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
2005/06/07
93321530
游精允
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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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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• Dominating set:
Given a graph G = (V, E), a dominating set of G is a
subset D ⊆ V, such that V  N [ x].

xD
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• 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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Lower Bound on Message
Complexity
• Theorem 1: [2] In asynchronous rings with point-topoint 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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Algorithms
•
•
•
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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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Das et al’s algorithm
• Greedily finds a minimal dominated set.
• Then finds a MST and output its internal nodes.
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k
n  k  2  2 2i 1  k  2 k 1
i 1
deg( vi )  2  2i 1  (k  1)  2  2i  k  1 1  i  k
k
deg( u j )   2i 1  k  1  2 k  k 1  j  2
i 1
deg(vk) = 2k+k+1
deg(u1) = 2k+k
deg(vk–1) = 2k–1
deg(u1) = 2k–1–1
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CDS: {v1, v2, … , vk}
optCDS: {u1, u2}
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• 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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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.
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|CDS| = n
|optCDS| = 2
ratio = n/2
• Message complexity O(n2).
• Time complexity O(3).
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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
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1. Each node has a unique rank parameter as the ID.
2. Each node which has the lowest rank among all neighbors
broadcasts a message declaring itself as a cluster-head.
3. 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.
4. Whenever a node has received the giving-up
messages from
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all of its neighbors with lower
ranks, if there is any, it
6 itself8 as a5 cluster-head.
1
broadcasts a message declaring
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5. After a node learns the status of all neighbors, it
joins the cluster centered at the neighboring clusterhead 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.
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|CDS| = n
|optCDS| = 1
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ratio = n
4
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• Message complexity O(n) ~ O(n2).
• Time complexity O(n) ~ O(n2).
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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.
3,
6 6
1,
2 2
5,
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• Message complexity O(nlogn).
• Time complexity O(n).
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5,
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5 5
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2,
3 3
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1 1
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• 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.
3, 6
1, 2
5, 10
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5, 11
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4, 8
2, 3
0, 1
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• 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.
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• Lemma 8: The size of any independent set in a unitdisk 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
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• Lemma 8: The size of any independent set in a unitdisk 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.
opt
| U |  | U i |  5  4(opt  1)  4opt  1.
i 1
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Main algorithm
(Dominating Tree)
• Message complexity O(n log n).
• Time complexity O(n).
ratio = 2|U| – 1
= 2(4opt + 1) – 1
= 8opt + 1
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Conclusion
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