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IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 18, NO. 6, DECEMBER 2010 1701
Approximation Algorithms for Wireless Link
Scheduling With SINR-Based Interference
Douglas M. Blough, Senior Member, IEEE, G. Resta, and P. Santi
Abstract—In this paper, we consider the classical problem of link
scheduling in wireless networks under an accurate interference
model, in which correct packet reception at a receiver node de-
pends on the signa-to-interference-plus-noise ratio (SINR). While
most previous work on wireless networks has addressed the sched-
uling problem using simplistic graph-based or distance-based
interference models, a few recent papers have investigated sched-
uling with SINR-based interference models. However, these
papers have either used approximations to the SINR model or
have ignored important aspects of the problem. We study the
problem of wireless link scheduling under the exact SINR model
and present the first known true approximation algorithms for
transmission scheduling under the exact model. We also introduce
an algorithm with a proven approximation bound with respect to
the length of the optimal schedule under primary interference. As
an aside, our study identifies a class of “difficult to schedule” links,
which hinder the derivation of tighter approximation bounds.
Furthermore, we characterize conditions under which scheduling
under SINR-based interference is within a constant factor from
optimal under primary interference, which implies that secondary
interference only degrades performance by a constant factor in
these situations.
Index Terms—Approximation algorithms, signal-to-inter-
ference-plus-noise ratio (SINR)-based interference model,
spatial-time-division multiple access (TDMA), wireless link
scheduling.
I. INTRODUCTION
THE classical problem of scheduling transmissions in
multihop wireless networks, first studied in [12], has
recently gained renewed interest from the networking research
community, mainly because of its potential application in wire-
less mesh networks [2], [3], [15], [17], [21], [22], where the
tight time synchronization between nodes needed to schedule
wireless transmissions is deemed technically feasible. In fact,
the IEEE 802.16 [1] standard for mesh networks is considering
time-division multiple access (TDMA)-based MAC imple-
mentation. Study of the transmission scheduling problem is
motivated by the likelihood that good scheduling techniques
can optimize mesh performance by maximizing throughput and
improving fairness properties.
Manuscript received September 22, 2008; revised September 23, 2009
and March 19, 2010; accepted March 29, 2010; approved by IEEE/ACM
TRANSACTIONS ON NETWORKING Editor E. Modiano. Date of publication April
26, 2010; date of current version December 17, 2010.
D. M. Blough is with the School of Electrical and Computer Engineering,
Georgia Institute of Technology, Atlanta, GA 30332 USA.
G. Resta and P. Santi are with the Istituto di Informatica e Telematica del
CNR, Pisa 56124, Italy (e-mail: paolo.santi@iit.cnr.it).
Digital Object Identifier 10.1109/TNET.2010.2047511
There are several different versions of the scheduling
problem for multihop wireless networks. In one version [6],
flows are given, flow demands are elastic, and the problem is
to construct a set of routes, a bandwidth assignment, and a
link schedule over a sufficiently long scheduling interval that
maximize the total throughput (i.e., the sum of individual flow
rates). In a second version of the problem [7], typically referred
to as link scheduling, link demands are given, and the problem
is to construct a schedule of minimum length that satisfies
all demands. If new demands are placed on the network as
soon as one set of demands is satisfied, minimizing schedule
length will also maximize throughput. While our algorithms are
formulated for the link scheduling problem, we discuss related
work on signal-to-interference-plus-noise ratio (SINR)-based
scheduling for the first version of the problem as well. Yet
another version, which is less common, specifies end-to-end
flow demands [6] instead of link demands. For this version, as
for the first version mentioned, routing and scheduling must
be jointly considered. Given a routing algorithm, this version
can be trivially converted to the link demand version. However,
note that this version still differs from the first in that demands
are fixed rather than elastic.
A crucial point to address when tackling any of the sched-
uling problems is to determine whether a certain set of wire-
less transmissions can occur in parallel without corrupting each
other. This, in turn, requires using appropriate models for radio
signal propagation and interference. In fact, in contrast to wired
networks, wireless communications share the same (radio) com-
munication channel, and interference between simultaneously
transmitting links must be considered. The interference model
has been shown to have a major impact on the complexity of op-
timal wireless link scheduling. In fact, while the first problem
version is solvable in polynomial time under the primary in-
terference model1[9], both problem versions are known to be
NP-hard under models that account for secondary interference
(e.g., graph-based models [20] and SINR-based models [7]).
Furthermore, while algorithms with constant-factor approxima-
tion bounds have been developed for geometric graphs under
some of the simpler secondary interference models, such as
k-hop interference [20] and protocol interference [21], no ap-
proximation bounds have yet been proven for the most accurate
SINR model.
Using accurate radio signal propagation and interference
models is fundamental to ensure that the schedules computed
by a certain algorithm do not lead to collisions in a practical
scenario. This explains the efforts in the research community to
1In the primary interference model, two links conflict if and only if they share
a common endpoint.
1063-6692/$26.00 © 2010 IEEE
1702 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 18, NO. 6, DECEMBER 2010
derive near-optimal scheduling algorithms using increasingly
accurate models. In particular, several interference models
have been considered in the literature, including, most recently,
the physical (or SINR-based) interference model used for the
first time in [8] to investigate asymptotical wireless network
capacity. The basic assumption of the SINR-based interference
model, i.e., that it is cumulative interference power, and not
the number of interfering signals, which determines whether
a packet is correctly received at the intended receiver, has
been recently experimentally validated in a low-power wireless
network [13]. Furthermore, in the same paper, the authors show
that graph-based and distance-based interference models are
inaccurate and lead to throughput degradation with respect to
SINR-based models. Thus, usage of the physical interference
model for determining feasible transmission schedules is moti-
vated by its practical relevance.
According to the physical interference model, a set of con-
current transmissions along links do not corrupt each
other if and only if the SINR at each receiver is at least a cer-
tain threshold , when all nodes at the transmitter end of links
are concurrently transmitting.
Designing and analyzing algorithms under this SINR-based
model is especially difficult since, contrary to what happens
with simpler models such as graph-based [20] and protocol in-
terference [8], interference under this model is not localized
(i.e., even faraway interferers can potentially corrupt a trans-
mission) and does not induce binary “conflict” relationships be-
tween links (e.g., transmission along a link might not corrupt a
transmission along link by itself, but it might corrupt when
a third transmission along link is also occurring). These in-
trinsic properties of the SINR model have hindered derivations
of computationally efficient wireless link scheduling algorithms
under this model. Only very recently have these problems been
approached in the literature [3], [4], [6], [7], [14]. However,
these prior works are either based on unrealistic assumptions
(unbounded transmit power) [14], or use approximate SINR
models in which the effect of noise [7] or of faraway inter-
ferers [3], [4] is ignored, or ignore certain “difficult to schedule”
links [6]. Thus, to the best of our knowledge, the problem of de-
signing computationally efficient wireless link scheduling algo-
rithms with proven, deterministic approximation bounds under
the exact SINR model remains open for both versions of the
wireless scheduling problem.
