other. k=3 Assume transmission strength of k-1. Then red nodes are a maximum set of nodes that can use the same frequency.. For transmission strength k/2 red nodes are a maximum set that will not cause any interference when using the same frequency. With M. Mjelde: SS-algorithm that solves the problem in O(n 3 ) moves.

15 Ongoing projects Practical experiments with ss-algorithms for list coloring on general graphs. With K.J. Hole and T. Martens (student). Developing distance-2 coloring ss- algorithm for general graphs.

16 Future plans Develop ss-algorithms for various frequency allocation problems –Heuristic algorithms for general graphs –Optimal algorithms for special graphs Verify algorithms through simulations