1. CPSC 689: Discrete Algorithms for Mobile and Wireless Systems Spring 2009 Prof. Jennifer Welch

2. Discrete Algs for Mobile Wireless Sys2 Lecture 22  Topic: Distributed Dominating Sets  Sources: Jia, Rajaraman, Suel MIT 6.885 Fall 2008 slides

3. Discrete Algs for Mobile Wireless Sys3 Finding a Destination by Flooding

4. Discrete Algs for Mobile Wireless Sys4 Finding a Destination Efficiently

5. Discrete Algs for Mobile Wireless Sys5 Dominating Set  Dominating Set Subset S of V s.t. each v in V is in S or is a neighbor of a node in S

6. Discrete Algs for Mobile Wireless Sys6 Connected Dominating Set  Connected Dominating Set A dominating set inducing a connected topology  Connected DS can be used as routing backbone!

7. Discrete Algs for Mobile Wireless Sys7  Use synchronous message passing model  Network = graph (nodes: devices, edges: direct comm. links)  Node have unique IDs  Time is divided into rounds: Distributed Communication Model time complexity = number of rounds Each node sends message to each of its neighbors

8. Discrete Algs for Mobile Wireless Sys8 Jia, Rajaraman, Suel  Finding smallest possible dominating set is NP- complete  Sequential approximation algorithm: greedy algorithm approximation ratio is logarithmic in maximum degree  of the graph  This paper: randomized distributed approximation algorithm: O(log n log  ) rounds w.h.p. approximation ratio is O(log  ) in expectation and O(log n) w.h.p.

9. Discrete Algs for Mobile Wireless Sys9 Sequential Greedy Algorithm  Given graph G, start with empty dominating set S  For any node v, define its span, denoted d v, to be number of uncovered nodes in {v} U N(v)  Greedy algorithm: Add node v with maximal span d v to S until all nodes are covered

10. Discrete Algs for Mobile Wireless Sys10 Greedy Algorithm Performance  Well-known theorem:  Greedy algorithm is H(  +1)-approximation  : max degree H(n) = 1+1/2+1/3+…+1/n  ln(n)

11. Discrete Algs for Mobile Wireless Sys11 Distributed Greedy Algorithm  Proceed in rounds, initially no node is in S  Each round has 3 steps: each node calculates its span each node sends (span,ID) to all nodes within distance 2 each node joins the dominating set S iff its (span,ID) is lexicographically higher than all others within distance 2

12. Discrete Algs for Mobile Wireless Sys12 Distributed Greedy Algorithm  Distributed algorithm with same approximation ratio as the greedy algorithm  However, algorithm can be very slow …

13. Discrete Algs for Mobile Wireless Sys13 Caterpillar Graph  Caterpillar graph (path of decreasing degrees):  Nodes along the "backbone" add themselves to S sequentially from L to R  (  n) rounds

14. Discrete Algs for Mobile Wireless Sys14 A Modification  Problem with caterpillar graph can be solved with this technique: each node rounds up its span to next smallest power of 2 (relaxing the greedy condition) node add itself to S if its (rounded span, ID) are largest among its 2-hop neighborhood Break ties randomly Max chain in caterpillar now has length O(log n)  But still a problem…

15. Discrete Algs for Mobile Wireless Sys15 Star-Complete Network  First, a node of the clique joins S, leaves remain uncovered  Leaves could be covered simultaneously but will be covered sequentially because all degree 2 nodes are neighbors  We need to be able to add nodes to S simultaneously if they cover sufficiently disjoint sets of nodes

16. Discrete Algs for Mobile Wireless Sys16 Local Randomized Greedy (LRG) Algorithm  Steps by node v in each round: 1.Compute span d(v) (d(v) = # of uncovered neighbors of v, including v if v is not covered) 2.d’(v) = d(v) rounded up to next smallest power of b 3.v is candidate if d’(v) ≥ d’(w) for all w in 2-neighborhood 4.If v not covered, support s(v) = # of candidates that cover v 5.med(v) = median of s(w) for all uncovered neighbors w (include v if uncovered) 6.If v is candidate, join dominating set with probability 1/med(v) b is parameter, trade off time vs. approx. ratio

17. Discrete Algs for Mobile Wireless Sys17 Notes on LRG Algorithm  Candidate selection: Greedy condition with constant factor relaxation to avoid long chains  Choosing median of support of neighbors: Half of the neighbors are covered by at most 1/med(v) candidates  max possible parallelism without ‘over- covering’ these nodes