A. Summary of Contributions
In this paper, we present computationally efficient sched-
uling algorithms for the schedule-length-minimization version
of wireless link scheduling under the assumption that nodes
use the same, fixed, transmission power . Through analysis
of these algorithms, we prove, for the first time under the exact
SINR model, deterministic approximation bounds on schedule
length. By combining our main algorithm with algorithms for
scheduling under the primary interference model in a novel
way, we are also able to prove an approximation bound that
is relative to the optimal schedule computed under primary
interference. Since the optimal schedule under primary in-
terference is, in general, shorter than the optimal schedule
under SINR-based interference, this approximation bound
is even stronger than approximation bounds relative to the
optimal SINR-based solution. Based on near-optimal primary
interference scheduling algorithms and, under certain condi-
tions, constant factor approximation for our SINR scheduling
algorithm, the combined algorithm produces schedules that are
within a constant factor of the optimal schedule for primary
interference under these conditions. This result shows that,
under certain conditions, the impact of secondary interference
on schedule length is minimal, and that efficient algorithms
exist to derive schedules that are provably close to optimal
primary interference schedules.
As a second major contribution of this work, we present
insights into which factors affect the complexity of optimally
scheduling links under the SINR model. In particular, we
identify a paradox that we call the “black-gray link paradox,”
which leads to the identification of a class of links that are
“difficult to schedule” (gray links). We show that gray links
are directly responsible for problems that prior work had in
producing approximation results for this problem. Gray links
have an impact on our results as well. As our analysis shows, the
higher the fraction of gray links in the set of links to schedule,
the looser the approximation bounds we are able to prove for
our algorithms. In case no gray link occurs in the network,
our algorithms are proven to be within a constant factor from
optimal (Corollary 1). The same constant approximation bound
can be obtained if nodes are deployed in a region of constant
diameter, independently of the number of black/gray links to
schedule (Theorem 4). Furthermore, if density of nodes per unit
area is assumed to be constant, one of our algorithms is proven
to be within a constant factor from optimal under primary
interference (Corollary 2). To the best of our knowledge, ours
is the first characterization of conditions under which wireless
link scheduling under the SINR model asymptotically achieves
the same performance as that achievable under primary in-
terference. This result indicates that under some (reasonable)
assumptions, the impact of secondary interference on network
throughput is asymptotically marginal. The approximation
bounds of the various algorithms considered in this paper, as
well as of existing algorithms, are summarized later in Table I.
The rest of this paper is organized as follows. In Section II,
we survey recent works related to our study. In Section III, we
introduce notation and define the problem considered in the rest
of the paper. In Section IV, we describe a phenomenon that we
call the “black-gray link paradox,” which hinders the derivation
of tight lower bounds to wireless link scheduling complexity
under the exact SINR-based model. In Section V, we present
our approximation algorithms based on the exact SINR-based
model, prove their properties in terms of performance bounds
and time complexity, and discuss guidelines for distributed im-
plementation. Section VI concludes the paper.
II. RELATED WORK
The problem of transmission scheduling in multihop wireless
networks has been deeply investigated in the literature following
the seminal work by Nelson and Kleinrock [12]. It has been in-
vestigated either in isolation, known as link scheduling [7], or
BLOUGH et al.: APPROXIMATION ALGORITHMS FOR WIRELESS LINK SCHEDULING WITH SINR-BASED INTERFERENCE 1703
as part of the more general problem of investigating network ca-
pacity limits, which encompasses also finding optimal routes be-
tween source destination pairs, transmission power assignment,
and so on. Herein, we focus primarily on the link scheduling
version of the problem.
Given the shared nature of the communication medium
in a wireless network, how to model interference between
transmissions occurring on different, spatially separated links
is a fundamental component of the network model, which con-
siderably affects computational and algorithmic complexity. A
first fundamental distinction is between models that consider
only primary interference and those considering also effects
of secondary interference. Primary interference, according to
which two links interfere with each other if and only if they
share a common endpoint, is a mandatory constraint to model
single-radio-per-node networks. On the other hand, secondary
interference models a peculiar feature of wireless communica-
tions, i.e., usage of a shared communication medium, implying
that two links can interfere with each other even if they do not
share endpoints. Several models accounting for both primary
and secondary interference have been considered in the liter-
ature, which can be roughly classified into graph-based (e.g.,
hop-based [20]), distance-based (e.g., the homogeneous pro-
tocol model of [8]), and SINR-based models (e.g., the physical
model of [8]).
It is important to observe that there is a significant difference
in the complexity of computing a minimum-length schedule,
depending on whether secondary interference is considered or
ignored: While this problem can be solved in polynomial time
with primary interference only (which is equivalent to 1-hop,
graph-based interference) [9], it becomes NP-hard when con-
sidering models that include secondary interference, e.g., SINR-
based models [7].
A second important distinguishing feature of interference
models is whether only local, pair-wise interference is consid-
ered or all possible simultaneous communications occurring
in the network are taken into account. Models that can be rep-
resented by a conflict graph, e.g., hop-based interference and
protocol interference, belong to the former category, whereas
SINR-based interference models belong to the latter. Recently,
constant-factor approximation bounds have emerged for geo-
metric graphs under several graph-based interference models,
e.g., k-hop interference [20] and protocol interference [21].
However, the global nature of the SINR-based interference
models challenges the design of algorithms with proven ap-
proximation bounds with respect to optimal since under these
models it is not possible to spatially divide the deployment area
into smaller regions, with the property that link scheduling in
different regions can be done independently. Furthermore, the
global nature of SINR-based interference models hinders the
design of localized, distributed scheduling algorithms based on
these models.
Recently, a few papers have addressed some of the challenges
related to SINR-based interference models.
In [14], Moscibroda and Wattenhofer derive upper bounds
on the length of schedules built according to the exact SINR
model, under the assumption that nodes can use arbitrarily high
transmission power, which is unrealistic in practical scenarios.
The works that are more closely related to ours are [3] and [7],
in which the authors present computationally efficient sched-
uling algorithms with proven approximation bounds under
the assumption that nodes use the same constant transmission
power. However, the bounds proven in [3] are very loose, hold
only in a probabilistic sense under a random node distribution
assumption, and are obtained for an SINR-based interference
model in which interference from faraway transmitters is
neglected. The interference model used in [7] is also an ap-
proximation of the SINR model, in which the effect of noise is
neglected. As we will thoroughly discuss in the following, the
interference model considered in [7] is significantly different
from the exact SINR model, and the approximation bound
given in [7] does not hold in the exact model.
