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Brief Announcement:

Distributed Contention Resolution in Wireless Networks

∗

Thomas Kesselheim

Department of Computer Science

RWTH Aachen University

Aachen, Germany

thomask@cs.rwth-aachen.de

Berthold Vöcking

Department of Computer Science

RWTH Aachen University

Aachen, Germany

voecking@cs.rwth-aachen.de

ABSTRACT

We present and analyze simple distributed contention res-

olution protocols for wireless networks. In our setting, one

is given n pairs of senders and receivers located in a met-

ric space.Each sender wants to transmit a signal to its

receiver at a prespecified power level, e.g., all senders use

the same, uniform power level as it is typically implemented

in practice. Our analysis is based on the physical model in

which the success of a transmission depends on the Signal-

to-Interference-plus-Noise-Ratio (SINR). The objective is to

minimize the number of time slots until all signals are suc-

cessfully transmitted.

Our main technical contribution is the introduction of a

measure called maximum average affectance enabling us to

analyze random contention-resolution algorithms in which

each packet is transmitted in each step with a fixed probabil-

ity depending on the maximum average affectance. We prove

that the schedule generated this way is only an O(log2n)

factor longer than the optimal one, provided that the pre-

specified power levels satisfy natural monontonicity proper-

ties. By modifying the algorithm, senders need not to know

the maximum average affectance in advance but only static

information about the network. In addition, we extend our

approach to multi-hop communication achieving the same

appoximation factor.

Categories and Subject Descriptors: C.2.1 Computer-

Communication Networks Network Architecture and Design:

Wireless Communication, Distributed Networks

General Terms: Algorithms, Theory

Keywords: Wireless Network, Interference, Physical Model,

SINR, Distributed Scheduling

1.INTRODUCTION

We analyze distributed contention-resolutions protocols

for packet scheduling in wireless networks giving worst-case

guarantees. The interference constraints are modelled by

the physical interference model [3]. Between any two nodes

of the network u and v a distance d(u,v) is defined. If node

u transmits a signal at power level p then it is received by

v with strengthp/d(u,v)α, where the constant α > 0 is the

∗This work has been supported by the UMIC Research Cen-

tre, RWTH Aachen University.

Copyright is held by the author/owner(s).

PODC’10, July 25–28, 2010, Zurich, Switzerland.

ACM 978-1-60558-888-9/10/07.

so-called path-loss exponent1. The node v can successfully

decode this signal if the signal strength received from the

intended sender is at least β times as large as the signals

strengths by interfering transmissions made at the same time

plus ambient noise. This is, the Signal-to-Interference-plus-

Noise Ratio (SINR) is above some threshold β ≥ 0, the

so-called gain.

In our setting, we are given a set of n requests R ⊆ V ×V ,

corresponding to pairs of nodes from a metric space and a

power level p(ℓ) > 0 for each of them. We have to select a

time slot c(ℓ) ∈ {1,...,k} for each request ℓ ∈ R such that

for each ℓ = (u,v) ∈ R the SINR constraint

p(ℓ)

d(u,v)α≥ β

X

ℓ′=(u′,v′)∈R

c(ℓ)=c(ℓ′)

p(ℓ′)

d(u′,v)α+ N

!

is fulfilled. The constant N ≥ 0 expresses ambient noise

that all transmissions have to cope with. The objective is

to minimize the number of time slots k.

Our objective is to calculate a schedule whose length is

close to the optimal schedule length that could possibly be

achieved by an optimal schedule in the same instance. We

denote the optimal schedule length for R that uses some

fixed power assignment p by T(R,p). For the problem vari-

ant in which powers are subject to optimization, a similar

measure has been introduced by Moscibroda et al. [6] as

scheduling complexity T(R).

Powers might be given by hardware or by a scheme. Such

schemes for assigning the powers that have been used in re-

lated work include uniform [5], linear [2] and square-root

(or mean) power assignments [1, 4]. For each of them, there

are specialized algorithms, which are mostly centralized. So

far, de-centralized algorithms with a provable performance

guarantee are only known for linear power assignments [2].

Furthermore, most existing transceivers support only a rel-

atively small, fixed number of possible power levels so that

a practical implementation of both linear and square-root

power assignments remains a challenge. As a consequence it

is necessary to have more general algorithms which not only

work for a certain power scheme.

Our algorithms do not require a certain power scheme

but work for every power assignment satisfying the following

natural conditions. First, it has to be non-decreasing and

sublinear. That means if d(ℓ) ≤ d(ℓ′) for two requests ℓ,ℓ′∈

R then p(ℓ) ≤ p(ℓ′) and p(ℓ)/d(ℓ)α≥ p(ℓ′)/d(ℓ′)α. So the

1Typically it is assumed that 2 < α < 5. However, our

analysis works for any α > 0.

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transmission power of ℓ′has to be as least as large as the one

for ℓ. At the same time, the received power at the receiver of

ℓ′must not be larger than the one at the receiver of ℓ. This

monotonicity condition is very natural and is fulfilled by all

previously studied power assignments, particularly the ones

mentioned above. The second condition is that powers are

chosen sufficiently large so that ambient noise plays a minor

part compared to interference.