18. Discrete Algs for Mobile Wireless Sys18 Time Complexity  Theorem: LRG terminates in (log b  )((c+1)log 1/d (bn)+2) rounds with probability at least 1 – 1/(n c–1 ), for sufficiently large n  I.e., w.h.p.  Proof uses a potential function argument as well as standard probabilistic techniques

19. Discrete Algs for Mobile Wireless Sys19 Approximation Ratio  Theorem: The expected size of the dominating set S produced by LRG is 4bH  times the optimal.  Proof sketch: By def, E[|S|] =  i E[|S i |] over each round i Show  i E[|S i |] ≤ 4b  u E[cost(u)] over all nodes u, where cost(u) = 1/d'(v), where v is a node that covers u and d'(v) is v's rounded-up span in the round when v covers u Show  u cost(u) ≤ H  times the optimal

20. Discrete Algs for Mobile Wireless Sys20 Approximation Ratio  Theorem: The size of the dominating set produced by LRG is O(log n) times the optimal, with high probability  See paper for proof.

21. Discrete Algs for Mobile Wireless Sys21 Tightness of Analyses  The approximation ratio of O(H  ) is optimal (cf. Johnson paper)  Paper describes a family of graphs on which the algorithm takes  (log n log  ) rounds w.h.p.

22. Discrete Algs for Mobile Wireless Sys22 Related Papers  Kuhn & Wattenhofer, "Constant-Time Distributed Dominating Set Approximation" Distributed dominating set algorithm based on greedy algorithm and LP relaxation techniques Achieves trade-off between time complexity and approximation Non-trivial approximation ratio in a constant number of rounds  Kuhn, Moscibroda, Wattenhofer, “The Price of Being Near-Sighted” extends the idea and achieves a better trade-off

23. Discrete Algs for Mobile Wireless Sys23 Connected Dominating Set  Connected dominating set: A dominating set that induces a connected sub-graph  How to transform a DS into a CDS? Given: Dominating set S of a graph G Call nodes u and v in S close if there is a path of length at most 3 connecting u and v

24. Discrete Algs for Mobile Wireless Sys24 Connected Dominating Set  Lemma: If G is connected, for every u,v in S, there is a sequence of nodes u=u 0, u 1, …, u k =v such that u i-1 and u i are close for all i  Proof: Take any path u=v 0, v 1, …, v k =v connecting u and v  For every v i, either it is in S : set u i = v i, or it has a neighbor w i in S: set u i = w i  Take the sequence u = u 0, u 1, …, u k-1, u k =v of nodes in S

25. Discrete Algs for Mobile Wireless Sys25 Connected Dominating Set uvv1v1 v2v2 w1w1 w2w2 v3v3

26. Discrete Algs for Mobile Wireless Sys26 Small Connected Dominating Set  Given dominating set S  Construct auxiliary graph H=(V H, E H ): V H = S for u,v in S: {u,v} in E H iff u, v are close (at dist. at most 3 in G)  From previous, H is connected iff G is connected Any connected subgraph of H gives a connected DS of G  Take a spanning tree of H to obtain a small CDS!  Since two close nodes u, v in S can be connected by at most 2 nodes, size of resulting connected dominating set is at most 3|S|

27. Discrete Algs for Mobile Wireless Sys27 Dubhashi et al.  How to do this by a distributed algorithm?  Simple observations: A round of a distributed algorithm on the auxiliary graph H can be simulated in 3 rounds on G Given a small dominating set S, it is sufficient to compute a sparse sub-graph of H to obtain a good connected dominating set  Paper gives a solution to the remaining problem: How to compute a sparse sub-graph of H?

28. Discrete Algs for Mobile Wireless Sys28 “Fault-Tolerant” Dominating Set  Parameter r  For each u not in S, there are r neighbors in S  Studied in Jia, Rajaraman, Suel paper Kuhn, Moscibroda, Wattenhofer: Fault-tolerant clustering in ad hoc …  Similar techniques and results

29. Discrete Algs for Mobile Wireless Sys29 Capacitated Dominating Set  Intuition: Dominating set nodes have to carry out tasks for their neighbors and are thus more loaded  Assume that every node has a capacity the bounds the maximal number of neighbors it can serve  Kuhn and Moscibroda, "Distributed approximation of capacitated dominating sets"