Neighbor Discovery among Mobile Nodes Equipped with Smart Antennas
Martha E. Steenstrup
Stow Research L.L.C.
Flanders, NJ USA
in time and hence over the achievable network topology,
represented as a directed graph.
Usually, network managers impose certain objectives
and constraints on the network in terms of acceptable performance or particular characteristics that the network or
individual nodes and links must exhibit. Examples of objectives include minimizing the average rate of energy dissipation or the average amount of interference inflicted on
other nodes per transmission. Examples of constraints include bounds on network properties such as k-connectivity
and maximum diameter, node properties such as minimum and maximum degree, and link properties such as
maximum bit error rate. The constraints result in a set
of admissible network topologies, each of which is a subgraph of the achievable network topology. Note that it
might be impossible for the nodes to form an admissible
topology without relaxing any of the constraints.
Nodes may act independently or cooperatively to form
an admissible topology and may have access to complete
or only partial information about the current state of the
network when doing so. In the mobile wireless context,
we expect that each node will have only a partial view
of network state and that this view may be out-of-date
because of volatility of state and limitations on opportunities for communication of information, potentially leading to the formation of an inadmissible topology. Even if
each node were to have perfect information about the current state of the network, computing an admissible topology for a large network might be impractical, because
for many combinations of objectives and constraints the
topology formation problem is NP-hard. In such cases, efficient approximation or heuristic algorithms that yield a
“reasonable” topology should be employed instead of exhaustive search. Thus, in practice, the routing topology
formed by the nodes is not guaranteed to be admissible
with respect to the specified constraints or optimal with
respect to the specified objectives.
Abstract - In mobile wireless networks, individual
nodes with adaptive transceivers and antennas can
control the network topology in order to achieve desired properties of the network as a whole or to satisfy specific needs of individual sessions. We examine
the role of smart antennas in topology management,
where we define a “smart” antenna as one that can
form a relatively narrow (transmit or receive) beam
and steer it in an arbitrary direction. By focussing
energy in a specific direction, such antennas can offer benefits such as increased range and reduced interference and probability of detection, but they also
present challenges in terms of determining pointing
directions for communication among mobile nodes. In
this paper, we address the problems of forming and
maintaining neighbor relationships among nodes, a
key aspect of topology management, providing solutions that do not rely on prior information about node
locations to efficiently and effectively direct transmissions.
I. INTRODUCTION
In a mobile wireless network, the ability of two nodes
to communicate directly depends upon the fundamental
properties of radio signal propagation; the inherent characteristics of the environment in which the nodes exist,
including the presence of hostile entities; the capabilities of the nodes’ transceivers and antennas; and the current locations and relative mobility of the nodes. Terrain, foliage, weather, obstacles, node movement, and the
presence of emitters of electromagnetic radiation all affect communication between the nodes, resulting in problems such as distance-based attenuation, multipath fading, shadowing, and interference. An agile node can mitigate many of these effects by adapting its transceiver and
antenna parameters, such as frequency, power, modulation, transmission rate, error-control coding, and transmit and receive beam patterns, to the current situation.
Furthermore, an agile node may improve communication
by moving to a new location and by using reflective surfaces and barriers to its advantage to increase or limit
the range of its transmissions and to reduce interference.
Thus, each node can exercise control over the set of nodes
with which it can communicate directly at a given point
A. Routing Topology
The process of forming the routing topology begins with
neighbor discovery. During neighbor discovery, each node
first determines achievable links to other nodes in the network and then eliminates those that would likely result in
an inadmissible topology. The remaining links define the
set of neighbors. A node, X, is said to be a neighbor of
a node, Y , if X is currently reachable from Y via direct
This work was funded in part by the U.S. Army Research Laboratory under SBIR contract DAAD17-02-C-0114 awarded to San
Diego Research Center, Inc. and was performed when the author
was employed by SDRC.
1
communication and if inclusion of the link from Y to X
is not likely to result in an inadmissible topology.
Neighbor discovery produces a routing topology which
may be subsequently refined depending upon the expected
persistence of neighbor relationships and upon the service
needs of particular sessions. Persistent neighbor relationships result in links that are likely to last for more than a
few seconds and thus can be used effectively in transporting data traffic across the network. Such links may also
be advertised as part of the routing information associated with the node, depending upon the routing procedure
used in the network. In selecting persistent neighbors, a
node favors those that are less mobile and exhibit minimal outages. Nevertheless, in a mobile wireless network,
no link is expected to exist forever, and thus nodes must
attempt to adjust the routing topology toward admissibility when neighbors vanish.
