1. 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
游精允
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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
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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.
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4. Dominating set:
Given a graph G = (V, E), a dominating set of G is a subset D ⊆ V, such that
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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)
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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.
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7. Algorithms Das et al’s algorithm (1997) Wu and Li’s algorithm (1999)
Stojmenovic et al’s algorithm (2001)
Main algorithm (2002)
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8. Das et al’s algorithm Greedily finds a minimal dominated set.
Then finds a MST and output its internal nodes. 1 2
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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}
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10. 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.
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11. 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
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12. Message complexity O(n2). Time complexity O(3).
|CDS| = n
|optCDS| = 2
ratio = n/2
Message complexity O(n2).
Time complexity O(3).
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13. 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
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14. 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
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15. 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
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16. 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
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17. 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
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18. 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
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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  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.
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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
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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  i  opt, at least one node in v1, v2, …, vi-1 is adjacent to vi. This implies that |Ui|  4.
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22. 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
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23. Conclusion
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