On Calculating Connected Dominating Set for Efficient Routing in Ad
Hoc
Wireless Networks
Jie Wu and Hailan Li
Department of Computer Science and Engineering
Florida Atlantic University
Boca F&ton, FL 33431
{jie, hli}@cse.fau.edu
Abstract
Efficient routing among a set of mobile hosts (also called
nodes) is one of the most important functions in ad-hoc
wireless networks. Routing based on a connected dominat-
ing set is a frequently used approach, where the searching
space for a route is reduced to nodes in the set. A set is dom-
inating if all the nodes in the system are either in the set
or neighbors of nodes in the set. In this paper, we propose
a simple and efficient distributed algorithm for calculating
connected dominating set in ad-hoc wireless networks, where
connections of nodes are determined by their geographical
distances. Our simulation results show that the proposed
approach outperforms a classical algorithm. Our approach
can be potentially used in designing efficient routing algo-
rithms based on a connected dominating set.
Keywords:
ad hoc wireless networks, dominating sets,
mobile computing,
routing, simulation
1 Introduction
An ad hoc wireless network is a special type of wireless net-
work in which a collection of mobile hosts with wireless net-
work interfaces may form a temporary network, without the
aid of any established infrastructure or centralized adminis-
tration. If two hosts that want to communicate are outside
their wireless transmission ranges, they could communicate
only if other hosts between them in the ad hoc wireless net-
work are willing to forward packets for them.
We can use a unweighted graph G = (V, E) to represent
an ad hoc wireless network, where V represents a set of
wireless mobile hosts and E represents a set of edges. To
simplify our discussion, we assume all mobile hosts have
same transmission ranges, in other words an edge between
host pairs {v, u} indicates that both hosts v and u are within
their wireless transmitter ranges. Thus the corresponding
graph will be an undirected graph.
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Routing in ad hoc wireless networks requires fast conver-
gence and low communication overhead. Routing informa-
tion has to be localized to adapt quickly to network topologi-
cal changes. Connected-dominating-set-based routing can be
a solution to this kind of network environment. A subset of
the vertices (also name as gateways) of a graph is a domi-
nating set if every vertex not in the subset is adjacent to at
least one vertex in the subset.
The main advantage of connected-dominating-set-based
routing is that it centralizes the whole network into small
connected dominating set subnetwork, which means only
gateway hosts keep routing information, so that as long as
network topological changes do not affect this subnetwork
there is no need to recalculate routing tables.
Since finding a minimum connected dominating set is
NP-complete for most graphs, we propose a simple distribut-
ed approximation algorithm that can quickly determine a
connected dominating set in a given connected graph, which
represents an ad hoc wireless network. We show that pro-
posed approach outperforms a classical approach in terms
of complexity and average size of dominating set. We also
discuss ways to update and recalculate the dominating set
when the underlying graph changes with the movement of
mobile hosts. We also briefly describe efficient routing using
connected dominating set.
This paper is organized as follows: Section 2 overviews
related works. Section 3 presents the proposed approach on
finding a small connected dominating set. The connected
dominating set updating and recalculation are discussed in
Section 4. Section 5 briefly describes routing protocol in ad
hoc networks. Performance evaluation is done in Section 6,
where our algorithm is compared with the one proposed by
Das et al [3]. Finally, in Section 7 we conclude this paper
and discuss future work.
2 Previous Works
There are numerous routing protocols [l, 2, 4, 3, 5, 7, 8, 9,
10, 11, 121, particularly for wireless networks, are proposed.
Among them, [l, 5, 7, 10, 111 have been reviewed in detail
in [9]. In the rest of this section, we will concentrate on
approaches that are dominating-set-based [2, 4, 3, 8, 9, 121.
The cluster-based algorithm [9] divides a given unweigh-
ted graph into a number of overlapping, clusters. One (or
7
Figure 1: The example of ad hoc wireless network
more) representative node, called a boundary node, is se-
lected from each cluster to form a connected dominating
set subnetwork. The routing protocol is completed in two
phases: cluster formation and cluster maintenance. During
the cluster formation, the network is viewed as a dynami-
cally growing system, in other words, assume that mobile
hosts are inserted into the network sequentially. Therefore,
each node needs information of the entire network topology.