Tactical networks must exhibit a low probability of detection. Thus, the routing topology for such networks
will likely consist of a set of low-power links meeting a
minimum tolerated error rate that form a connected network. It may often be the case that the current routing
topology is unable to provide a particular session with the
desired quality of service, especially if the session needs
low delay, since paths through the low-power topology
are likely to comprise multiple hops. By including a longdistance link from the achievable topology, e.g., one that
directly connects the source and destination, the network
might be able to provide the quality of service desired
for the session, although the resulting topology might no
longer be admissible. A node decides whether to negotiate an ephemeral neighbor relationship with another node
to improve the performance of a session, depending upon
the importance and expected duration of the session and
the anticipated negative effects of using the corresponding link. This link is known only to the two nodes it
connects and is never advertised as part of the node’s
routing information. An ephemeral neighbor relationship
is terminated as soon as it is no longer needed.
B. Proposed Approaches
To date, most of the work on topology management in
mobile wireless networks has centered on the problem of
finding an assignment of transmit power levels to nodes
that minimizes a particular function of transmit power
(e.g., maximum level assigned to any one node or total
assigned to all nodes) while maintaining k-connectivity
of the network graph for some k ≥ 1. Results have included centralized and distributed algorithms, heuristics
for use when only partial information about the network is
available, and approximation algorithms for the NP-hard
problems. Refer to [9, 7, 10, 5] for detailed descriptions
and analyses of proposed algorithms. Given the complexity of apparently simple problems (such as the assignment
of transmit power levels to each node so as to minimize
the total power consumed in the network while keeping
the network graph 1-connected, which is shown to be NPhard [2,6]), it is not surprising that more complicated
topology management problems requiring satisfaction of
multiple objectives and constraints and involving adjustment of transceiver and antenna parameters in addition to
transmit power and beam direction, have not yet received
much attention. Moreover, techniques for adapting the
network topology to the needs of specific sessions are only
beginning to receive serious consideration as the technology for smart antennas matures.
In the remainder of this paper, we focus on neighbor
discovery in networks equipped with smart antennas. We
present and compare two simple algorithms that enable
nodes to detect each other, and thus form a connecting link in the routing topology, using directed transmissions and without the aid of geolocation information.
We also provide a rendezvous procedure that can be used
to monitor presistent neighbor relationships and establish
ephemeral ones. These algorithms form the foundation
on which a general scheme for topology management in
networks with smart antennas can be built.
II. DIRECTIONAL NEIGHBOR DISCOVERY
In a tactical network, communications are likely to take
place in three dimensions. Not only will there be communications between terrestrial and airborne nodes, but
also between terrestrial nodes located at various elevations and among airborne nodes flying at different altitudes. Ideally, a node should be able to detect a neighbor located in any direction (θ, φ) on the sphere, where
θ ∈ [0, 2π) is the horizontal angle and φ ∈ [ π2 , − π2 ] is
the vertical angle, in radians. The ability of two nodes
to detect each other’s presence through their transmissions depends upon the environment in which they are
operating, the settings of their transceiver and antenna
parameters, and whether neighbor discovery is executed
synchronously or asynchronously among nodes.
Most existing neighbor discovery algorithms operate
aynchronously as follows. Each node intermittently advertises its presence, thus offering itself as a potential
neighbor to other nodes within reception range. Repeated advertisements are necessary to provide information about a node’s presence to other nodes that have recently (re)entered its part of the network. A recipient of a
node’s advertisement learns that the advertising node can
communicate directly with it, but it should not infer that
it can communicate directly with the node. In wireless
networks, unidirectional communication between a pair of
nodes is not a rare event and may result from interference
affecting one node but not the other or mismatches in the
transmission and reception capabilities of the two nodes.
Node advertisements enable a recipient not only to learn
of a potential neighbor but also to track the reachability
of an existing neighbor. Failure to receive advertisements
2
from a known neighbor is an indication that the neighbor
is no longer directly reachable.
