Distributed Computing Column 38
Models for Algorithm Design in Wireless Networks
Idit Keidar
Dept. of Electrical Engineering, Technion
Haifa, 32000, Israel
idish@ee.technion.ac.il
The prosperity of research on wireless communication has not skipped the distributed computing community.
Wireless networks provide unique and challenging platforms for distributed computation, inspiring
researchers to develop many distributed algorithms for such environments. Any algorithmic work targeting
wireless networks must begin by defining an appropriate model. The first research results in this vein
employed simplified models, like the Unit Disk Graph (UDG), which facilitated algorithm design and were
readily amenable to analysis. Recently, models that more accurately capture the physical nature of wireless
networks were developed; such models are nowadays being gradually adopted.
This column’s contribution, by Zvi (Zvika) Lotker and David Peleg focuses on the promising signalto-interference
& noise ratio (SINR) model, and studies it from an algorithmic perspective. It surveys the
model’s structural properties as well as algorithms designed for this model. Along the way, some of the
similarities and differences between the SINR model and the UDG model are highlighted. Many thanks to
Zvika and David for their contribution!
Call for contributions: I welcome suggestions for material to include in this column, including news, reviews,
opinions, open problems, tutorials and surveys, either exposing the community to new and interesting
topics, or providing new insight on well-studied topics by organizing them in new ways.
73Structure and Algorithms in the SINR Wireless Model
Zvi Lotker∗ Dept. of Communication Systems Engineering
Ben Gurion University
Beer-Sheva, Israel
zvilo@cse.bgu.ac.il
David Peleg
Dept. of Computer Science and Applied Mathematics
Weizmann Institute of Science
Rehovot, Israel
david.peleg@weizmann.ac.il
Abstract
The signal-to-interference & noise ratio (SINR) model is one of the most commonly studied physical
(or fading channel) models for wireless networks. We survey some recent studies aiming at achieving
a better understanding of the SINR model and its structural properties and developing efficient design
algorithms and communication protocols for it.
Physical models and SINR
Traditional (wired, point-to-point) communication networks can be described satisfactorily using a graph
representation. A station s is able to transmit a message to another station s ′ if and only if there is a wire
connecting the two stations. This condition is independent of the locations, connections and activities of the
two stations or of other nearby stations 1 . Accurately representing a wireless network is considerably harder,
since it is nontrivial to decide whether a transmission by a station s is successfully received by another
station s ′ ; this may depend on the positioning and activities of s and s ′ , and on other nearby stations, whose
activities might interfere with the transmission and prevent its reception This means that a transmission from
s may reach s ′ in some settings but fail to reach it under other settings. Moreover, the question of successful
reception is more complex, since connections can be of varying quality and capacity.
In fact, there are many other relevant factors, such as the presence of physical obstacles, the directions
of the antennae at s and s ′ , the weather, and more. Obtaining an accurate solution taking all of those
factors into account involves solving the corresponding Maxwell equations. Since this is usually far too
complicated, the common practice is to resort to approaches based on approximation models. For instance,
one way to predict the behavior of wireless systems in an urban environment is using a ray tracing model,
based on the assumption that radio waves behave according to geometric optics where walls are modeled
as reflective mirrors. The uncertainly involved in the dynamics of a radio channel can be modeled using a
Markov process. For more information we refer the interested reader to Chapters 2 and 3 of [9].
For the purposes of the current discussion, we follow the approach of ignoring those complicating factors,
and assuming a relatively clean abstract setting where the only players are the transmitting and listening
∗
Partially supported by a gift from Cisco research center and by the Israel Science Foundation, grant 894/09.
1
excepting certain types of wired local area networks.