The algorithm runs O(Y) rounds, for each round its com-
plexities are O(A”) in terms of time and O(B + A) in terms
of messages, where v is the number of hosts in the network,
B is the number of boundary nodes, and A is the maximum
node degree.
Das et al proposed a series of routing algorithms [2, 4, 3,
121 for the ad-hoc wireless network. Similar to the cluster-
based routing [9], the idea is to identify a subnetwork that
forms a minimum connected dominating set (MCDS). Each
node in the subnetwork is called a spine node or backbone
node (we call it gateway in our proposed algorithm). Das
et al proposed a distributed version of Guha and Khuller’s
approximation algorithm [6] for calculating minimum con-
nected dominating set. This distributed algorithm has been
used in all the papers by Das et al [2, 4, 3, 121.
This MCDS calculation algorithm has two main advan-
tages over to the cluster-baaed approach [9]. First, each node
only needs 2-distance neighborhood information, unlike [9],
in which each node needs information of the entire network
topology. Second, the algorithm runs O(7) rounds. The
over all complexities are O(7A2 + v) in terms of time and
O(Av7 + m + Y log v) in terms of messages, where 7 is the
number of nodes in the resultant dominating set, m is the
number of edges, Y is the number of node, and A is the max-
imum node degree. This is an improvement compared to [9]
in terms of time complexity although it has higher message
complexity over [9] in the worst case. The main drawback of
this algorithm is that it still needs a non-constant number
of rounds to determine a connected dominating set.
Methods in [2, 4, 3, 121 differ in the way routing ta-
bles are constructed and propagated to non-MCDS nodes.
The requirement for shortest paths adds one additional di-
mension of complexity. Because the set of MCDS nodes
(dominating set) may not include ail intermediate nodes of
a shortest path, the routing process cannot be restricted to
MCDS nodes. In other words, in order to compute a rout-
ing table, each MCDS node needs to know entire network
topology. An all-pairs shortest path algorithm is actually
running on G, not on the reduced subnetwork of MCDS
nodes. Therefore, it may lose part of the original goal of
network centralization.
Another extreme approach, as proposed by Johnson [8],
uses dynamic source routing without constructing any rout-
ing tables. Normally, the resultant routing path is not the
shortest. However, this protocol adapts quickly to routing
changes when host movement is frequent, yet requires little
or no overhead during periods in which hosts move less fre-
quently. The approach consists of route discovery and route
maintenance. Route discovery allows any host to dynami-
cally discover a route to a destination host. Each host also
maintains a route cache in which it caches source routes that
it has learned. Unlike routing table approaches that have to
perform periodic routing updates, route maintenance only
monitors the routing process and informs the sender of any
routing errors.
3 Proposed Approach
As mentioned early, we will focus only on formation of a
dominating set. Some desirable features are: (1) The for-
mation process should be distributed and simple. Ideally, it
requires only local information and constant number of iter-
ative rounds of message exchanges among neighboring hosts.
(2) The resultant dominating set should be connected and
close to minimum. (3) The resultant dominating set should
include all intermediate nodes of any shortest path. In this
case, an all-pair shortest paths algorithm only needs to be
applied to the subnetwork containing the dominating set.
Marking process.
We propose a marking process that
marks every vertex in a given connected and unweighted
graph G = (V,E). m(v) is a marker for vertex v E V,
which is either T (marked) or F (unmarked). We will show
later that marked vertices form a connected dominating set.
We assume that all vertices are unmarked initially. N(v) =
{ul{v, u} E E} represents the open neighbor set of vertex v
and v has N(v) initially. The marking process is following:
(1) Initially assign marker F to every v in V. (2) Every v
exchanges its open neighbor set N(v) with all its neighbors.
(3) Every v assigns its marker m(v) to T if there exist two
unconnected neighbors.