A node’s advertisement usually includes a list of its
known neighbors. This information enables a recipient to
determine whether its own advertisements have reached
the advertising node. Presence of the recipient on the
node’s advertised list of neighbors indicates that the node
has indeed received the recipient’s advertisements, but
the absence of the recipient on the advertised list does
not necessarily indicate that the node has failed to receive the recipient’s advertisements, since formation of a
neighbor relationship depends upon not only receipt of advertisements but also whether the associated link is likely
to be admissible. A node’s advertisement may also contain information about transmissions and receptions (e.g.,
transmit power, received signal-to-noise-plus-interference
ratio) that might be used to adjust parameters to improve
communications with the recipient.
Both omnidirectional and directional neighbor discovery techniques (e.g., [8,5]) have been proposed for use
in networks with smart antennas. Omnidirectional techniques are intended to bootstrap the process of determining antenna pointing directions for data transmissions. In
most cases, each node is assumed to be able to ascertain
its geolocation and includes this information in its advertisements, thus permitting a recipient to compute the
direction in which the node lies and hence where to point
an antenna beam for future directional data transmssions
to or from the node. Directional techniques are likely
to detect more potential neighbors than omnidirectional
techniques, because of the higher gains and hence longer
range associated with directional transmission and reception. Most of the directional techniques proposed have
been designed for use with sectored antennas, in which
the antenna transmits advertisements out one or more
of its sectors and listens on all sectors but only selects
the sector with the strongest signal, ignoring the signals
received from the other sectors. Thus, a node can determine the general direction from which a advertisement is
received, without geolocation information.
In contrast to the above techniques, our neighbor discovery algorithm is specifically designed for beam-steering
antennas. It requires synchronous operation among nodes
but does not require that a node know its geolocation or
that of any other node in the network. A node must possess certain capabilities in order to successfully execute
this algorithm. These capabilities are outlined below, but
the particular technologies for realizing them are not discussed. Each node can form a single transmit or receive
beam of arbitrary width ω < π radians and steer the beam
in an arbitrary direction. Thus, each node has the ability
to transmit and receive directionally. Furthermore, each
node has the ability to receive omnidirectionally and to
determine direction of arrival. All nodes agree on the cur-
rent time and hence can synchronize their activities based
on a common global clock. Time is divided into frames
and a portion of each frame is dedicated to neighbor discovery.
A. Scanning for Neighbors
With our algorithm, each node is in one of two modes,
scanning or listening, during the neighbor discovery portion of each frame. When scanning, a node probes a sequence of directions for potential neighbors, by transmitting an advertisement in each specified direction. When
listening, a node waits for advertisements, which are received omnidirectionally. If a listening node receives an
advertisement, it determines the direction of arrival, responds directionally with its own advertisement, and expects to receive an acknowledgement in return, all within
a short time interval. This exchange of advertisement
messages may also be used to negotiate future rendezvous
to reassess reachability, if both nodes tentatively agree to
become neighbors. Depending upon the amount of time
required to step through the sequence of directions and
the proportion of time allotted to neighbor discovery during a frame, only a subset of a sequence may be probed
in each frame.
This algorithm enables scanning nodes to use very narrow beams and hence to detect distant neighbors reachable only with the additional gain provided by the transmitting antenna. Replacing omnidirectional with directional listening increases the gain of the receiving antenna
and hence the size of the region in which potential neighbors can be detected, but it causes the neighbor discovery
algorithm to become less tolerant of relative movement
among nodes and of discrepancies in beam alignment and
clock synchronization among nodes and hence is better
suited to quasi-stationary networks. Scanning sequences
with directional transmission and reception have been
proposed for scheduling transmissions among nodes with
directional antennas, once neighbor relationships and directions have been established. According to the approach
in [1], nodes listen directionally for data transmissions according to a predetermined scanning pattern known to
all nodes. If the current direction of the receive beams
is (θ, φ), each node knows that it can point its transmit
beam in direction ((θ + π) mod 2π, −φ) and communicate
with a neighbor lying in that direction. We note that
our analysis of the algorithms for neighbor discovery, described in the remainder of the paper, is independent of
whether nodes listen directionally or omnidirectionally for
scanning nodes and hence applies in both cases.