ACM SIGACT News 74 June 2010 Vol. 41, No. 2stations, and the antennae are omni-directional. In this setting, the rules governing the reception quality of
wireless transmissions can be described schematically by physical or fading channel models. Among those,
one of the most commonly studied is the signal-to-interference & noise ratio (SINR) model. In this model,
the energy of a signal fades with the distance to the power of the path-loss parameter α. When a station s
transmits, the message is successfully received by a listening station s ′ if and only if the strength of the signal
received by s ′ , divided by the strength of the interferences from other simultaneous transmissions (plus
the background noise N), exceeds some threshold β. Hence for a collection S = {s1, . . . , sn} of simultaneously
transmitting stations in the plane, it is possible to identify with each station si a reception region
H(si) around it, consisting of the points where the transmission of si is received correctly. More precisely,
denote by dist(p, q) the Euclidean distance between p and q, and assume that each station si transmits with
power Ei. At an arbitrary point p, the transmission of station si is correctly received if
Ei · dist(p, si) −α
N + ∑
j̸=i Ej
≥ β .
· dist(p, sj) −α
This formula represents a rather general model concerning the allowed transmission power, referred to as
the power control model, in which each station can control the power with which it transmits. A simpler
(and weaker) model is the uniform wireless network model, which assumes that all transmissions use the
same amount of energy, i.e., Ei = 1 for every i.
S1
S3
S3
S1
S1
�
�
�
S2
S2
S2
(a) (b) (c)
Figure 1: An SINR network with three simultaneously transmitting stations s1, s2, s3 and one receiver,
marked by the solid black square. (a) The receiver can hear s2. (b) Station s1 is relocated and now the
receiver cannot hear any of the transmissions. (c) If, at the same locations as in (b), s3 remains silent while
s1 and s2 transmit, then the receiver can hear s1.
The dependence of reception on the stations’ locations and activities is illustrated in Figures 1 and 2.
Fig. 1 depicts a uniform wireless network consisting of three stations s1, s2, s3 and their reception regions
under various settings. Figure 2 illustrates more involved configurations that may arise in the non-uniform
case or in the uniform case with β < 1.
Graph-based models and UDG
A fair amount of research exists on the SINR model and other variants of the physical model, yet as far as
algorithms are concerned, progress has been rather slow. Among the reasons for this are the fact that in physical
models it is nontrivial to decide some basic questions on a given setting, and it is much harder to develop
ACM SIGACT News 75 June 2010 Vol. 41, No. 2S3
�
p1
�
p2
S2
S1
S1
S3
S2
(a)
(b)
Figure 2: Reception regions in the SINR model. (a) A non-uniform network where station s1 transmits with
a higher power level than the other two stations. The white regions (including the crescent-like internal one)
represent “no-reception” areas. (b) A uniform network with path-loss parameter α = 2, reception threshold
β = 0.3 and background noise N = 0.05.
and analyze communication protocols and network design algorithms. Consequently, previous research of
wireless multi-hop networks, including the study of issues such as transmission scheduling, frequency allocation,
topology control, connectivity maintenance, routing, and related design and communication tasks,
has relied on simplified graph-based models rather than on physical models.
Graph-based models represent the network by a graph G = (S, E) such that a station s successfully
receives a message transmitted by a station s ′ if and only if s and s ′ are neighbors in G and s does not have
a concurrently transmitting neighbor in G. In particular, the model of choice for many protocol designers
is the unit disk graph (UDG) model [14]. In the general disk graph model (a.k.a. the protocol model), the
stations are represented as points in the Euclidean plane, and the transmission of a station si is received (in
the absence of collisions with other simultaneous transmissions) by every other listening station v located
within a disk centered around si, whose radius ri depends on the energy Ei with which si transmits. Hence
the reception relationships in a wireless network can be modeled by a disk graph, namely, a directed graph
whose vertices correspond to the stations, with a directed edge leading from si to sj if the two stations are
at distance at most ri from each other. Most of the literature on graph-based models for wireless networks
concentrates on the uniform variant of this model, known as the UDG model, where all disk radii are assumed
to be 1, hence the network is represented by an (undirected) unit disk graph [5], with an edge connecting
any two stations whose corresponding points are at distance at most one from each other.
