Xiaofeng Gao et al.: A Distributed Design for Minimum 2-Connected m-Dominating Set 555
1 Related Works
The research of virtual backbone have developed for
more than a decade. In 1997 Das and Bharghaven
[9]
firstly proposed the concept of virtual backbone. Two
years later, McKee and McMorris
[10]
discussed the
connected dominating set (CDS) and considered it
as the virtual backbone for a given network. From
then on, hundreds of research papers were proposed
to deal with minimum CDS problems under various
situations. Gandhi and Parthasarathy
[11]
summarized
the distributed algorithm for connected domination up
to 2007 and Zheng et al.
[12]
proposed a more detailed
survey for CDS in wireless ad-hoc networks.
However, most literatures for fault-tolerant virtual
backbone construction are completed after 2005. In
2005, Dai and Wu
[13]
designed three different
constructions for (k,k)-CDS and led the research to
the fault-tolerant virtual backbone construction. Then
Shang et al.
[14, 15]
gave the first centralized algorithm
of (k,m)-CDS. Almost during the same period, Thai et
al.
[7]
provided another centralized algorithm for (k,m)-
CDS, and Wu et al.
[3]
designed the first distributed
algorithm of (k,m)-CDS. In the next year, say, 2008,
this research group proposed an improvement of
their algorithm and gave another new construction
for (k,m)-CDS
[2]
. In the same year, Feng et al.
[16]
discussed about the construction of (2,m)-CDS and
proposed a centralized algorithm. Sun et al.
[17]
gave
two different algorithms for (2,2)-CDS in 2009. In
the same year, Wang et al.
[18]
presented a 2-connected
CDS in wireless networks. Bian et al.
[4]
discussed a
new MIS based distributed algorithm and improved
the algorithm for (1,m)-CDS. Kim et al.
[19]
proposed a
complex algorithm for (3,m)-CDS and figured out some
mistakes in previous researches. Li et al.
[5]
discussed
two algorithms for (1,m)-CDS and (2,m)-CDS and
gave some performance analysis. Li et al.
[1]
gave the
analysis of the distributed algorithm named Distributed
Deterministic Algorithm (DDA) in 2012, which is
the latest work in this field. Kim et al.
[6]
also did the
research of k-Center problem, which is close related to
(k,m)-CDS. Recently, some researches
[12, 20-22]
started
to focus on another variation of VB, which tries to
find a r-hop CDS with k-connectivity. Instead of
discussing undirected graph, Tiwari et al.
[23]
dealt with
the fault-tolerant VB based on the strongly connected
graph.
As shown in Table 1, we divide the algorithms
of (k,m)-CDS into four different categories after a
Table 1 Construction of (k, m)-CDS
Strategy Description
Dominating!Loop First construct a CDS and union (m
1) disjoint MIS’s. Then expand the
connectivity to k. Denoted as .1; 1/ !
.1; m/ ! .k; m/
Loop!Dominating First construct a CDS and then expand
new loops to CDS until all dominatees
are m-dominated. Denoted as
.1; 1/ ! .k; 1/ ! .k; m/
Probability-based Construct a (k,k)-CDS in unstable
methods like coloring or probability.
Pruning-based Construct a (k,m)-CDS by removing
the redundancy nodes.
careful study in this field. To construct a .k; m/-CDS,
most literatures like Refs. [1, 4, 5, 7, 17, 19] expend the
dominant number of CDS first and then improve the
connectivity, which becomes the first kind of methods
for .k; m/-CDS. We call it “Dominating!Loop”
method, denoted as .1; 1/!.1; m/!.k; m/. Briefly
speaking, all the works in this category first merge a
CDS with another (m1) disjoint MIS’s to get a (1,m)-
CDS, (some researchers also discuss the improvement
to make the MIS size smaller), and then they use
different ideas to expand the k-connectivity, such as
finding k different neighbors or finding the shortest
pathes between pairs of nodes in CDS. Finding shortest
path could only expand the connectivity one by one,
while finding different neighbors can directly get a
k-connected graph.
On the other hand, Ref. [17] tried to expand the
connectivity first and then expand the dominant
number. We call such a procedure as “Loop !
Dominating”, denoted as .1; 1/!.k; 1/!.k; m/. It
also constructs a CDS first and then adds new loops
into CDS to expand the dominant number until all
the dominatees are m-dominated (The construction
also guarantees 2-connectivity). Till now, Ref. [17]
is the only research of this category and it only fits
the situation when kD2. If we further generalize this
approach, we can firstly form a .k; 1/-CDS and then
expand the dominant number into m.
In some researches
[13]
probability methods were
used to construct a (k,k)-CDS. We call this strategy
“Probability-based” CDS construction. One of these
ideas is dividing the graph into k disjoint subgraphs
randomly, constructing CDS for each subgraph and
then connecting them together. The other idea is that
computing the probability of the appearance of each