A scan is a sequence of antenna-pointing directions,
S = {(θi , φi ) : 1 ≤ i ≤ n}. A full scan induces a minimal covering of the unit sphere by caps formed by the
intersection of the sphere and beams of width ω centered
over each of the pointing directions. Use of a minimal
covering ensures that each point on the sphere is covered
3
by at least one beam and that the length of the sequence
is the shortest possible. The area of each cap is:
Z 1
ω
Acap =
2πdx = 2π(1 − cos ).
ω
2
cos 2
scanning at any point in time, entering the current scan,
t−t
αj (S), at the appropriate element, k, where k = d δ j e
t
and j = d ∆ e.
B. Mode Selection Algorithms
The lower and upper limits on the number of scans
necessary for all nodes in the network to discover all of
their potential neighbors depends on the characteristics
of the achievable network graph and the algorithm a node
uses to select its mode, scanning or listening, for each
scan. At least one scan is required if the graph contains
no cycles of odd length, and at least two scans are required
if the graph contains at least one cycle of odd length.
The following deterministic algorithm for selecting the
mode of each node yields an upper limit of dlg N e scans
necessary for all nodes to detect all of their neighbors,
where N is the number of nodes in the network and
minimal node movement and interference and obstruction
among signals is assumed. Each node is initialized with
the parameters N , j ∈ {0, ..., N −1} (its unique identifer),
and mj0 = j. For each scan i, 1 ≤ i ≤ dlg N e, node j comN
putes mji = mj(i−1) mod d 2i−1
e, and then chooses scanN
ning mode, if mji < d 2i e, or listening mode, if mji ≥ d N
2i e.
Thus, during the first scan, a node has the opportunity of
detecting one-half of the nodes in the network. Comparing two successive scans from the perspective of a given
node, half of the nodes currently sharing the same mode as
the given node are forced to choose the opposite mode for
the next scan. Thus, during the next scan, a node has the
opportunity of detecting one-half of the nodes that were
undetectable during the previous scan. This algorithm
guarantees that all nodes will detect all potential neighbors within dlg N e scans and may detect them in fewer
scans. In fact, dlg N e is the lowest limit on the maximum
number of scans required for any algorithm in which nodes
have no other prior information about the network, other
than the three initialization parameters listed above.
Depending on the assignment of identifiers to nodes, the
number of scans required for all nodes to detect all of their
potential neighbors may be considerably more than two
using this deterministic algorithm, even with no cycles of
odd length in the achievable network graph. For example,
if there are 20 nodes in the network and node 1 is the
only node within range of node 0, nodes 0 and 1 will not
detect each other until the fifth and final scan. In this
case, the number of scans required for nodes 0 and 1 to
detect each other is equal to the maximum number of
scans required for all nodes in the network to detect all
potential neighbors.
The following stochastic algorithm for selecting the
mode of each node has no upper limit on the number
of scans necessary for all nodes to detect all of their potential neighbors, but is amenable to analysis of expected
behavior. For each scan, each node randomly chooses its
Thus, a lower bound on the cardinality of a minimal covering of the unit sphere by caps defined by ω is:
Asphere
4π
2
=
=
.
Acap
2π(1 − cos ω2 )
1 − cos ω2
The general problem of finding a minimal covering of
the sphere by equal caps defined by angle ω appears to
be open, although coverings have been found for specific
cases (see [4]). One can use stochastic search techniques,
such as simulated annealing, to find coverings of the unit
sphere that are likely to be close to minimal. For most
situations encountered in practice, a scan, S, need only
cover a hemisphere or an equatorial band and hence is a
proper subset of the full scan.
If two nodes are within reception range of each other,
one scanning and one listening, they are guaranteed to detect each other within a single scan, provided that they do
not move during that time and that no external entities
interfere with or obstruct their communications. The reason is that the scanning node’s transmit beam must point
toward the listening node at some time during the scan.
If at least one of the nodes is mobile, however, the two
nodes might never detect each other. Consider the following pathological example in the plane. Suppose that
node X is scanning and node Y is listening, and that the
scan, S = {θi : 1 ≤ i ≤ n}, is a monotonically increasing
sequence starting at θ1 = 0. Furthermore, suppose X and
Y are located on a circle with X at 0 and Y at π initially,
and both move counterclockwise around the circle at the
same angular speed as the angle changes in the scan. In
this case, X always points its transmit beam in a direction
diametrically opposed to that in which Y lies.