2.MAXIMUM AVERAGE AFFECTANCE

We introduce a new measure called maximum average af-

fectance¯ A that depends on the request set R and the power

assignment p. This measure extends a so-called measure of

interference for linear power assignments [2] in a non-trivial

way towards general power assignments satisfying the above

conditions.

For two requests ℓ = (u,v) and ℓ′= (u′,v′), and a power

assignment p, we define the affectance of ℓ on ℓ′by

ap(ℓ,ℓ′) = min

(

1,β

p(ℓ)

d(u,v′)α

,„

p(ℓ′)

d(u′,v′)α− βN

«)

.

The notion of affectance was introduced by Halld´ orsson and

Wattenhofer [5], which we extended to arbitrary power as-

signments and bounded by 1. When taking the noise out

of consideration, it indicates which amount of interference

ℓ induces at ℓ′, normalized by the signal strength from the

intended sender of ℓ. As a consequence the sum of affectance

is at most 1 for a request set that may be assigned to same

time slot.

To get the maximum average affectance¯ A(R,p), we take

the maximum over all subsets of requests and consider the

average affectance a link is exposed to from all other requests

in this subset:

¯ A = max

M⊆Ravg

ℓ′∈M

X

ℓ∈M

ap(ℓ,ℓ′) = max

M⊆R

1

|M|

X

ℓ′∈M

X

ℓ∈M

ap(ℓ,ℓ′) .

The maximum average affectance is the key to analyze

random contention-resolution based algorithms and compar-

ing the perfomance to the optimum. In our basic algorithm

each sender transmits with a certain probability q in each

step until one of the transmissions has successfully been re-

ceived. We first prove that if q ≤1/4¯

successful within O(log n/q) time slots whp2. Thus choosing

q =1/4¯

We complement this result by proving¯ A is at most a factor

O(logn) larger than the optimal schedule length T(R,p).

In combination, this yields the schedule generated by the

algorithm has length O(T(R,p) · log2n).

A all transmissions are

A, we generate a schedule of length O(¯ A·logn) whp.

3.TOWARDS DISTRIBUTED

ALGORITHMS

An algorithm that is applicable in a realistic environment

has to work in a distributed fashion with as few informa-

tion as possible. In order to achieve this goal, we present

two modifications. These do not affect the schedule length

vitally and we still get schedules of length O(¯ A·logn) whp.

On the one hand, we extend it such that the network nodes

do not have to know¯ A anymore but adapt the transmission

2with high probability: with probability 1−ncfor each con-

stant c

probability q on their own. On the other hand, we present

a way to inform each sender if a transmission has success-

fully been received by transmitting acknowledgement pack-

ets. This is not a trivial task because these acknowledgement

packets may also interfere.

Altogether, this is the first distributed algorithm to the in-

terference scheduling problem with a guaranteed approxima-

tion ratio. It requires only static information on the network

that can be spread at the time of deployment. Particularly,

the number of network nodes, the clock synchronization and

the power assignment can be seen as such static informa-

tion. In contrast, no information about the current state

of the network will be necessary. For example, communica-

tion requests arise after the deployment and an algorithm

has to work without knowledge on which requests have to

be served by the network and which of them were already

successfully served. Our algorithm can be run on all senders

and receivers of a network such that during the execution

no central entity is needed.

As a further result, we adapt the ideas to a distributed

multi-hop algorithm that allows packets to use intermediate

relay nodes. For a fixed choice of paths and powers we get

an O(log2n) whp approximation for this problem as well.

4. DISCUSSION AND OPEN PROBLEMS

While previous algorithms are mostly centralized, the al-

gorithms and analyses we present seem to be much closer

to realistic scenarios as the scheduling protocol only needs

static information. Nevertheless, it is an interesting ques-

tion which performance can still be achieved without this

knowledge. Unfortunately, we cannot get rid of any of these

assumptions in a non-trivial way. However, concerning the

number of nodes and the clock synchronization there are

various results in other scenarios that could possibly trans-

ferred.

For the power assignment problem the best solution up

to know is to take distance-based power schemes such as

the square-root power assignment. Up to now there is no

known way to calculate a power assignment achieving an

approximation ratio that is close to optimal in all instances,

even not in a centralized way. This leaves much space for

future research.

5.ACKNOWLEDGEMENTS

We like to thank Alexander Fangh¨ anel for valuable dis-

cussions and comments.

6.REFERENCES

[1] Alexander Fangh¨ anel, Thomas Kesselheim, Harald R¨ acke, and

Berthold V¨ ocking. Oblivious interference scheduling. In

PODC, pages 220–229, 2009.

[2] Alexander Fangh¨ anel, Thomas Kesselheim, and Berthold

V¨ ocking. Improved algorithms for latency minimization in

wireless networks. In ICALP, pages 447–458, 2009.

[3] Piyush Gupta and P. R. Kumar. The capacity of wireless

networks. IEEE Transactions on Information Theory,

46:388–404, 2000.

[4] Magn´ us M. Halld´ orsson. Wireless scheduling with power

control. In ESA, pages 361–372, 2009.

[5] Magn´ us M. Halld´ orsson and Roger Wattenhofer. Wireless

communication is in APX. In ICALP, pages 525–536, 2009.

[6] Thomas Moscibroda, Roger Wattenhofer, and Aaron Zollinger.

Topology control meets SINR: The scheduling complexity of

arbitrary topologies. In Mobihoc, pages 310–321, 2006.

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