In the example of Figure 1, N(A) = {B,D}, N(B) =
{A,C,D}, N(C) = {B,E}, N(D) = {A,B}, and N(E) =
{C}. After the the Step 2 of the marking process. Vertex A
has N(B) and N(D), B has N(A), N(C), and N(D), C has
N(B) and N(E), D has N(A) and N(B), and E has N(C).
Based on Step 3, only vertic,eT B and C are marked T.
Properties.
Assum: V 1s the set of vertices that are
marked T in V, i.e.,
V = {v(v E V,m(v) = 2’). The
re,duced eaph G’ is the subgraph of G induced by V’) i.e.,
G = G[V 1. The following two theorems show that G is a
dominating set of G and it is connected.
THEOREM 1:
Given a G = (V, E) that is connected, but not
completely connected, the vertex subset V’, derived from the
marking process, forms a dominating set of G.
PROOF:
Ran$omly select a vertex v in G. We show that v
is either in V (a set of vertices in V that are marked T) or
adjacent to a vertex in V’ . Assume v is marked F, if there
is at least one neighbor marked T, the theorem is proved.
When all its neighbors are marked F, we consider the follow-
ing two cases: (1) All the other vertices in G are neighbors of
8
v. Based on the marking process and the fact that m(v)=F,
all these neighbors must be pairwise connected, i.e., G is
completely connected. This contradicts to the assumption
that G is not completely connected. (2) There is at least one
vertex u in G that is not adjacent to vertex v. Construct
a shortest path, {v,vi,vz, . . . .
u}, between vertices v and u.
Such a path always exists since G is a connected graph. Note
that 212 is u when v and u are 2-distance apart in G, i.e.,
do(v,u) = 2. Also, v and vz are disconnected; otherwise,
{v,
212,
..I
u} is a shorter path between v and u. Based on
the marking process, vertex vi, with both v and 212 as its
neighbors, must be marked T. Again this contradicts to the
assumption that v’s neighbors are all marked F.
0
When the given G is completely connected, all vertices
are marked F. This make sense, since if all vertices are di-
rectly connected, there is no need of gateway hosts.
THEOREM 2: The reduced graph G’ = G - V’ is a connected
mph.
PROOF: We prove this theorem by contradiction. Assume
G’ is disconnected and v and u are two disconnected vertices
inG’. AssumedisG(v,u) =L+l > land{v,vi,vr,...,v~,u}
is a shortest path between vertices v and u in G. Clearly, all
vr,vz, . . ..vk are distinct and among them there is at least
one vi such that m(vi) = F (otherwise, v and u are con-
nected in G’). On the other hand, the two adjacent ver-
tices of v;, vi-1 and vi+i, are not connected in G (other-
wise, {v, vi, 02, . . . .
vk,u} is not a shortest path). Therefore,
m(vi) =T based on the marking process.
cl
Vertices in a dominating set are called gateway nodes
and vertices outside a dominating set are called non-gateway
nodes. The next theorem shows that, except for source and
destination vertices, all vertices in a shortest path are con-
tained in the dominating set derived from the marking pro-
cess.
THEOREM 3: The shortest path between any two nodes does
not include any non-gateway node as an intermediate node.
PROOF: We prove this theorem also by contradiction. As-
sume a shortest path between two vertices v and u includes
a non-gateway node vi as an intermediate node, in other
words, this path can be represented as {v, . . . . vi-i, vi, v;+i,
..*,
u}. We label the vertex that precedes vi on the path
a~ vi-i, similarly, the vertex that follows vi on the path
a.3 Vi+l.
Because vertex v; is a non-gateway node, i.e.,
m(vi)=F, there must be a connection between vi-1 and ui+i.
Therefore, a shorter path between v and u can be found as
{v, . . .
j Vi-l, Vi+lj ...)
u}. This contradicts to the original as-
sumption.
0
Since we proposed an approximation algorithm, in some
cases, the resultant dominating set is trivial, i.e., V = V’.
For example, any vertex-symmetric graph will generate a
trivial connected dominating set using the proposed marking
process. However, the marking process is efficient for the ad
hoc wireless mobile network in average case. Our simulation
results (to be discussed later) confirm this observation.
