Performance-Guaranteed Strongly
Connected Dominating Sets in
Heterogeneous Wireless Sensor Networks
Chunyan Liu∗ , Hejiao Huang∗†‡, Hongwei Du∗ and Xiaohua Jia∗ ∗Harbin Institute of
Technology Shenzhen Graduate School, China. †Shenzhen Key Laboratory of Internet
Information Collaboration, China. ‡Corresponding author:[email protected]
IEEE INFOCOM 2016
Presentation prepared by Rakefet Menda
The general goal
find a special Strongly Connected Bidirectional Dominating
Set within minimum routing cost for each pair of nodes in
directed graphs.
Efficient routing
CDS- connected dominating set
To achieve efficient broadcasting and routing, the size of CDS should be
kept as small as possible while the length of routing path does not
increase a lot through the nodes in the CDS. However, sometimes, CDS
with smaller size may result in some terrible long routing paths.
Minimum connected dominating set
(virtual backbone- VB)
The shortest path through VB from 1 to 10 is
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
length - 10
Minimum connected dominating set
(virtual backbone- VB)
The shortest path through VB from 1 to 10 is
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
length - 10
connected dominating set
If some nodes (10) are added to the CDS, the
shortest path from 1 to 10 based on the VB is
{1, 10}
length - 2
Minimum connected dominating set
(virtual backbone- VB)
The shortest path through VB from 1 to 10 is
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
length - 10
connected dominating set
If some nodes (10) are added to the CDS, the
shortest path from 1 to 10 based on the VB is
{1, 10}
length - 2
Longer routing paths will cause a lower delivery ratio. If the size of the CDS is too large, too
many nodes are involved to forward messages.
Problem statement
Our main goal is to solve the α-MOC-SCBDS problem(α Minimum rOuting Cost Strongly Connected Bidirectional Dominating Set)
We want to find set S of graph G which is a SCBDS with routing cost less than
α*Minimum routing cost.
 finding minimum connected dominating set is a np hard problem.
Some assumptions and definitions
• G = (V, E) Direct graph
• S⊆V
•
• S is a α-MOC-SCBDS if and only if:
• Each node v ∈ V has a transmission range rv ∈ [rmin, rmax]
Centralized algorithm
Nin(v) - the set of nodes from which node v can receive messages
Nout(v) - the set of nodes which can receive messages from v
N(v) = Nin(v) ∩ Nout(v)
N +(v) = N(v) ∪ {v}
Centralized algorithm
Nin(v) - the set of nodes from which node v can receive messages
Nout(v) - the set of nodes which can receive messages from v
N(v) = Nin(v) ∩ Nout(v)
N +(v) = N(v) ∪ {v}
• First stage - find the Bidirectional Dominating Set (BDS) C of G using a
greedy method (Algorithm 1):
1: Step1: Select a node v ∈ V with the maximal transmission range (or minimal node
ID) join the C: C = C ∪ {v}.
2: Step2: V = V − N +(v)
3: Step3: repeat Step1 and Step2 until V = ∅
4: Output: C
Centralized algorithm
Note that, for any dominated node v, if it is dominated by u, then u ∈ N +(v).
Therefore, a bidirectional dominating set is constructed
Lemma In a directed disk graph G = (V, E), the size of Dominating Set got by
Algorithm 1 is upper bounded by (2ρ + 1)^2 opt
where ρ = rmax / rmin and opt is the size of the optimal solution of the Minimum
Strongly Connected Bidirectional Dominating Set (MSCBDS) problem.
Centralized algorithm
• Second stage - connect the BDS as a SCBDS (Algorithm 2) :
For any pair nodes u, v ∈ C, if d(u, v) ≤ 4, all nodes on the path join the
SCBDS S. Then S forms a SCBDS of G.
For any nodes u, v ∈ V \ C, dS(u, v) ≤ αd(u, v), when α = 5.
