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

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### 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

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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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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-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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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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Das et al’s algorithm- Greedily finds a minimal dominated set.
- Then finds a MST and output its internal nodes.

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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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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.

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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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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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- 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.

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- 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.

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|CDS| = n

|optCDS| = 1

ratio = n

- Message complexityO(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.
- Message complexityO(nlogn).
- Time complexity O(n).

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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 ik} where ui is the ith node which is marked red. For any 1 jk, let Hj be the graph over {ui: 1 ij} 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.

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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 unit-disk graph G = (V, E) is at most 4opt + 1.

(opt = |MCDS|)

- Proof(1/2): Claim: Any independent set size is at most 5opt.

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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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 iopt, 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 iopt, at least one node in v1, v2, …, vi-1 is adjacent to vi. This implies that |Ui| 4.

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Main algorithm (Dominating Tree)- Message complexityO(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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