CS547: Wireless Networking

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CS547 Wireless Networking

• Lecture 6 Virtual Backbone

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UDG Model of Communication Topology

• Distributed over a plane
• Equal maximum transmission range
• Communication topology unit-disk graph

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Virtual Backbone Connected Dominating Set

• Dominating Set (DS)

• Connected Dominating Set (CDS)

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Minimum Connected Dominating Set (MCDS)

• NP-hard in UDGs
• Logarithmic-approximality in general graphs
• Allows PTAS in UDGs

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Basic Properties

• ? domination number
• ?c connected domination number
• ? independence number
• MIS (arbitrary) maximal IS
• CDS is a DS ? ? ? ?c
• MIS is a DS ? ? ? MIS ? ?

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From MIS to CDS

• The distance between any pair of complementary
  subsets of a MIS is at most and may be three
  hops.
• ? Any MIS can be connected by at most 2(MIS-1)
  nodes
• ? ?c ? 3MIS-2 ? 3?-2

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Tighter Relation

• There exists an MIS s.t. the distance between any
  pair of its complementary subsets of a MIS is
  exactly two hops.
• can be constructed easily (see later slides)
• ? Such MIS can be connected by at most MIS-1
  nodes
• ? ?c ? 2MIS-1 ? 2?-1

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MIS Induced by A Ranking

• Ranking strict total ordering.
• Static ID only
• Dynamic (degree, ID), (degree, location).
• MIS induced by a ranking
• U ?
• While (V ? ?)
• Add the node u in V with lowest rank to U
• Remove u and all its neighbors from V
• Update the ranks of nodes in V if the ranking is
  dynamic

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Ranking Based on Tree-Level and ID

• Given a rooted spanning tree, the rank of a node
  is the ordered pair of its tree level and its ID.

Root/leader
Level 0
Level 1
Level 2
Level 3
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Properties of This Ranking

• Static
• The distance between any pair of complementary
  subsets of the MIS induced by this rank is
  exactly two hops.

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Two-phased heuristic

• Construct a maximal independent set U satisfying
  that any two complementary subsets are separated
  by two hops
• Add additional nodes W to connect U, i.e., U?W is
  a CDS

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Analysis on Approximation Ratio

• WU-1 ? U?W2U-1
• Upper bound on U in terms of size of MCDS ?
• Densest packing of independent nodes in the
  union of unit-disks centered at the nodes of a
  connected UDG

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(Loose) upper bounds on independence number

• Any unit-disk can pack at most 5 independent
  nodes ? ? ? 5? ? 5?c
• The union of any two unit-disks whose centers are
  apart by at most one can pack at most 9
  independent nodes ? ? ? 4?c 1

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Distributed Construction of MIS

• Initially, each node with the lowest rank in its
  neighborhood becomes black and broadcasts a
  DOMINATOR message.
• Whenever a node receives a DOMINATOR message for
  the first time, it becomes gray and broadcasts a
  DOMINATEE message.
• Whenever a node has received the DOMINATEE
  messages from all lower-ranked neighbors, it
  becomes black and broadcasts a DOMINATOR message.
• Dynamic ranking require additional rank updating
  and/or synchronization.

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Color Markup

• rank (level,ID).
• Initially all nodes white.
• The leader first marks itself black and send out
  a DOMINATOR message.
• All other nodes act according to the following
  diagram.

white
DOMINATEE messages from all low-rank neighbors
DOMINATOR message from a neighbor
gray
black
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Distributed Construction of CDS

• Phase 1 Leader election and and spanning-tree
  construction
• using the existing algorithm with O(nlogn)
  messages in O(n) time.
• Phase 2 Level calculation along the spanning
  tree.
• Phase 3 MIS construction.
• Phase 4 Dominating tree (T) construction. All
  internal nodes of T form a CDS.

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Phase 4 Construction of Dominating Tree T

• Select the root of T the leader choose its
  (gray) neighbor adjacent to most black nodes.
• The root sends an INVITE2 message.
• Upon receiving an INVITE2 message for the first
  time, a dominatee puts the sender as its parent
  in T, replies to the sender with a JOIN message,
  and broadcasts an INVITE1 message.
• Upon receiving an INVITE1message for the first
  time, a dominatee dominator puts the sender as
  its parent in T, replies to the sender with a
  JOIN message, and broadcasts an INVITE2 message.
  
• Upon receiving a JOIN message towards itself, a
  node puts the sender as its child in T.

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An Example MIS Construction
0
0
4
12
5
2
8
10
6
11
9
3
1
7
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An Example Dominating Tree Construction
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Case 1 OPT contains a black node

• no. of black nodes ? 14(opt-1) 4opt- 3
• no. of internal gray nodes in T ? 4opt- 3
• no. of internal nodes in T ? 8opt - 6

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Case 2 OPT contains no black node

• k no. of black nodes adjacent to the root of T
• no. of black nodes ? k4(opt-1)
• no. of internal gray nodes in T ? 1 4(opt-1)
• no. of internal nodes in T ? 8opt k-7? 8opt -2

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Performance analyses

• The size of the CDS is at most 8opt - 2
• Approximation Factor is 8.
• Message complexity O(nlogn)
• Dominated by Phases 1 the leader election.
• All other phases uses linear messages.
• Time complexity linear

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Stojmenovic et al's Algorithm Description

• Construct an MIS U.
• Construct a clustering
• Every node not in U joins the cluster centered at
  the neighboring cluster-head with the lowest
  rank.
• The border-nodes are those which are adjacent to
  some node from a different cluster.
• The CDS consists of U and border-nodes.

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Stojmenovic et al's Algorithm Performance

• Rank is ID only
• Approximation factor n
• Message complexity ?(n)
• Time complexity ?(n)
• Dynamic Rank
• Approximation factor n/2
• Message complexity O(n2)
• Time complexity O(n2)

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Das et al's Algorithm Description

• Construct a dominating set U
• Translation of Chvatal's greedy algorithm for Set
  Cover
• Construct a spanning forest F
• Each dominatee picks a unique node in U as its
  neighbor in F
• Union of stars centered at the dominators
• Expand the forest F into a spanning tree T
• Add only edges across the tree components
• All internal nodes of T form a CDS

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Das et al's Algorithm Performance

• Upper bound 3H(?) by Das et al
• Lower bound
• Message complexity O(n2)
• Time complexity O(n2)

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Wu and Li's Algorithm Description

• A node is locally redundant if it has
• either a neighbor with larger ID which dominates
  all its other neighbors,
• or two adjacent neighbors with larger IDs which
  together dominate all its other neighbors
• All nodes but the locally redundant ones form a
  CDS
• Applicable to only non-complete topology

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Wu and Li's Algorithm Performance

• Trivial upper bound n/2
• Lower bound n/2
• Worst possible!
• Message complexity O(m)
• Time complexity O(?3)
• Instead of O(?2) claimed by Wu and Li