Another work that is closely related to ours is [6], in which
the authors consider the throughput maximization version of the
link scheduling problem under the exact SINR-based interfer-
ence model. The authors present polynomial time algorithms
that achieve a throughput within certain nontrivial bounds from
optimal. Interestingly, the approximation bounds are valid only
under the assumption that the transmit power of nodes is slightly
decreased (by a constant, multiplicative amount , where
is an arbitrary constant) with respect to the transmit power
used in the original problem instance. A consequence of this is
that certain links, including the so-called black and gray links
that we have identified as critical, are ignored by the algorithm.
Thus, the approximation bounds hold only when no such links
are present in the network.
It is also worth mentioning two recent papers in which lo-
calized distributed algorithms working under SINR-based in-
terference models have been designed. In [4], Brar et al. present
a localized distributed implementation of the GreedyPhysical
scheduling algorithm presented in [3], which thus achieves the
same approximation bounds (under the approximate SINR in-
terference model described above). In [19], Scheideler et al.
present a localized distributed algorithm for building a domi-
nating set under the exact SINR interference model. Both algo-
rithms make extensive use of physical carrier sensing to achieve
local computation at the nodes, while guaranteeing algorithm
operations under the global SINR interference model.
III. BACKGROUND
A. Problem Formulation
Given is a weighted communication graph ,
where edge represents a directed wireless link
between a sender and a receiver in the network. Graph
is geometric, i.e., nodes in can be thought of as points in the
two-dimensional plane, and edge implies that
, where is the Euclidean distance between
nodes and . In other words, a directed edge between nodes
and can exist only if is within distance from , where
is the maximum transmission range (see below for a formal
definition). Directed edges in represent directed wireless
communication links used to carry traffic between network
nodes. For this reason, in the following we will use terms
“edge” and “link” interchangeably.
1704 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 18, NO. 6, DECEMBER 2010
Note that in general is a subset of the set of all possible
links between nodes in . This is to model situations in which a
routing algorithm has selected a subset of all possible commu-
nication links to actually carry traffic (e.g., a set of trees routed
at the gateways in case of wireless mesh networks). Again, we
stress that the focus of this paper is on optimal link scheduling,
hence we assume that a routing algorithm has already been ex-
ecuted, and the set of links to schedule with relative demands
(see below) is given.
For each edge , integer weight represents the (cur-
rent) traffic demand on link . In case of time-varying traffic de-
mands, we can think of periodically reexucuting the presented
algorithms in order to adapt to the new traffic distribution. In
some places, we assume all edges have a weight of one. This
is referred to as the unit demand case, as has been done in
some previous work [7], [14]. The problem is then to construct
a schedule, , where edge is assigned to exactly
of the slots, the set of edges assigned to slot is a
feasible transmission set with respect to the interference model
under consideration, and is as small as possible. A feasible
transmission set is a set of wireless links for which, if all of
nodes at the transmitter end of edges in are transmitting con-
currently, all of the receivers will correctly receive their intended
packets, according to some model of the interference caused
by one communication on reception of another communication.
We discuss interference models later in this section. A schedule
is feasible if and only if all the ’s are feasible
transmission sets (w.r.t. the considered interference model). We
are interested in constructing minimum-length feasible sched-
ules so as to maximize the overall throughput of the network.
This is the problem considered, e.g., in [3], [7], [9], [14], and
[20]. Note that the dimension of our problem is the number
of links to schedule, i.e., , and not the number of network
nodes.
Summarizing, the problem considered in this paper is the
following.
Input: Parameters , , , and . A set of wireless nodes
and for each , its location . A set of links,
where each connects two nodes , where
the Euclidean distance between and is no greater than the
transmission range (given by (3) in the following). For each
, its demand , where is the
maximum link demand.
Problem: Find a schedule of slots
satisfying the following:
1) for , ;
2) every link appears in slots of ; and
3) for and for every , inequality (1)
(the SINR inequality) is satisfied at when all such that
are transmitting;
and having the minimum possible .
B. Radio Signal Propagation
We adopt the classical model for radio signal propagation
in wireless networks, which is referred to as the log-distance
path loss model. In this model, the radio signal strength (power)
at a distance from the transmitter is given by , where
is the transmission power and is the path loss co-
efficient [16] (the actual value of the constant depends on
the environment—e.g., indoor or outdoor).2Up to technical de-
tails, our results can be extended to more general radio propaga-
tion models that account for irregular radio coverage area, such
as the cost-based model proposed in [19], which is shown to
closely approximate log-normal shadowing propagation. In the
following, we assume all nodes use the same transmit power, an
arbitrary constant .
C. Interference Models
The simplest interference model for single-radio, half-du-
plex wireless networks (i.e., networks in which each node is
equipped with a single, half-duplex radio) is the primary inter-
ference model, according to which two links cannot transmit
simultaneously if and only if they share an endpoint.
More accurate interference models consider also secondary
interference, which accounts for the fact that all nodes in a wire-
less network share the same radio communication channel. In
particular, in this paper we use the SINR interference model
[8] (a.k.a. physical interference model), according to which the
successful reception of a packet sent by node and destined to
node depends on the SINR at . To be specific, denoting by
the received power at node of the signal transmitted by
node , a packet along link is correctly received if and
only if
(1)
where constant is the background noise, is the subset of
nodes in that are transmitting simultaneously, and is
a constant threshold (the SINR threshold).3In the SINR model,
every concurrent transmission in the network (including trans-
missions between very distant nodes) must be explicitly consid-
ered when evaluating whether any single given transmission is
successful.
Combining (1) with the formula for radio signal propagation,
we have that a packet sent along link is correctly received
if and only ifc
(2)
We stress that the SINR interference model with log-distance
path loss, under the assumption that , implicitly accounts
for primary interference. In fact, it is easy to see that (2) cannot
be satisfied in each of the following situations leading to pri-
mary interference: i) node transmits simultaneously to two
receiver nodes and ; ii) node receives simultaneously
2Note that
>
2
is a standard assumption in literature, justified by the fact
that area (and approximately the number of interferers) grows as a power of 2;
hence, the only way of having bounded aggregate interference is to have radio
signal decay larger than 2.
3In practice, the exact value of
depends on the desired data rate on the
wireless channel, the modulation scheme, etc.