Extensions. In the following, we propose two rules to
reduce the size of a connected dominating set generated from
the marking process. We first assign a distinct id, id(v), to
each vertex v in G’. N[v] = N(v)
U
{v} is a closed neighbor
set of v, as oppose to the open one N(v).
Figure 2: Two samples.
RULE 1: Consider two vertices v and u in G’. If N[v] C
N[u] in G and id(v) < id(u), change the marker of v to F
if
node v is marked, i.e., G’ is changed to G’ - {v},
The above rule indicates when the closed neighbor set
of v is Fevered by the one of u, vertex v can be removed
from G if v’s id is smaller than u’s. It is easy to prove
that G - {v} is still a connected dominating set of G. The
condition N[v] c N[u] implies v and u are connected in G’.
In Figure 2 (a), since N[v] C N[u], vertex v is removed
from G’ if id(v) < id(u) and vertex u is the only dominating
node in the graph. In Figure 2 (b), since N[v] = N[u], either
v or u can be removed from G’ . To sure one and only one
is removed, we pick the one with a smaller id.
RULE 2: Assume u and w are two marked neighbors of
marked vertex v in G’ .
If
N(v) C N(u)
UN(w)
in G and
id(v) = min{id(v), id(u), id(w)}, then change the marker of
v to F.
The above rule indicates when the closed neighbor set
of v is covered by the neighbor sets of two of its marked
neighbors, u and 20, if v has the minimum id of the three,
it can be removed from G . The condition N(v) c N(u)
U
N(w) in Rule 2 implies that u and w are connected. The
subtle difference between Rule 1 and Rule 2 is the use of
open and close neighbor sets. Again, it is easy to prove
G - {v} is still a connected dominating set. Obviously, to
apply Rule 2, an additional step needs to be added at the
end of the marking process: If a host v is marked (m(v)=T),
send its status to all its neighbors.
Consider the example in Figure 3. Clearly, N(v) c
N(u)
U
N(w). If id(v) = min{id(v),id(u), id(w)}, vertex
v can be removed from G’ based on Rule 2. If id(u) =
min{id(v),
id(u), id(w)}, then vertex u can be removed based on Rule
1, since N[u] C N[v]. If id(w) = min{id(v),id(u),id(w)},
no vertex can be removed. Therefore, the id assignment also
decides the final outcome of the dominating set. Note that
Rule 2 can be easily extended to a more general case where
the open neighbor set of vertex v is covered by the union of
open neighbor sets of more than two neighbors of v in G’.
However, the connectivity requirement for these neighbors
is more difficult to specify at vertex v.
4 Update/Recalculation of Connected Dominating Set in
Mobile Networks
We can summarize topological changes of an ad hoc wireless
network into three diierent types: mobile host’s switch on,
mobile host’s switch off, and mobile host’s movement. In the
9
---
bmkcm link
Figure 3: One additional sample.
- new link
Figure 4: Mobile host v switches on
rest part of this section we will discuss update/recalculation
of connected dominating set for the above cases. Without
lost of generality, we assume that the underlying graph of
an ad hoc wireless network is always connected.
Mobile host v’s switch on. When a mobile host v
switches on, only its non-gateway neighbors along with host
v need to update their status, because any gateway neighbor
will still remain as gateway after a new vertex v is added. For
example, in Figure 4 (a), when host v switches on gateway
neighbor host u is not affected. Because at least two of the
U’S neighbors ur,u2, and ‘1~s are not connected originally,
and these connections will not be affected by host v’s switch
on. On the other hand, in Figure 4 (b), host v’s switch on
might lead non-gateway neighbor host w to mark itself as
gateway, depending on the connection between host v and
w’s neighbors wr , ~2, and ws.
The corresponding marking process can be the following:
(1) Mobile host v broadcasts to its neighbors about its switch
on. (2) Each host w E N[v] exchanges its open neighbor set
N(w) with its neighbors. (3) Host v assigns its marker m(v)
to T if there are two unconnected neighbors. (4) Each non-
gateway host w E N(v) assigns its marker m(w) to T if it
has two unconnected neighbors. (5) Whenever there is a
newly marked gateway, host v and all its gateway neighbors
apply Rule 1 and Rule 2 to reduce the number of gateway
hosts.
