New Distributed Algorithm for Connected Dominating Set in Wireless Ad Hoc Network

1. New Distributed Algorithm for Connected Dominating Set in Wireless Ad Hoc Network Khaled M. Alzoubi, Peng-Jun Wan, Ophir Frieder Proceedings of the 35th Hawaii International Conference on System Sciences, Jan. 2002 93321530 游精允 2005/06/07

2. Outline • Introduction • Lower Bound on Message Complexity • Algorithms • Das et al’s algorithm • Wu and Li’s algorithm • Stojmenovic et al’s algorithm • Main algorithm • Conclusion Eugene

3. Introduction • Unit-disk graph: A geometric graph in which there is an edge between two nodes if and only if their distance is at most one. Eugene

4. Dominating set: Given a graph G = (V, E), a dominating set of G is a subset D ⊆ V, such that . Eugene

5. Connected dominating set: (1) CD is a dominating set of G. (2) G[D], the subgraph induced by D is connected. Minimum connected dominated set (MCDS) Eugene

6. Lower Bound on Message Complexity • Theorem 1: [2] In asynchronous rings with point-to-point transmission, any distributed algorithm for leader election in sends at least (n log n) messages. • Theorem 2/3/4: In asynchronous wireless ad hoc networks whose unit-disk graph is a ring, any distributed algorithm for leader election / spanning tree / nontrivial CDS sends at least (n log n) messages. Eugene

7. Algorithms • Das et al’s algorithm (1997) • Wu and Li’s algorithm (1999) • Stojmenovic et al’s algorithm (2001) • Main algorithm (2002) Eugene

8. 1 1 1 1 1 2 2 1 1 1 1 2 2 1 1 1 1 1 Das et al’s algorithm • Greedily finds a minimal dominated set. • Then finds a MST and output its internal nodes. Eugene

9. deg(vk) = 2k+k+1 deg(u1) = 2k+k deg(vk–1) = 2k–1 deg(u1) = 2k–1–1 CDS: {v1, v2, … , vk} optCDS: {u1, u2} Eugene

10. Since n = k + 2k+1, the lower bounds is (lg n)/2–1 of the algorithm. (ratio = O(lg n)) • Message complexityO(n2). • Time complexity O(n2). • The implementation lacks lack mechanisms to bridge two consecutive stages. Eugene

11. 6 2 10 9 7 5 1 3 11 8 4 Wu and Li’s algorithm • The initial connected dominating set U consists of all nodes which have at least two non-adjacent neighbors. • Locally redundant: It has either a neighbor in U with larger ID which dominates all other neighbors of u, or two adjacent neighbors with larger IDs which together dominates all other neighbors of u. Eugene

12. |CDS| = n |optCDS| = 2 ratio = n/2 • Message complexityO(n2). • Time complexity O(3). Eugene

13. 6 2 10 9 7 5 1 3 11 8 4 Stojmenovic et al’s algorithm • Independent set: Given a graph G = (V, E), a independent set of G is a subset S ⊆ V, such that no two vertices of S are adjacent in G. • A maximal independent set is a independent dominating set Eugene

14. 7 2 10 6 8 5 1 3 11 9 4 • Each node has a unique rank parameter as the ID. • Each node which has the lowest rank among all neighbors broadcasts a message declaring itself as a cluster-head. • Whenever a node receives a message for the first time from a cluster-head, it broadcasts a message giving up the opportunity as a cluster-head. • Whenever a node has received the giving-up messages from all of its neighbors with lower ranks, if there is any, it broadcasts a message declaring itself as a cluster-head. Eugene

15. 7 2 10 6 8 5 1 3 11 9 4 • After a node learns the status of all neighbors, it joins the cluster centered at the neighboring cluster-head with the lowest rank by broadcasting the rank of such cluster head. The border-nodes are those which are adjacent to some node from a different cluster. Eugene

16. 1 5 2 6 4 3 |CDS| = n |optCDS| = 1 ratio = n • Message complexityO(n) ~ O(n2). • Time complexity O(n) ~ O(n2). Eugene

17. 6 2 10 9 7 5 1 3 11 8 4 3, 6 1, 2 5, 10 4, 9 3, 7 2, 5 0, 1 2, 3 5, 11 4, 8 1, 4 Main algorithm (MIS) • The distributed leader election algorithm. (1998) • O(n) time complexity and O(n log n) message complexity, to construct a rooted spanning tree T rooted at a node v. • Each node identifies its tree level with respect to T. • The ranks of all nodes are sorted in the lexicographic order. • Message complexityO(nlogn). • Time complexity O(n). Eugene

18. 3, 6 1, 2 5, 10 4, 9 3, 7 2, 5 0, 1 2, 3 5, 11 4, 8 1, 4 • Theorem 7: The distance between any pair of complementary subsets of U is exactly two hops. Proof(1/2): Let U = {ui: 1 ik} where ui is the ith node which is marked red. For any 1 jk, let Hj be the graph over {ui: 1 ij} in which a pair of nodes is connected by an edge if and only if their graph distance in G is two. Since H1 consists of a single vertex, it is connected trivially. Eugene

19. Theorem 7: The distance between any pair of complementary subsets of U is exactly two hops. • Proof(2/2): Assume that Hj-1 is connected for some j 2. When the node uj is marked red, its parent in T must be already marked orange. Thus, there is some node ui with 1 i < j which is adjacent to uj ’s parent in T. So (ui, uj) is an edge in Hj. As Hj-1 is connected, so must be Hj. Eugene

20. Lemma 8: The size of any independent set in a unit-disk graph G = (V, E) is at most 4opt + 1. (opt = |MCDS|) • Proof(1/2): Claim: Any independent set size is at most 5opt. Let U be any independent set of V , and let T* be any spanning tree of an MCDS. Consider an arbitrary preorder traversal of T given by v1, v2, …, vopt. U U1 U2 …… Uopt Eugene

21. Lemma 8: The size of any independent set in a unit-disk graph G = (V, E) is at most 4opt + 1. (opt = |MCDS|) • Proof(2/2): Let U1 be the set of nodes in U that are adjacent to v1. For any 2 iopt, let Ui be the set of nodes in U that are adjacent to vi but none of v1, v2, …, vi-1. |U1|  5, For any 2 iopt, at least one node in v1, v2, …, vi-1 is adjacent to vi. This implies that |Ui|  4. Eugene

22. 3, 6 1, 2 5, 10 4, 9 3, 7 2, 5 0, 1 2, 3 5, 11 4, 8 1, 4 Main algorithm (Dominating Tree) • Message complexityO(n log n). • Time complexity O(n). ratio = 2|U| – 1 = 2(4opt + 1) – 1 = 8opt + 1 Eugene

23. Conclusion Eugene