BLOUGH et al.: APPROXIMATION ALGORITHMS FOR WIRELESS LINK SCHEDULING WITH SINR-BASED INTERFERENCE 1705
from two sender nodes and ; iii) node receives from node
while transmitting to node . In case i), at both and ,we
would have a SINR below 0 dB (at least one interferer, node
itself, is at the same distance as the intended transmitter), which
is not sufficient to correctly receive a packet. In case ii), at least
the packet sent by the farther transmitter cannot be correctly
received due to negative SINR value. Similarly, in case iii),
at least the packet transmitted to node cannot be correctly
received due to negative SINR value— an interferer colocated
with the receiver. Note that the log-distance path loss model
implies the use of omnidirectional antennas and, hence, we do
not consider the use of specific technologies such as directional
antennas and multiple-input–multiple-output (MIMO), wherein
primary interference could possibly occur without violating
SINR constraints.
Equation (2) leads to the notion of maximum transmission
range, denoted , which is defined as the maximum distance
up to which a packet can be correctly received in absence of
interference, i.e., when set is empty. Formally
(3)
In [7], the SINR model is considered without noise, i.e., it is
assumed that in inequality (1). We refer to this as the
SIR model. Observe that in the SIR model, a link has a positive
budget up to infinite distances in absence of interference, i.e., the
transmission range is infinite. This is a major difference w.r.t.
to the SINR model in which the link budget is positive up to
afinite distance even in absence of interference. As we shall
see, this difference between the two models has a significant
impact on the complexity of deriving efficient approximation
algorithms for the scheduling problem. More specifically, the
fact that the transmission range is finite in the SINR model gives
rise to what we call the “black-gray link paradox,” which we
carefully describe in the next section.
IV. BLACK-GRAY LINK PARADOX
Consider a transmission from a node to a node and as-
sume that is exactly at the border of ’s maximum transmission
range (i.e., the SINR at node in absence of interference—the
SNR—is exactly ). In this situation, independently of the size
of the deployment region, no other transmission concurrent with
that along link is possible since even a very tiny contribu-
tion to the interference is sufficient to drive the SINR at below
. Hence, if we call black those links such that the sender–re-
ceiver distance is exactly equal to , then sequentially sched-
uling black links is the best that even the optimal algorithm can
do (i.e., black links are “easy to schedule”). However, let us now
consider a link (call it a gray link) such that the SINR at
in absence of interference is , for some arbitrarily small con-
stant . If the deployment region is unbounded, then we can
put a concurrent transmitter at a large enough distance from
in such a way that the SINR at when is transmitting is some
value , for some . Since we can repeat this ar-
gument over and over, if the deployment region is unbounded,
we can have an infinite number of transmissions going on in
parallel to the transmission along link without impairing
correct message reception at . We thus have the following ap-
parent paradox, which we call the black-gray link paradox:If
is a black link, it must be scheduled sequentially (i.e., only one
simultaneous transmission is possible) even if the deployment
region is unbounded; on the other hand, if is a gray link, then
an infinite number of transmissions can potentially be scheduled
in parallel with transmission along (i.e., simultaneous
transmissions are, in principle, possible). This huge difference
in potential concurrency comes despite an arbitrarily small dif-
ference in SNR values for black and gray links.
As we shall see, the “black-gray link paradox” hinders the
derivation of tight lower bounds to the number of time slots
needed to optimally schedule gray links. As a consequence, the
quality of our derived approximation bounds for the wireless
link scheduling problem under the SINR model is heavily af-
fected by the black-gray link paradox. In particular, gray links
are “difficult to schedule”: If few or no gray links are present,
we can prove an approximation bound for our scheduling
algorithm; however, if relatively more gray links are present in
the network, the approximation bound becomes looser. In the
extreme case in which all the links to schedule are gray, we can
prove only the trivial approximation bound.
A possible way to limit the extent of the “black-gray link
paradox” is to make some assumptions on the size of the deploy-
ment region. In particular, if the deployment region is assumed
to have bounded diameter, we can prove an approxima-
tion bound for the problem of optimally scheduling transmis-
sions under the SINR model regardless of the number of gray
links to schedule.
We close this section with a brief analysis of how the black-
gray link paradox manifests itself also in the work presented in
[6], which, similarly to our work, is based on the exact SINR
interference model. In fact, the approximation bounds reported
therein (which are of the form for the case of homo-
geneous transmit power assignment, where , similarly to
[7], is defined as the number of link length classes) are valid
up to the following technical trick. Let be a
problem instance, where is the set of nodes, is the set of
possible links, is the vector of source/destination pairs, and
is the nodes transmit power.4Furthermore, let be the
power-reduced instance of ( is an arbitrary constant ),
i.e., the problem instance in which all nodes use transmit power
, and the set of all possible links is reduced
accordingly. The authors of [6] show that the solution of a
certain linear program built on is feasible also for the orig-
inal problem instance , and that the throughput achieved by
is within a factor from the optimal solution of , not
of the original instance . Technically speaking, the one pre-
sented in [6] is not an approximation bound since no relation-
ship between the optimal throughput for instance and that for
instance is proven. Note that the introduction of the
margin to the nodes transmission power is needed to keep pos-
sible communication links (the set ) sufficiently above the re-
quired threshold for correct message reception, i.e., to avoid
occurrence of gray links.
4We recall that the authors of [6] consider the more general problem of ap-
proximating network capacity, hence routes between source/destination pairs
are not specified, but are part of the computed solution.
1706 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 18, NO. 6, DECEMBER 2010
Fig. 1. 4-coloring used in step 6 of Algorithm GOW*.
V. A PPROXIMATION ALGORITHMS
A. Analysis of Algorithm GOW
Our first approximation algorithm, referred to as GOW*, is
presented in Section V-B. GOW* builds on the algorithm of [7],
which is referred to as GOW in the rest of this paper. In order
to understand GOW*, we first review the operation of Algo-
rithm GOW, and we also evaluate GOW’s performance under
the true SINR model in light of the black-gray link paradox.
Algorithm GOW is designed for the case of unit link de-
mands, i.e., for each , and is based on the idea of
partitioning links into length classes , with the prop-
erty that . For each such class
, a proper square cell partitioning of the (possibly infinite) de-
ployment region (i.e., of the portion of the plane where nodes in
lie) is defined. The analysis of GOW in [7] ignores the effect
of noise, and thus, given that in the SIR model, the pos-
sible number of length classes is infinite as well. Cells are then
4-colored in such a way that no two adjacent cells have the same
color (see Fig. 1). A fundamental property of algorithm GOW
is that if the side of the square cells is chosen properly, links
whose receivers are in different cells with the same color can
be scheduled in the same slot without corrupting each other’s
transmissions. A crucial point in the derivation of the approx-
imation factor reported in [7] is that this property holds if the
cell side for length class is set to a constant multiple of the
minimum length of links in . The value of this constant is de-
noted . For each of the cell partitionings corresponding to the
various length classes, the authors then prove an upper bound
on the maximum number of feasible transmissions whose re-
ceivers are in the same cell, which is shown to be a certain con-
stant depending on . This upper bound is used to prove
the approximation bound: The basic argument here is that algo-
rithm GOW schedules all links whose receivers are in the same
length class and same cell sequentially, while the optimal al-
gorithm can schedule at most of them in a single slot, re-
sulting in an overall approximation bound, where
is the number of length classes (which can be as high as
if link lengths grow as a geometric series).