Mobile host v’s switch off. When a mobile host v
switches off, only gateway neighbors of that switched off
host need to update their status, because any non-gateway
neighbor will still remain as non-gateway after vertex v is
deleted. For example, in Figure 5 (a), when v switches off
non-gateway neighbor w is not affected. The host w’s neigh-
bors wi, ~2, and wa are pairwise connected originally, and
these pairwise connections will not be affected by host v’s
switch off. On the other hand, in Figure 5 (b), host v’s
switch off might change a gateway neighbor u to a non-
gateway, depending on the connection between its neighbor
hosts ur,u2, and
‘1~3.
Figure 5: Mobile host v switches off
The corresponding marking process can be the follow-
ing: (1) Mobile host v broadcasts to its neighbors about
its switching off. (2) Each gateway neighbor w E N(v) ex-
changes its open neighbor set N(W) with its neighbors. (3)
Each gateway neighbor ‘~1 changes its marker m(w) to F if
all neighbors are pairwise connected.
Note that since the underlying graph G is connected, we
can easily prove by contradiction that the resultant dominat-
ing set (using the above marking process) is still connected
when a host (gateway or non-gateway) switches off.
Mobile host v’s movement. A mobile host v’s move-
ment can be viewed as several simultaneous or non simul-
taneous link connections and disconnections. For example,
when a mobile host moves, it may lead several link discon-
nections with its neighbor hosts, and at the same time, it
may have new link connections to the hosts within its wire-
less transmission range, these new links may be disconnected
again depending on the way host v moves.
The challenge here is when and how each vertex should
update/recalculate gateway information. The gateway up-
date means that only individual mobile hosts update their
gateway status. The gateway recalculation means that the
entire network recalculates gateways/non-gateway status. If
many mobile hosts in the network are in movement, gateway
recalculation might be a better approach, i.e., the connected
dominating set is recalculated from scratch. On the other
hand, if only few mobile hosts are in movement, then gate-
way information can be updated locally. In the remaining
discussion, we will focus on the latter case. In order to
synchronize mobile host’s movement with gateway updates,
just before mobile host v starts to move, it sends out a spe-
cial signal {id(v), Start}, then during its movement host v
continuously sends out signal {id(v), Heart-Beat} at every
r time interval. and when it stops, host v sends out signal
{id(v), Stop}.
When a host u receives sianal {id(v), Start}, it starts to
monitor host v’s movement. If host u continuously receives
signal {id(v), Heart-Beat} at every r time interval, and at
the end, it receives signal {id(v), Stop}, then no action is
needed at host u. On the other hand, if host u does not
receive a {id(v), Heart-Beat} or {id(v), Stop} signal after r
time interval since last time it received a {id(v), HeartBeat}
or {id(v), Start} signal, then host u immediately concludes
it has a broken link to host v, and it will perform certain
actions (to be discussed later) for broken link {u, v}.
When a host u receives signal {id(v), He&Beat} with-
out receiving signal {id(v), Start} previously, host u can
conclude that it has a new link to host v, and it imme-
IO
diately performs certain actions (to be discussed later) for
new link {u, u}. At the same time, host u continuously
monitors host ‘u’s movement. If host u continuously receives
signal {id(v), Heart-Beat} at every T time interval, and at
the end, it receives signal {id(v), Stop}, then host u simply
does nothing further. On the other hand, if host u does not
receive a {id(v), HeartBeat} or {id(v), Stop} signal after 7
time period since last time it received a {id(v), Heart-Beat}
or {id(v), Start} signal, then host u immediately concludes
it has a broken link to host u, and similar actions are needed
as the case discussed earlier.
For mobile host v that is in movement, at every r time in-
terval during its movement, it performs the following mark-
ing process: (1) Mobile host u compares its new neighbor
set N’(v) with the original neighbor set N(v). If they are
the same, host v simply does nothing further; otherwise, it
continues the following steps. (2) Host v exchanges the open
neighbor set with its neighbors. (3) Host v marks itself as
gateway if it have two unconnected neighbors; otherwise, it
marks itself as non-gateway. (4) Further, if host v marked
itself as gateway, host u and all its gateway neighbors apply
Rule 1 and Rule 2 to reduce the number of gateway hosts in
the network.