Comparisons between the SINR and UDG models
Let us illustrate some of the differences between the UDG and SINR models, with respect to the way they
deal with interference. The UDG model provides a rather rigid representation for the reception region of a
station s, taking it to be a unit disk around s. Hence, considering for example the configuration depicted in
Fig. 3(a), we observe that a listening station v located at the solid black square is adjacent to the transmitting
station s1, but not to any of the other simultaneously transmitting stations, hence the UDG model predicts
that v will receive the message in this configuration. In reality, the presence of several nodes slightly outside
the unit disk around the receiver station v might still generate enough cumulative interference to prevent
ACM SIGACT News 76 June 2010 Vol. 41, No. 2v from successfully receiving a message from s1, hence the UDG model might yield a “false positive”
indication of reception. Indeed, the prediction of the SINR model in this same configuration, depicted in
Fig. 3(b), is that the reception regions will be smaller than unit disks, and subsequently, v will not hear any
of the transmitting stations.
S1
S1
�
�
S3
S4
S3
S4
S2
S2
(a)
(b)
Figure 3: (a) In the UDG model, p can hear s1. (b) In the SINR model, the cumulative interference of
stations s2, s3, s4 prevents p from hearing s1.
Conversely, a simultaneous transmission by two or more neighbors should not always end in collision
and loss of the message; in reality this depends on other factors, such as the relative distances and the
relative strength of the transmissions. Thought provoking experimental results presented in [20] show that
even basic wireless stations can achieve communication patterns that are impossible in disk graph-based
models. One such example is the pattern depicted in Figure 2(a), restricted to the stations s1 and s2 and
the points p1 and p2. This configuration extends to two dimensions a 1-dimensional example of [20], and
illustrates a possible setting in the SINR model in which p1 can hear s1 and p2 can hear s2 simultaneously.
In contrast, it can be shown that in the disk graph model there can be no assignment of transmission powers
that will achieve this effect. In the same spirit, certain situations are shown in [17] in which it is possible to
apply routing / transport schemes that may break the theoretical throughput limits of any protocol that obeys
the laws of a graph-based model.
An additional such scenario is illustrated in Figure 4, which compares the reception regions of the UDG
and SINR models with four transmitting stations s1, s2, s3, s4 and one receiver p (represented as a solid
black square). Suppose that initially, only station s1 transmits, and all others remain silent. In this case, both
models provide the same picture, in which p can hear s1. Figure 4 illustrates the process of gradually adding
s2, s3 and s4 to the transmitting set.
Bridging between the models
Graph-based models are attractive for higher-layer protocol design, as they conveniently abstract away some
of the complications of the wireless model. For example, issues of topology control, scheduling and allocation
are handled more directly, since notions such as adjacency and overlap are easier to define and test,
in turn making it simpler to employ also some useful derived concepts such as domination, independence,
clusters, and so on. On the down side, it should be realized that graph-based models ignore or oversimplify
a number of important physical aspects of real wireless networks, such as the laws of interference. This lim-
ACM SIGACT News 77 June 2010 Vol. 41, No. 2(a)
(b)
S2
�
S1
S2
�
S1
(c)
(d)
S3
S3
S2
�
S1
S2
�
S1
(e)
(f)
S3
S3
S2
�
S1
S2
�
S1
S4
S4
Figure 4: Reception regions in the UDG model (left) and SINR model (right). (a)-(b): When both s1 and s2
transmit simultaneously, p hears neither station in the UDG model, but it does hear s1 in the SINR model,
hence the UDG model yields a “false negative” indication. (c)-(d): When s3 joins the transmitting stations,
p still cannot hear any station in the UDG model, but now it can hear station s3 in the SINR model. (e)-(f):
Station s4 starts to transmit as well.
ACM SIGACT News 78 June 2010 Vol. 41, No. 2its the applicability or accuracy of many of the algorithms presented in the literature for wireless networks
under the UDG model.
In summary, while the existing body of literature on models and algorithms for wireless networks represents
a significant base containing a rich collection of ideas, paradigms, tools and techniques, the limitations
described above leave us in the unfortunate situation where the more practical graph-based models (such
as the UDG model) are not sufficiently accurate, and the more accurate physical models (such as the SINR
model) are not sufficiently practical or well-understood. This state of affairs has led to recent interest in the
study of some of the structural and algorithmic aspects of the SINR model, as well as of the relationships between
physical and graph-based models, in the hope of bridging the gap between these models. Ultimately,
the motivation is to identify more suitable “hybrid” models, attaining some desirable tradeoff points between
practicality and accuracy, and subsequently develop better and more efficient algorithmics for fundamental
problems and tasks in wireless networks.