To factor noise into the analysis, we first observe that links
have a maximum possible length, which equals , as defined
in (3). Another effect of considering noise is that the value of
used to determine the cell partitioning is no longer a constant,
but is instead a function of the considered length class .
In particular, is unbounded for the last length class (the
one including links whose length is within a factor from the
transmission range). Since for the last length class is un-
bounded, also the number of feasible concurrent trans-
missions in the same cell becomes unbounded for this length
class. On the other hand, Algorithm GOW schedules all links in
the last length class sequentially (because is unbounded),
implying a trivial approximation bound (even if is a
constant) for the performance of the scheduling algorithm. Note
that this last length class, which causes the problem with the ap-
proximation bound, contains all gray links.
In summary, due to the difficulty of scheduling gray links,
the result of directly applying Algorithm GOW in the true SINR
model is a trivial approximation bound. Next, we show how Al-
gorithm GOW can be extended to handle gray links and provide
nontrivial approximation bounds even in the true SINR model.
B. Algorithm GOW*
Similarly to [7], we assume unit link demands throughout
this section. Algorithm GOW* is reported in Fig. 2. There are
two main differences between Algorithms GOW and GOW*:
1) links are grouped according to a SNR-based, instead of dis-
tance-based, criterion; and 2) links with the smallest SNR values
(black and gray links according to the terminology introduced)
are treated separately.
Grouping links according to SNR instead of distance reflects
the fact that what is relevant when scheduling a link is its
strength (expressed in terms of the SNR), rather than the length.
Furthermore, by combining usage of SNR-based link grouping
with the exact SINR interference model, we have that, indepen-
dently of the actual link lengths, there are a constant number of
classes to consider. This is because: 1) the minimum possible
SNR value for a feasible link is , by the very same definition
of SINR model; and 2) the maximum possible SNR value is
, corresponding to the situation in which the receiver is
within a distance known as near-field [16] from the transmitter,
where the received signal power is the same as the transmitted
power.5It is worth noting that situations in which the minimum
link length converges to 0 (e.g., when an increasing number
of nodes is distributed uniformly at random in a region of unit
area; see, e.g., the analyses of [3] and [8]) are also handled by
our model since the maximum possible SNR value of is
assigned to arbitrarily short links.
Note that, under our working assumption of log-distance
radio propagation with path loss exponent , links in the
th SNR class have length
and that black and gray links are included in class (this
amounts to classifying as gray those links whose SNR is below
and greater than , for a given value of ).
5By fundamental laws of physics, the received signal power can be at most
as large as the transmitted power.
BLOUGH et al.: APPROXIMATION ALGORITHMS FOR WIRELESS LINK SCHEDULING WITH SINR-BASED INTERFERENCE 1707
Fig. 2. The GOW* Algorithm.
When considering links in class , the deployment region
is divided into square cells of side , where constant is
defined as follows6:
Cells in the same class are then 4-colored in such a way that
no two adjacent cells have the same color (recall Fig. 1). Then, at
steps 7–12 links are greedily scheduled in successive slots, with
the property that only links with the same color whose receivers
are in different cells are assigned with the same slot.
We first prove that the schedule computed by Algo-
rithm GOW* is feasible under the SINR model. We recall
that usage of the SINR interference model ensures that links
scheduled in the same slot are not subject to primary inter-
ference. Then, we show an upper bound to the length of the
schedule constructed by our algorithm w.r.t. optimal.
Theorem 1: Assume that . Then, the schedule com-
puted by Algorithm GOW* is feasible under the SINR model.
Proof: We first observe that links in class are scheduled
sequentially, thus the resulting slots are obviously feasible under
the SINR model. Let us now consider a slot containing links in
class , for some .Wenow
upper-bound the interference experienced by a receiver in a
certain cell in the partitioning obtained for class . Once we
focus on a receiver in specific cell , the cells containing re-
ceivers of the interfering links can be arranged in circumcentric
square frames around . The inner frame contains
cells, the second frame contains cells, and in gen-
eral the th frame will contain
cells. The generic receiver contained in the th frame will be at
least apart from . Considering that in class
6We recall that we assume
>
2
, so constant
is well defined.
all links have a length smaller than , the minimum distance
between and a sender relative to frame is
Hence, the total interference experienced by can be upper-
bounded by
(4)
(5)
(6)
(7)
(8)
(9)
where (5) follows because for and indeed
is always greater than 2, and (9) follows
from a known bound on Riemann’s zeta function.
The SINR for the receiver can thus be bounded by
since and .
Definition 1: Given a problem instance , the max-
imal SNR density is the maximal number of receivers in
a cell of class , for some .
Constant is called the SNR diversity of instance .
We now prove an upper bound on the length of the schedule
computed by Algorithm GOW*.
Theorem 2: The schedule computed by Algorithm GOW*
has length, where is the number of links
in class .
Proof: Links in class are scheduled sequentially, which
brings the term in the big-O notation. Links in class ,
for , whose receivers are in a cell of color, say, ,
are scheduled in parallel if they are in different cells; hence, the
number of slots needed to accommodate all links in class is
the number of receivers in the maximally occupied cell, which
we denote , times the number of colors. We then have that
the total schedule length is upper-bounded by
.
We are now ready to prove the approximation bound for Al-
gorithm GOW*.
Theorem 3: Algorithm GOW* computes a schedule whose
length is within a factor:
i) from optimal when ;
1708 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 18, NO. 6, DECEMBER 2010
ii) from optimal when and
;
iii) from optimal when and
.
Proof: We first consider case i). In this case, we can use the
trivial lower bound 1 on optimal schedule length and Theorem 2
to conclude that Algorithm GOW* is within a factor
from optimal, which is since implies that all
links are in class .
Assume now that . We start by proving an upper
bound on the number of feasible transmissions with receivers
belonging to the same cell relative to length class , for
. In particular, we prove that no more than
links with receiver belonging to the same cell can be scheduled
together, where link class is considered. The value of is
obtained by solving the following inequality:
(10)
which leads to
from which the above value of is obtained. Inequality (10)
comes from assuming the largest possible received power at the
numerator, and the minimum possible contribution to interfer-
ence from links whose receiver end is in cell .
