Khaled M. Alzoubi, Peng-Jun Wan, Ophir Frieder 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

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 2005/06/07 Eugene

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. 2005/06/07 Eugene

Dominating set: Given a graph G = (V, E), a dominating set of G is a subset D ⊆ V, such that . 2005/06/07 Eugene

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) 2005/06/07 Eugene

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. 2005/06/07 Eugene

Algorithms Das et al’s algorithm (1997) Wu and Li’s algorithm (1999) Stojmenovic et al’s algorithm (2001) Main algorithm (2002) 2005/06/07 Eugene

Das et al’s algorithm Greedily finds a minimal dominated set. Then finds a MST and output its internal nodes. 1 2 2005/06/07 Eugene

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} 2005/06/07 Eugene

Message complexity O(n2). Time complexity O(n2). Since n = k + 2k+1, the lower bounds is (lg n)/2–1 of the algorithm. (ratio = O(lg n)) Message complexity O(n2). Time complexity O(n2). The implementation lacks lack mechanisms to bridge two consecutive stages. 2005/06/07 Eugene

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. 1 2 3 4 5 6 7 8 9 10 11 2005/06/07 Eugene

Message complexity O(n2). Time complexity O(3). |CDS| = n |optCDS| = 2 ratio = n/2 Message complexity O(n2). Time complexity O(3). 2005/06/07 Eugene

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 1 2 3 4 5 6 7 8 9 10 11 2005/06/07 Eugene

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. 1 2 3 4 5 7 8 9 6 10 11 2005/06/07 Eugene

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. 1 2 3 4 5 7 8 9 6 10 11 2005/06/07 Eugene

Message complexity O(n) ~ O(n2). Time complexity O(n) ~ O(n2). |CDS| = n |optCDS| = 1 ratio = n Message complexity O(n) ~ O(n2). Time complexity O(n) ~ O(n2). 1 2 3 4 5 6 2005/06/07 Eugene

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 complexity O(nlogn). Time complexity O(n). 0, 1 1, 2 2, 3 1, 4 2, 5 3, 6 3, 7 4, 8 4, 9 5, 10 5, 11 1 2 3 4 5 6 7 8 9 10 11 2005/06/07 Eugene

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. 0, 1 1, 2 2, 3 1, 4 2, 5 3, 6 3, 7 4, 8 4, 9 5, 10 5, 11 2005/06/07 Eugene

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. 2005/06/07 Eugene

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 2005/06/07 Eugene

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. 2005/06/07 Eugene

Main algorithm (Dominating Tree) Message complexity O(n log n). Time complexity O(n). ratio = 2|U| – 1 = 2(4opt + 1) – 1 = 8opt + 1 0, 1 1, 2 2, 3 1, 4 2, 5 3, 6 3, 7 4, 8 4, 9 5, 10 5, 11 2005/06/07 Eugene

Conclusion 2005/06/07 Eugene