A node might detect all of its potential neighbors within
a single scan, but it will usually require multiple scans
to find them. Mobile nodes within range of each other
can increase the likelihood of mutual detection by choosing a different sequence for each scan. Returning to the
pathological example above, if X now chooses a scan,
S 0 = {θi0 : 1 ≤ i ≤ n} such that θi0 = (θi + π) mod 2π, X
will point its transmit beam at Y throughout the entire
scan and thus X and Y will detect each other within a
single scan. We recommend that the scanning algorithm
be initialized with a pseudorandom sequence of permutations, {αi : i ≥ 0}, to be applied to the initial scan S and
a partitioning of time into equal intervals Ti = [ti , ti+1 ) of
duration ∆, such that a node applies permutation αi at
time t if t ∈ Ti . Let δ be the time required to execute a
single direction of a scan sequence. Thus, a node can start
4
mode according to a uniform distribution. The expected
number of scans required for nodes 0 and 1, described
above, to detect each other with this algorithm is two,
as follows. Suppose that a new node comes within range
of L > 0 other nodes. The expected number of scans
required for the node to discover at least one of its L potential neighbors is:
Eany
µ
¶ ∞
µ ¶i
1 X
1
2L
= 1− L
(i + 1)
=
.
2
2L
2L − 1
i=0
its potential neighbors to form a connected routing topology. Thus,
Etree =
k=1
=
=
=
∞
X
i=1
∞
X
(1)
i(Pi − Pi−1 )
à µ
¶L µ
¶L !
1
1
i
1− i
− 1 − i−1
2
2
i=1
µ ¶µ
¶
L
X
L
2k
(−1)k+1
.
k
2k − 1
(2)
k=1
Another useful expectation is the number of scans required to form a connected routing topology following
node deployment, since once such a topology is formed,
communication can theoretically take place among any
nodes in the network. To compute this expectation, in
the case of the stochastic mode-selection algorithm, we
begin with the achievable network graph, in which the
presence of a link between two nodes indicates that the
two nodes are able to communicate directly. We consider
only connected achievable graphs since only these can lead
to connected routing topologies. Although we have been
unable to generate a closed-form expression for this expectation for an arbitrary N -node achievable graph, we can
compute it exactly when the achievable graph is a complete graph or a tree. In the case of a complete graph,
which is (N − 1)-connected, as long as all nodes do not
choose the same mode for a scan, neighbor discovery will
produce a connected routing topology. Thus,
Ecomplete =
(−1)k+1
µ
N −1
k
¶µ
2k
k
2 −1
¶
,
which can be derived directly from (2). Any connected
N -node achievable graph must be k-connected, where 1 ≤
k ≤ N −1, and thus the expected number of scans required
to form a connected routing topology given such a graph
must be at least Ecomplete and at most Etree .
We have relied on Monte Carlo simulation to compute
estimates of the expected number of scans required to
form a connected routing topology, given an arbitrary N node achievable network graph, for both the stochastic
and deterministic mode-selection algorithms. Achievable
graphs were constructed by generating random graphs
such that a link was added between two nodes according
to a fixed probability. If the resulting achievable graph
was not initially connected, randomly-selected links were
added one at a time between the components, until the
graph was connected. For the example illustrated here,
twenty different 20-node achievable graphs were generated. For each graph, twenty different random permutations of node identifiers and twenty different random
settings of mode were selected for the deterministic and
stochastic algorithms respectively. Results are shown in
table 1. We provide the average number of scans computed from the simulated networks; the maximum and
minimum number of scans exhibited by all 400 simulations from which each average was computed; and the expectations, Etree and Ecomplete , derived for N = 20. We
also give, for the randomly-generated achievable graphs,
the initial probability of including a link and the average
degree of a node.
We conclude that in networks with static membership,
neighbor discovery with deterministic mode selection requires significantly fewer scans than with stochastic mode
selection, both in the worst case and on average. In networks with dynamic membership, however, the outcome
may be different. The deterministic algorithm must be
initialized with the maximum number of nodes anticipated rather than the actual number of nodes currently in
the network, and the larger the difference in these two values the higher the number of scans that may be required.
The stochastic algorithm, however, does not rely on any
knowledge about the network, and hence the number of
scans required is independent of the difference between
anticipated and actual membership.