More elaborate graph-based models may employ two separate graphs, a connectivity graph GC =
(S, EC) and an interference graph GI
= (S, EI), such that a station s successfully receives a message
transmitted by a station s ′ if and only if s and s ′ are neighbors in the connectivity graph GC and s does
not have a concurrently transmitting neighbor in the interference graph GI. Protocol designers sometimes
consider special cases of this general model. For example, it may be assumed that GI is GC augmented with
all edges between 2-hop neighbors in GC. An alternative variant of the UDG model handling transmissions
and interference separately, named the Quasi Unit Disk Graph (Q-UDG) model, is introduced in [15]. In
this model, two concentric circles are associated with each station, the smaller representing its reception
region and the larger representing its area of interference. Another interference model, also based on the
UDG model, is proposed in [24].
One way to bridge the gap between the two models is using emulation. Lebhar et al. [16] consider the
case of α > 2 and nodes that are deployed uniformly at random in a given area. For this setting, they show
how a UDG protocol can be emulated when the network operates under the SINR model, with an emulation
cost factor of O(log 3 n). Note, however, that this does not resolve all questions related to the SINR model,
for three reasons. First, the paper only talks about randomly and uniformly deployed transmitters. Second,
there is an overhead of O(log 3 n) in time. Finally, the algorithm works only for α > 2.
A natural question concerns the difference between the power control model and the uniform model. It
is possible to prove that if some resource (e.g., the used energy or the general area in which the network
resides) is bounded, then the ratio between the two models is proportional to log B, where B is the bound
on the resource; see [1, 3, 13, 22].
Capacity and scheduling in the SINR model
Some recent studies aim at achieving a better understanding of the SINR model. In particular, in their
seminal work [12], Gupta and Kumar analyze the capacity of wireless networks in the physical and protocol
models. Specifically, they consider a setting where n transmitter-receiver pairs are located in the unit square.
In this setting, they show that the capacity of the transmitters under any deployment where the average
distance is constant is the same in both models. In contrast, Moscibroda [17] analyzes the worst-case
capacity of wireless networks in arbitrary deployments (specifically, allowing arbitrary average distances),
and establishes an exponential gap between the SINR model and the UDG model, by showing that there are
transmitter deployments for which the capacity in the SINR model is exponentially greater than in the UDG
model. (Those deployments involve some very short and some very long links, and the average distance
when all stations are located in the unit square is O(1/n).) In both studies, the deployments are not required
to ensure connectivity. (Actually, it may be impossible to ensure connectivity if one considers only a single
ACM SIGACT News 79 June 2010 Vol. 41, No. 2radio frequency and a single time step.)
Transmission scheduling (so as to prevent collisions) is a central issue in wireless communication theory.
Given a set of transmitter-receiver pairs, the goal is to select a schedule for the transmissions, so that all the
receivers correctly decode their messages. Several variants of the scheduling problem were studied in the
literature. Some of those variants assume a given power assignment for the transmitters (which in some
settings is required to adhere to a certain policy, such as uniformity or distance-proportionality, and in other
cases is assumed to be arbitrary). Other variants require the algorithm to decide also on the power assignment
for the transmitters (again, possibly adhering to some specific policy).
Schedules for basic network structures, namely, strongly connected networks, are studied in [19, 21].
It is shown that allowing arbitrary adjustments to the transmission power gives an exponential advantage
over the uniform or linear power assignment schemes. (The latter require a transmitter s sending a message
to a receiver r to use power proportional to d(s, r) α , the distance to power α.) This gives an interesting
complement to the capacity bounds of [12, 17] discussed above.
A measure called disturbance, capturing the intrinsic difficulty of finding a short schedule for a problem
instance, is defined in [18], along with an algorithm that achieves provably efficient performance (in terms
of schedule length) in any network and request setting that exhibits low disturbance.
For the special case of many-to-one communication with data aggregation in relaying nodes (e.g., summing
up some counters maintained at the nodes), a scaling law describing the achievable rate in arbitrarily
deployed sensor networks is derived in [17]. It is shown that for a large number of aggregation functions, a
sustainable rate of 1/ log 2 n can be achieved.