For neighbor u of host ‘u, we consider two subcases: mo-
bile host u recognizes a new link {u,v} and mobile host u
recognizes a broken link {u, v}.
Mobile host ‘1~ recognizes a new link {u,~}. When
a mobile host u recognizes a new link {u,‘v}, two types of
mobile hosts need to recalculate their own gateway status.
One is mobile host u itself if it is non-gateway originally; the
other ones are common neighbors of mobile hosts u and v.
For example, in Figure 6 (a), non-gateway host u may mark
itself as gateway after a new link to v is established, depend-
ing on the connection between v and u’s neighbors ur , uz, us,
and 2~4. On the other hand, in Figure 6 (b), at least two of
gateway host u’s neighbors are unconnected originally, new
link {u, V} has no effect on these unconnected neighbors;
therefore, u still remains as gateway. In the same figure,
common gateway neighbor u4 of mobile host u and 21 may
unmark itself as non-gateway after new link {u, V} is estab-
lished.
The corresponding marking process can be the following:
(1) Mobile host u detects a new link to V, and it exchanges
the open neighbor set with its neighbors. (2) Upon receiving
the open neighbor set N(u) from host u, gateway host w
recalculates its status, if it is a common neighbor of hosts
u and v. (3) If host u is gateway, it simply does nothing
further; otherwise, host u marks itself as gateway if it has
two unconnected neighbors. (4) Whenever there is a newly
marked gateway, the newly marked gateway host and its
gateway neighbors apply Rule 1 and Rule 2 to reduce the
number of gateway hosts.
Mobile host u recognizes broken link {u, v}. When
a mobile host u recognizes broken link {u,~}, two types
of mobile hosts need to recalculate their gateway status.
One is mobile host u itself if it is a gateway originally; the
other ones are common neighbors of mobile hosts u and
V. For example, in Figure 7 (a), neighbors of non-gateway
host u are all pairwise connected, and they remain so after
link {u, V} is broken. On the other hand, in Figure 6 (b),
Figure 6: Mobile host u recognizes new link {u, V}
Figure 7: Mobile host u recognizes broken link {u, V}
depending on the connections between neighbors ui, uz, ‘1~3,
and 214, gateway host u may unmark itself as non-gateway
after link {u,v} is broken. In the same figure, the mobile
hosts u and V’S common non-gateway neighbor 214 may mark
itself as gateway after link {u, V} is broken.
The corresponding marking process can be the following:
(1) Mobile host u detects a broken link to V, and it exchanges
the open neighbor set with its neighbors. (2) If host u is non-
gateway, it simply does nothing further; otherwise, host u
will assign its marker m(u) to F if its neighbors are all pair-
wise connected. (3) Upon receiving the open neighbor set
N(u) from host u, non-gateway neighbor (excluding host V)
w recalculates its status if it is a common neighbor of hosts
u and V. (4) Whenever there is a newly marked gateway,
the newly marked gateway host and its gateway neighbors
apply Rule 1 and Rule 2 to reduce the number of gateway
hosts.
The above marking processes are effective only if few
nodes have new/broken links with host v. When there are
many such nodes, a better way of updating gateways can
be similar to the one for a host’s switch on/off. During its
movement, host v continuously sends out its open neigh-
bor set N(v) along with signal {id(v), Heart-Beat}. This
triggers each host w E v
U
N(v) to update gateway status.
The corresponding marking process can be the following:
(1) Mobile host v periodically exchanges its open neighbor
set with its neighbors at every r time interval. (2) Each
host w E v
U
N(v) assigns its marker m(w) to T if it has
two unconnected neighbors. (3) Whenever there is a newly
marked gateway, the newly marked gateway host and its
gateway neighbors apply Rule 1 and Rule 2 to reduce the
number of gateway hosts.
For those hosts u that have broken link to host v, their
gateway status are updated as soon as they recognize link
11