From the above fact, it follows that links belonging to a cell
for which link occupancy is are scheduled in at least
slots in the optimal schedule. This leads to an
lower bound on the length of the optimal schedule. By The-
orem 2, we have an approximation ratio with respect to optimal
of . In case ii), we have that
, and the approximation bound becomes .In
case iii), we have , and the ap-
proximation bound becomes . This concludes the proof of
the theorem.
Corollary 1: If , then the approximation bound
of algorithm GOW* becomes .
As observed above, the GOW* approximation bound depends
on the number of long links (black and gray links) in the net-
work: The higher this number, the worse the approximation
bound.
The relationship between number of black/gray links and
quality of approximation bound lies in the fact that when SNR
is arbitrarily close to , it is not possible to define a bounded
grid size that is able to spatially separate links. In other words,
when scheduling links whose SNR is arbitrarily close to ,
the deployment region can be considered to be composed of
a single cell comprising the whole region, and links have to
be scheduled sequentially according to the GOW/GOW* ap-
proach. A possible way of improving the approximation bounds
in presence of relatively many black/gray links would be to de-
fine more accurate scheduling policies within a single cell, thus
possibly allowing spatial reuse within the same cell. However,
this would essentially entail redesigning the entire scheduling
approach. On the other hand, we stress that in most practical
situations the approximation bound is indeed accurate since:
1) the approximation bound becomes independently of
the number of black/gray links when the deployment region is
bounded; and 2) the number of black/gray links to schedule can
be reduced by appropriate routing decisions and/or transmit
power control.
Given the relationship between number of black/gray links
and approximations bounds, the network designer might want
to select links with a relatively high SNR value for routing net-
work traffic (i.e., to include in link set ) so that the perfor-
mance of the scheduling algorithm becomes close to optimal.
Note that wireless links are typically subject to link quality fluc-
tuations due to the time-varying nature of radio signal propaga-
tion. Hence, selecting links with relatively high SNR values is
very reasonable in practice since they provide good transmis-
sion quality also in presence of radio signal fluctuations.7For
instance, if we consider typical settings of mW
dBm, dBm, dB, and ,wehavea
nominal transmission range of 2.154 km; if we set when
defining the SNR classes, we get an approximation bound
for GOW-PI by excluding links of length larger than 2.060 km
from the set of links used to route traffic.
Another possible way of limiting the number of black/gray
links is to use transmit power control. In principle, it is possible
to slightly increase the transmit power used at some transmitters
in such a way that the SNR at receivers is sufficiently above
, so to reduce/avoid the occurrence of black/gray links. Note
that up to properly redefining SNR link classes and constant ,
all the results presented in the paper remain valid also when
transmitters use different transmit powers as long as the ratio
between the highest and the lowest transmit power used by the
transmitters is (i.e., an arbitrary constant).
We now prove a stronger approximation bound for the case
where network nodes are deployed in a region with
bounded diameter.
Theorem 4: Assume nodes in are deployed in a region of
constant bounded diameter . Then, Algorithm GOW*
computes a schedule whose length is within a factor from
optimal, regardless of the number of links in class .
Proof: To prove the theorem, it is sufficient to show that,
under the assumption that is a constant, at most a
constant number of links in class can be scheduled con-
currently by the optimal algorithm. From this observation, we
have a lower bound of on the optimal schedule
length, and the theorem follows by Theorem 2.
The above fact can be easily shown by solving inequality
where is the maximum distance between a pair of
nodes in , which results in , which
is a constant under the theorem assumptions.
7Time-varying radio signal propagation is not accounted for in the log-dis-
tance path loss model, which can be thought of as representing the time-aver-
aged intensity of the radio signal at a certain distance from transmitter.
BLOUGH et al.: APPROXIMATION ALGORITHMS FOR WIRELESS LINK SCHEDULING WITH SINR-BASED INTERFERENCE 1709
To the best of our knowledge, the bounds proved in Theo-
rems 3 and 4 are the first nontrivial, deterministic approxima-
tion bounds for the problem of wireless link scheduling under
the SINR model.
Before ending this section, we formally prove that GOW* has
polynomial time complexity.
Theorem 5: Algorithm GOW* has ,
where is the maximum number of cells in a partitioning of
the deployment region computed at step 5 of the algorithm.
Proof: The outer for cycle (steps 4–12) is executed times.
At each iteration, for each of the four colors considered, all the
squares of current color are scanned ( operations), and
possibly a link is selected in each square ( operations).
This repeat-until cycle is repeated times. Hence,
GOW* time complexity is .
C. Approximation Bounds With Respect to Primary
Interference Model
Here, we present a two-phase approach to prove approxi-
mation bounds with respect to optimal scheduling under the
primary interference model. We recall that a transmission set
is feasible under the primary interference
model if and only if no two links share a common endpoint.
Furthermore, throughout this subsection we allow arbitrary in-
teger link demands.8
Proving approximation bounds with respect to optimal under
primary interference is important to isolate the effects of radio
interference, which, we recall, is responsible for secondary in-
terference, on the wireless link scheduling problem. In partic-
ular, in this section we will prove that, under some (reason-
able) conditions, accounting for secondary interference when
building the schedule incurs no schedule length increase with re-
spect to the primary interference model (in an asymptotic sense).
Thus, our results indicate that, in some conditions, radio in-
terference has no (asymptotic) effect on schedule length, and
the wireless link scheduling problem can be accurately char-
acterized in asymptotic terms even using the simple primary
interference model. It is also worth observing that since the
optimal schedule under primary interference cannot be longer
than that under SINR interference, the approximation bounds
proved in this section can be considered as relatively stronger
than those reported in Section V-B and in existing literature on
the SINR-based interference model.
The reader might wonder why we do not directly use algo-
rithm GOW*, which is able to produce schedules free ofprimary
interference, to derive performance bounds with respect to op-
timal scheduling under primary interference. The difficulty here
lies in the fact that, by directly applying algorithm GOW*, it is
not easy to relate the length of the optimal schedule under pri-
mary interference to the upper bound on the length of the sched-
ules produced by GOW* (which accounts for both primary and
secondary interference).
To get around this difficulty, we present a two-phase ap-
proach, which we call PRIMARYSINR. The key idea is to
separate primary and secondary interference contributions
8We only assume that the sum of the link demands remains polynomial in the
number of links, so that when expanding the demand graph to a multigraph with
unit demands, the problem size remains polynomial in the number of links.