Goussevskaia [10] studies some additional algorithmic aspects of scheduling in the SINR model, and in
particular presents an O(log(n)) approximation algorithm for scheduling wireless requests. On the negative
side, some impossibility results are proved in [11], and the NP-hardness of the scheduling problem in the
SINR model is established in [10, 23].
In summary, the literature reviewed above presents several examples where the SINR model allows
better performance than the UDG model, typically through solutions involving exponential gaps in transmission
power or in distances between different transmitter-receiver pairs. We next discuss some further
results derived specifically in the SINR model.
Several recent papers consider oblivious transmission scheduling schemes. In oblivious scheduling, the
transmission power is entirely determined by the distance between the transmitter and the intended receiver.
(In particular, E(r) = c for constant c is a function determined by the distances, so uniform networks
employ an oblivious scheme.) In [7], Fanghänel et al. prove that there is an exponential gap between the
directed and bidirectional versions of the scheduling problem. Specifically, they show that in the case where
each pair of points consists of a dedicated transmitting device and a dedicated receiving device for any
oblivious power assignment scheme, there exists an instance with n directed communication requests that
requires Ω(n) time steps. In contrast, when the communication is symmetric and both endpoints use the
same transmission power, it is shown that there exists a universally good oblivious power assignment. This
means that the capacity increases when we allow transmission power to vary between the different sides of
a communication channel.
The problem of optimizing a single scheduling step was studied in [1, 6]. Given a collection of senderreceiver
pairs (si, ti) in the plane, the goal is to assign each transmitter si a power (possibly 0) so as to
maximize the number of successful pairs. Let ∆ = maxi d(si, ti)/ mini d(si, ti), i.e., the “aspect ratio”
restricted to the sender-receiver pairs. NP-hardness of the problem is established in [1], as well as an
O(log ∆)-approximation algorithm and a proof that in an appropriately defined game every Nash equilibrium
has an expected number of successful connections that is within O(∆2α ) of optimal. The main result
ACM SIGACT News 80 June 2010 Vol. 41, No. 2of [6] is that if all transmitters use no-regret algorithms to play the game defined in [1], then the average
number of successful connections (over a sufficient number of rounds) will be within O(∆2α ) of optimal.
This is the first distributed algorithm (in some sense) for this problem.
In [8], Fanghänel et al. introduce an instance-based measure I of interference, and prove for general
power assignments in the plane a lower bound of Ω( ) steps for α > 2. They also show that when
restricted to linear power assignments, the bound can be improved to Ω(I); in this case they also present
an efficient algorithm for computing an O(I log n) step schedule for linear power assignments. They also
extend these results towards multi-hop scheduling and routing. In the multi-hop scheduling problem, the
request consists of a sequence of pairs, referred to as paths, rather than a single pair of nodes.
In the context of routing, the problem of constructing minimum delay end-to-end schedules for a given
set of routing requests is studied in [4]. In this problem, each node is assigned a distinct power level, the
paths for all requests are determined, and all message transmissions are scheduled to guarantee successful
reception in the SINR model. A polynomial-time algorithm with provable worst-case performance for the
problem in this setting is presented in [4].
Another line of research, in which known results from the UDG model are analyzed under the SINR
model, is explored in [21], which studies the problem of topology control. The stations are assumed to be
embedded in the Euclidean plane, and are required to simulate a given underlying graph topology. For each
station v, let rv denote the length of the longest outgoing edge of v, and consider a ball of radius rv centered
at v. Denote the number of nodes contained in this ball by Iin(v), and let Iin = maxv∈V Iin(v). The main
result of [21] is that there exists a schedule that allows the communication to flow on the original topology
and completes in time O(Iin log 2 (n)).
SINR diagrams
An issue of increasing significance is that of obtaining a “reception map” depicting the behavior of a multistation
wireless network. Such a reception map characterizes the reception regions of the stations, namely,
partitions the plane into n reception regions H(si), 1 ≤ i ≤ n, and a region H ∅ where no station can
be heard. (Each of these n + 1 regions may possibly be composed of several disconnected sub-regions).