Centralized algorithm
Lemma 8. Given a directed graph G = (V, E) and a Bidirectional Dominating Set C of
G. ∀u, v ∈ V with d(u, v) = 2, find a shortest path p(u, v) through their dominating
nodes and let all nodes on the path join the set S. S is a strongly connected
dominate set and S is with the upper bound
where ρ = rmax /rmin
Distributed Algorithm
In the distributed environment, each node can only sense its incoming
neighbors but outgoing neighbors. For an arbitrary vertex v ∈ V ,
let N(v) denote the neighbors of v that are within the coverage of v
while v is within the coverage of each node in N(v).
It is also that: N(v) = {u|dist(u, v) ≤ min{ru, rv}}.
Distributed Algorithm
First step - Distributed algorithm to construct a BDS (Algorithm 3) :
• Each node in the network is assigned a positive number ID and marked
white.
• Firstly, each node generates a neighbor list through a handshake of
broadcasting and replying.
• Next, in each iteration, each white node compares its ID with those of
white neighbors. The one with the smallest ID among all its white
neighbors is marked black and all its neighbors are marked gray.
• Until all nodes are marked either black or gray. Then all black nodes form a
Dominating Set.
Distributed Algorithm
Second step - Distributed algorithm to construct SCBDS with guaranteed
routing cost (Algorithm 4) :
The nodes are connected in order to form a SCBDS.
• Each node is located in the state of dominator or dominated and different
ID. The state is assigned according to the result of the Algorithm 3.
• Similarly with Algorithm 3, each node generates a neighbor list through a
handshake of broadcasting and replying.
Distributed Algorithm
• Next, each node with the state of Dominator broadcast its ID to all its
neighbors to find a shortest path which the length is not larger than 4 to
those nodes with the state of Dominator.
• Let all nodes on the paths join the set S.
• Until all node pairs with the state of Dominator have been traversed while
the shortest path length between them is no more than 4.
S is a SCBDS
Distributed Algorithm
• Next, each node with the state of Dominator broadcast its ID to all its
neighbors to find a shortest path which the length is not larger than 4 to
those nodes with the state of Dominator.
• Let all nodes on the paths join the set S.
• Until all node pairs with the state of Dominator have been traversed while
the shortest path length between them is no more than 4.
S is a SCBDS
Distributed Algorithm
The Algorithm 3 and 4 are the distributed version of Algorithm 1 and 2.
Therefore, S constructed in Algorithm 4 forms a SCBDS, and the
distributed algorithms have the same performance ratio with the
centralized algorithms
Distributed Algorithm
The Algorithm 3 and 4 are the distributed version of Algorithm 1 and 2.
Therefore, S constructed in Algorithm 4 forms a SCBDS, and the
distributed algorithms have the same performance ratio with the
centralized algorithms.
Lemma The distributed algorithms α-MOC-SCBDS-D produces a SCBDS
with the upper bound
where ρ = rmax /rmin
Simulation
• In the simulation, we model a wireless sensor network as a set of
nodes randomly deploy in a 100×100 Euclidean plane. The number of
nodes varies among 100, 150, 200, 250, 300, 350 and 400.
• Each node has a fixed transmission range in the range of [rmin, rmax].
rmin changes among 15, 20, 25 and 30. rmax=ρ*rmin where
ρ=rmax/rmin varied among 1.25, 1.5, 1.75 and 2.
Simulation
Compared results:
• CDS-BD-D is a approximate algorithm that generates a CDS whose size and
average path length are bounded.
The author improve CDS-BD-D as SCBDS-BD-D using forward tree and
backward tree to work on directed graphs.
Simulation
Compared results:
• CDS-BFS, algorithm under unidirectional graph. SCDS size is bounded, but
dominated nodes may not communicate with its dominating nodes.
Improve CDS-BFS as SCBDS-BFS through constructing bidirectional
dominating set to work on directed graphs.
Simulation
Simulation
Simulation
Simulation
Questions?