Fig. 3. Algorithm PRIMARYSINR.
when building the schedule: In the first phase, links are sched-
uled according to some algorithm considering only primary
interference as the feasibility criterion; then, transmission sets
corresponding to each slot computed by are considered
and are possibly split into several slots in order to satisfy
also requirements of the SINR interference model. For this
purpose, a second scheduling algorithm is considered. This
two-phase approach allows us to easily derive bounds with
respect to optimal schedule length under primary interference,
if approximation bounds for and upper bounds on the length
of the schedule built by are known. The PRIMARYSINR
scheduling approach is summarized in Fig. 3.
The following proposition shows how to turn approximation
bounds for and upper bounds on schedule length for into
an approximation bound for PRIMARYSINR with respect to the
optimal schedule under primary interference.
Proposition 1: Let be a scheduling algorithm for primary
interference with approximation bound with respect
to optimal, and let be a scheduling algorithm for SINR in-
terference that builds schedules of length at most , for
some functions . Algorithm PRIMARYSINR builds a
schedule whose length with respect to optimal under primary
interference is upper bounded by .
Proof: It is sufficient to observe that the number of slots
computed by is at most , where is the length
of the optimal schedule under primary interference, and that
each of these slots is divided into at most slots by al-
gorithm in the second phase of the algorithm. Hence, the
total length of schedule computed by PRIMARYSINR is within a
factor from the optimal schedule under primary
interference.
An immediate consequence of Proposition 1 is that whenever
is an optimal algorithm for primary interference scheduling
and produces a schedule of length, the PRIMARYSINR
approach builds a feasible schedule under SINR interference
whose length is within a constant factor from the length of the
optimal schedule under primary interference. In particular, if
both and are polynomial time algorithms, the schedule
can be computed efficiently and used, e.g., for benchmarking
more practical SINR interference scheduling approaches such
as GreedyPhysical [3]. This brings us to the question of whether
1710 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 18, NO. 6, DECEMBER 2010
TABLE I
COMPARISON OF THE SCHEDULING ALGORITHMS
such algorithms for primary and SINR interference scheduling
exist.
We first observe that building an optimal schedule under pri-
mary interference consists in finding a minimum edge coloring
of the multigraph obtained from by replacing each edge
with demand with multiedges between the same pair of
nodes. Thus, an edge-coloring algorithm for multigraphs can be
used in phase 1 of PRIMARYSINR. The minimum edge coloring
problem is known to be NP-hard even for simple graphs [11].
However, the optimal solution can be computed in polynomial
time if the graph is bipartite.9If graph is not bipartite, a 4/3
approximation of the optimal schedule can be computed using,
for instance, the algorithm of [10], yielding the same asymp-
totical approximation bounds as in the case where graph is
bipartite.
On the other hand, Theorem 2 states that the length of the
schedules computed by SINR interference scheduling algorithm
GOW* is , which implies that GOW* builds
schedules of length whenever the number of black/gray
links to schedule is , and we have a constant node density
per unit area. Thus, we can conclude this section with the fol-
lowing corollary.
Corollary 2: Assume is an arbitrary graph with integer
link demands, is a polynomial time edge-coloring algorithm
for multigraphs that is at most a constant factor from optimal,
and is the GOW* algorithm. Furthermore, assume the
number of black/gray links in the network is , and the
node density per unit area is constant. Under these assumptions,
algorithm PRIMARYSINR builds in polynomial time a schedule
that is feasible under SINR interference and whose length is
within a factor from the optimal schedule under primary
interference.
Proof: The proof is an immediate implication of Proposi-
tion 1, Theorem 2, and of the assumptions on link length and
node density.
To the best of our knowledge, this is the first characteriza-
tion of conditions under which wireless scheduling under SINR
interference achieves a performance comparable to that under
primary interference presented in the literature. This result is
very interesting since it shows that, under certain conditions,
secondary interference reduces network throughput by at most
a constant factor with respect to primary interference. It is also
worth observing that the conditions for the above to happen are
indeed satisfied in most realistic application scenarios.
9Note that the multigraph to edge color is bipartite if and only if the original
graph
G
is bipartite.
D. Summary of Approximation Bounds
Here, we summarize the approximation bounds of the various
scheduling algorithms introduced in the previous sections. We
first observe that algorithms for unit link demands (i.e., GOW
and GOW*) can be extended to the case of arbitrary integer link
demands by replacing a link with demand into copies of
the same link with unit demand. The resulting approximation
bounds are for algorithm GOW and
for algorithm GOW*, where is the sum of the demands on
links in class , and is the maximal sum of demands of
links whose receivers are in the same cell.
The approximation bounds of the various algorithms consid-
ered in this paper are summarized in Table I. The bounds for the
PRIMARYSINR approach refer to the case in which a constant
factor to optimal approximation algorithm for primary interfer-
ence scheduling is used, and GOW* is used as the SINR inter-
ference scheduling algorithm. In the table, “interference model”
refers to the model used to build the schedule, while “reference
model” refers to the interference model used to build the optimal
schedule to which performance of the considered algorithm is
compared.
In the table, we have reported a generalized version of the ap-
proximation bound for PRIMARYSINR stated in Corollary 2 that
can be easily derived from Theorem 2. This bound holds for net-
works not necessarily satisfying the link length and node den-
sity conditions stated in the corollary, and is defined as ,
where is defined as , is
the number of links in slot belonging to class , and
is the maximum number of links in slot whose receivers are
in the same cell (for some link class , with ).
It is worth observing that ours are the only bounds using the
exact SINR model to build the schedule and possibly using pri-
mary interference as a reference model. While bounds of the
various algorithms depend on specific quantities defined therein
(e.g., number of link length classes for GOW, or maximal
SNR density for GOW* and PRIMARYSINR), all the considered
bounds (excluding those for GREEDYPHYSICAL) have the prop-
erty that they can be as tight as in some conditions, but they
can be as loose as the trivial bound in worst-case condi-
tions. The design of scheduling algorithms based on SINR in-
terference models with provable, nontrivial worst-case bounds
is, to the best of our knowledge, still open.
E. Distributed Implementation of Scheduling Algorithms
For some types of wireless multihop networks, centralized
operation of scheduling algorithms is quite realistic. For ex-
BLOUGH et al.: APPROXIMATION ALGORITHMS FOR WIRELESS LINK SCHEDULING WITH SINR-BASED INTERFERENCE 1711
ample, in small- to medium-sized mesh networks, gateway
nodes serve as natural centralization points for collecting
network information, computing schedules, and disseminating
the schedules to the other network nodes. However, large mesh
networks and other types of networks might require distributed
scheduling algorithms. Here, we briefly discuss the suitability
of Algorithms GOW* and PRIMARYSINR for distributed
execution.