The map may change dynamically with time, as the stations may choose to transmit or keep silent, or even
change their location from time to time.
In the UDG model, the basic (static) underlying UDG diagram of a given wireless system can be constructed
in linear time in a straightforward manner. Moreover, for a given time step in which only some of
the stations transmit simultaneously, and for a given point p, it is simple to decide which station (if any) is
heard at p. The task becomes more challenging in the SINR model, where it is not completely clear what
shapes the reception regions may take; in fact, it is not easy to construct an SINR diagram even in a static
setting. Figures 1 and 2 illustrate the complexity of understanding SINR diagrams.
SINR diagrams appear to be fundamental to understanding the dynamics of wireless networks, and play
a key role in the development of suitable algorithmics for such networks, analogous perhaps to the role
played by Voronoi diagrams in the study of proximity queries and related issues in computational geometry.
In fact, the analogy between SINR diagrams and Voronoi diagrams runs deeper; it can be shown that the
SINR diagram of a set of points S converges to the Voronoi diagram of S when the path loss parameter α
tends to infinity and N = 0. The case where N > 0 is a little more complicated and has a nice interpolation.
In this case, one may compute the Voronoi cell of each station and intersect the cell with the a unit disk
center at the transmitter. The resulting structure consists of regions known as alpha complexes / shapes. One
use of the SINR diagram is to bridge between the Voronoi diagram and the structure of alpha complexes, as
can be seen in Figure 5.
I
log ∆
ACM SIGACT News 81 June 2010 Vol. 41, No. 2Voronoi (a)
alpha (b)
Figure 5: As α → ∞, the SINR diagram converges (a) to the Voronoi diagram, when the background noise
N = 0; (b) to the structure of alpha complexes, when N > 0.
SINR diagrams were introduced in [2], where it is shown that in uniform networks with α = 2 and
β ≥ 1, every reception region in the SINR diagram is convex and fat 2 . This “niceness” of the reception
regions may potentially make it easier to handle them by searching, sampling and location algorithms. In a
certain sense, this result lends support to the model of Quasi Unit Disk Graphs suggested by Kuhn et al. in
[15]. In contrast, when 0 < β < 1, the reception regions may be nonconvex and even overlapping, as can
be seen in figure 2(b).
The characterization of the reception regions as convex and fat is subsequently exploited in [2] for developing
an efficient structure for answering point location queries, by preprocessing the station locations
(in polynomial time) and providing each station with an additional data structure of size O(1/ɛ) that approximates
the reception region boundaries, so that for every point outside the ɛ vicinity of the boundary (for
performance parameter ɛ > 0) it is possible to answer a point location query in log(1/ɛ) time. Specifically,
given a collection of simultaneously transmitting stations S, a station s ∈ S, and a point p, the data structure
responds with one of three possible answers: (1) s is heard at p; (2) s is not heard at p; or (3) cannot say
for sure. Letting H be the reception region of s and F be the region for which answer (3) is returned, the
algorithm guarantees that Area(F) ≤ ɛ · Area(H).
Conclusions
In this survey we discussed some recent studies of the SINR model, its structural properties and algorithms
for it. Special attention was given to studies focusing on identifying the similarities and differences between
the SINR model and the graph-based UDG model, and drawing analogies and algorithmic ideas in both
directions.
While this area has attracted considerable attention in the last decade or so, much remains to be done,
especially when it comes to algorithms for the SINR model. It is hoped that improving our understanding of
the SINR model may lead to more efficient algorithms for a variety of design and communication problems.
In particular, the study of SINR reception diagrams and their structure and algorithms is still in its infancy.
Hence a challenging direction involves broadening our scope of understanding concerning SINR
2
i.e., the radius ratio between its largest enclosed circle and smallest enclosing circle is O(1).
ACM SIGACT News 82 June 2010 Vol. 41, No. 2diagrams, allowing us to generate, interpret and utilize such diagrams for non-uniform power models. This
may lead to the development of more accurate, yet practically usable, abstract models for wireless communication,
and possibly to bridging the gap between physical and graph-based models.
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ACM SIGACT News 84 June 2010 Vol. 41, No. 2