Thus, Eany ∈ (1, 2], with Eany = 2 when L = 1 and
Eany → 1 as L → ∞. Therefore, the expected number of
scans required for a node to join an existing network is no
more than two, independent of the number of potential
neighbors. The expected number of scans, Eall , required
for a node to detect all L of its potential neighbors depends upon Pi , the probability that at most i scans are
required to discover all potential neighbors. Thus, with
some rearrangement of terms,
Eall
N−1
X
2N−1
,
2N −1 − 1
C. Rendezvous
The technique of scanning is useful not only for discovering potential neighbors but also for monitoring the
persistence of links and for establishing ephemeral links.
which can be derived directly from (1). In the case of a
tree, which is 1-connected, each node must discover all of
5
link
prob.
avg.
degree
tree
0.05
1.92
0.1
2.41
0.2
4.12
0.4
7.95
complete graph
deterministic
connect
all nbrs
4.36
3.77
2.34
1.08
[2,5]
[2,5]
[1,5]
[1,3]
4.38
4.53
4.84
4.99
[2,5]
[3,5]
[3,5]
[4,5]
stochastic
connect
all nbrs
5.62
5.57
[2,12] 5.62
[2,12]
4.62
[2,10] 6.01
[3,12]
2.76
[1,9] 6.76
[3,13]
1.17
[1,3] 7.73
[4,14]
1.00
Table 1: Expected number of scans required to form a connected routing topology and to find all potential neighbors
in various 20-node networks, using deterministic and stochastic mode selection during neighbor discovery.
When two nodes detect each other during a scan, they
may elect to form a neighbor relationship conditional
upon the observed persistence of their direct link. Specifically, when exchanging advertisements, the nodes negotiate a set of future rendezvous to reassess the link before including it in the routing topology. The listening
node always starts the negotiation by including in its advertisement sent to the scanning node a set of candidate
rendezvous times (expressed as particular frames within
a block of frames), and the scanning node responds with
an acknowledgement containing its selection of rendezvous
times from the set offered (or an indication that it does not
wish to rendezvous). Rendezvous are always conducted
during the neighbor-discovery portion of a frame and supersede the sequence of directions usually scanned during
that time.
Beginning at the negotiated rendezvous time, the two
nodes attempt to reestablish communications using a
directionally-restricted spiralling scan whose duration is
no longer than the portion of the frame allotted to neighbor discovery. This approach enables the two nodes to
quickly detect each other, provided that they have not
moved a large angular distance relative to each other since
their last contact. Each node maintains a special scanning
sequence for rendezvous, comprising a set of directions
starting with (0, π2 ) and spiralling outward counterclockwise to cover an angle ψ . To produce a spiral sequence
starting in a particular direction, the appropriate rotation
can be applied to this sequence.
The node, X, that requested the rendezvous executes
a spiral scan for the other node, Y , starting with its last
known direction for Y . Y listens for X ’s advertisement
beginning at the rendezvous time negotiated. If X and
Y detect each other at some point during the scan, the
link is still intact. Moreover, through the exchange of advertisements during this scan, each node has the opportunity to determine direction of arrival and hence update
its directional information for the other. The two nodes
may require multiple successful rendezvous before including their link in the routing topology, depending upon the
specified criteria for persistence.
Rendezvous are also used in forming ephemeral neighbor relationships with distant nodes to improve performance of specific sessions. In this case, a node, X , at-
tempts to form an ephemeral link with a distant node,
Y , only if X can already communicate with Y over a
multiple-hop path composed of persistent links and if X
has information about Y ’s direction, implying that X and
Y must have detected each during a prior scan. Since X
and Y do not currently have a neighbor relationship, X
must solicit a rendezvous with Y using multihop communication across a path connecting the nodes. Specifically,
X sends a message to Y containing a set of candidate
rendezvous times, and Y responds with a message containing the rendezvous times selected. (The general technique of communicating over multiple-hop paths to bootstrap direct communications between distant nodes has
previously been proposed for CSMA/CA medium access
control in the context of directional antennas [3].) X and
Y attempt to rendezvous at the negotiated time, with X
executing a spiral scan starting with its last known direction for Y and Y listening for X’s advertisement. If X
and Y detect each other at some point during the scan,
the ephemeral link can be successfully established.
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