For Algorithm GOW*, many of the parameters can be deter-
mined prior to, or at the beginning of, deployment. The length
classes for the links, along with the associated cell sizes, are de-
pendent on the transmission power , the SINR threshold ,
the noise floor , and the path loss exponent . Parameters
and are functions of the devices and technology used in the
network, while and can be easily measured in the deploy-
ment environment [5]. Prior to network operation, these param-
eters can be distributed to all nodes along with the location of a
fixed reference point so that every node can calculate the grids
used for each length class. We also assume that nodes know their
approximate locations, so that they know in what cell they re-
side for every length class grid. Location could be determined
from built-in GPS, from GPS measurements done during de-
ployment, or through estimation techniques such as triangula-
tion from known points. The grid colorings can also be done in
a deterministic way so that every node can reproduce them.
The remaining values that need to be known prior to schedule
construction are the link demands. These can be estimated peri-
odically by sampling queue lengths at transmitters, and for dis-
tributed execution need only to be communicated from the trans-
mitter of every link to the associated receiver, since scheduling
is done at the receiver side. For every cell within every length
class, all nodes that serve as a receiver for at least one link in the
class must coordinate to determine a schedule. This can be done
in several ways. One possibility is to elect a cell leader, send all
of the link demands for the cell to the leader, and then have the
leader calculate and distribute the cell schedule to all nodes in
the cell. The nodes who are receivers for scheduled links would
then simply communicate the schedule slots to their associated
transmitters. It is also possible to have every receiver broadcast
its demand to all nodes in the cell, and then have every node
independently calculate in a deterministic fashion the same cell
schedule. The important points to emphasize are that: 1) coordi-
nation is limited to within individual cells, so there is no global,
network-wide communication performed; and 2) the number of
length classes is constant, so the number of grids for which these
distributed scheduling computations must be performed does
not increase with the size of the network. As a result of these
two points, we believe that distributed implementation of Algo-
rithm GOW* is realistic, even for quite large networks.
VI. DISCUSSION AND CONCLUDING REMARKS
In this paper, we have investigated the problem of wireless
link scheduling under the exact SINR interference model. First,
we have shown that even tiny contributions to the SINR at the
receivers (e.g., the noise or the interference from faraway trans-
mitters) cannot be ignored when building the schedule nor when
evaluating the approximation bounds. In view of this obser-
vation, we have provided the first known algorithm for wire-
less link scheduling with proven, deterministic approximation
bounds under the SINR model and identified a class of links that
are “difficult to schedule” and hinder the derivation of tighter ap-
proximation bounds. We have also introduced an algorithm with
proven approximation bounds with respect to primary interfer-
ence and identified conditions under which scheduling under
SINR interference is within a constant factor from optimal under
primary interference. These conditions might serve as a guide-
line in the design of wireless networks (e.g., avoid routing along
links whose SINR value is close to the threshold for correct
message reception).
The study reported in this paper leaves several avenues open
for further research on this intriguing problem. In particular, the
problem of finding better lower bounds on the length of optimal
schedules for the class of “difficult to schedule” links, which
might lead to the derivation of nontrivial worst-case approxi-
mation bounds, remains open. In view of this, it is interesting to
observe that the “black-gray link paradox” is a consequence of
using a thresholded interference model, according to which the
packet reception rate on a link is 100% if the SINR value at the
receiver is , and it is 0 if the SINR is even slightly below
. Indeed, such a sharp SINR threshold for correct packet re-
ception is unlikely to occur in practical settings, where transi-
tion between near-zero packet reception rates and near-100%
rates spans a few dBs (see, e.g., [13]). Hence, there exists a
gray SINR area in which packet reception is still possible, al-
though with a rate significantly below 100% (and above 0). A
promising direction for future work is studying the wireless
link scheduling problem using a nonthresholded SINR-based
interference model and proving approximation bounds under
this model. Usage of nonthresholded SINR-based interference
model has the potential to improve (expected) throughput (e.g.,
30% throughput improvement with respect to thresholded SINR
model has been observed in [13] in an experimental testbed),
while at the same time (possibly) countering the “black-gray
link paradox” described in this paper. Initial steps in this direc-
tion can be found in [18].
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Douglas M. Blough (SM’00) received the B.S. de-
gree in electrical engineering and the M.S. and Ph.D.
degrees in computer science from The Johns Hopkins
University, Baltimore, MD, in 1984, 1986, and 1988,
respectively.
Since Fall 1999, he has been Professor of Elec-
trical and Computer Engineering with the Georgia In-
stitute of Technology, Atlanta, where he also holds
a joint appointment with the College of Computing.
From 1988 to 1999, he was on the faculty of Elec-
trical and Computer Engineering, University of Cal-
ifornia, Irvine. His research interests include distributed systems, dependability
and security, and wireless multihop networks.
Dr. Blough was Program Co-Chair for the 2000 International Conference
on Dependable Systems and Networks (DSN) and the 1995 Pacific Rim Inter-
national Symposium on Fault-Tolerant Systems. He has been on the program
committees of numerous other conferences, was Associate Editor for the IEEE
TRANSACTIONS ON COMPUTERS from 1995 through 2000, and was Associate
Editor for the IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS
from 2001 through 2005.
G. Resta received the M.S. degree in computer sci-
ence from the University of Pisa, Pisa, Italy, in 1988.
In 1996, he became a Researcher with the Isti-
tuto di Matematica Computazionale of the Italian
National Research Council (CNR), Pisa. He is now
a Senior Researcher with the Istituto di Informatica
e Telematica (CNR), Pisa. His research interests
include computational complexity (especially in
relation to linear algebra problems), parallel and
distributed computing, and the study of structural
properties of wireless ad hoc networks.
P. Santi received the Laura and Ph.D. degrees in
computer science from the University of Pisa, Pisa,
Italy, in 1994 and 2000, respectively.
He has been a Researcher with the Istituto di Infor-
matica e Telematica del CNR, Pisa, Italy, since 2001,
and is now a Senior Researcher. During his career,
he visited Georgia Institute of Technology, Atlanta,
in 2001 and Carnegie Mellon University, Pittsburgh,
PA, in 2003. He has contributed more than 40 papers
and a book in the field of wireless ad hoc and sensor
networking, His research interests include fault-tol-
erant computing in multiprocessor systems (during Ph.D. studies) and, more
recently, the investigation of fundamental properties of wireless multihop net-
works such as connectivity, lifetime, capacity, mobility modeling, and cooper-
ation issues.
Dr. Santi is a Member of the IEEE Computer Society and a Senior Member
of the Association for Computing Machinery (ACM) and SIGMOBILE. He is
Associate Editor of the IEEE TRANSACTIONS ON MOBILE COMPUTING. He has
been General Co-Chair of ACM VANET 2007 and 2008, and he is involved in
the organizational and technical program committee of several conferences in
the field.
