Coded Splitting Tree Protocols

0.26 MB·English

by  Jesper H. Sørensen

Coded Splitting Tree Protocols
Jesper H. Sørensen, Cˇedomir Stefanovic´, and Petar Popovski
Aalborg University, Department of Electronic Systems, Fredrik Bajers Vej 7, 9220 Aalborg, Denmark
E-mail: {jhs,petarp,cs}@es.aau.dk
Abstract—This paper presents a novel approach to multiple themainconceptsofthecodedsplittingtreeprotocol,followed
access control called coded splitting tree protocol. The approach by an analysis in Section IV. Numerical results are presented
buildson the known tree splitting protocols, code structure and
in Section V. Finally, Section VI concludes the paper.
successiveinterferencecancellation(SIC).Severalinstancesofthe
tree splitting protocol are initiated, each instance is terminated
3 II. SYSTEMMODEL
prematurely and subsequently iterated. The combined set of
1
leaves from all the tree instances can then be viewed as a We consider a multiple access channel with a single re-
0
2 graph code, which is decodable using belief propagation. The ceiver and N transmitters, also referred to as users, with the
maindesignproblemisdeterminingtheorderofsplitting,which
population size N being unknown to all of them. Link time
n enablessuccessfuldecodingasearlyaspossible.Evaluationsshow
is organized in fixed-size slots; the duration of the slot is
a that the proposed protocol provides considerable gains over the
J standard tree splittingprotocol applying SIC. The improvement equal to the duration of user messages. Every time i−th user

0 comes at the expense of an increased feedback and receiver transmits, 1 ≤ i ≤ N, he sends a a replica of its message
3 complexity. X (we assume thateach replica containspointersto all other
i

replicas, as this is required for the execution of SIC). The
] I. INTRODUCTION
T transmissions from different uses are assumed synchronized
.I Distributed random access control schemes, like ALOHA andthe receivedsignalin slot j, Yj = i∈Aj Xi isnoiseless,
s [1] and splitting-tree protocols [2], represent a simple but where Aj is the set of active (transmittPing)usersin slot j and
c popularchoice forthe multiple access channels.In theirbasic theirnumberisdenotedby|A |.Slotj isreferredtoas: idleif
[ j
variants, these protocols treat packet collisions as waste, but |Aj|=0, a single if |Aj|=1 and a collision if |Aj|>1. The
1 the recent research has shown that successive interference receiveris onlyable to distinguishbetween these three events
v cancellation(SIC)cansignificantlyincreasethethroughputof in a given slot, but is not able to determine the multiplicity
2
random access protocols. The use of SIC was explored in the of a collision, also referred to as the collision degree. It is
3
splitting-tree framework [3], doubling the throughput of the assumed that the receiver is able to store any number of Y
2 j
7 binary version of the scheme. Recently, SIC has been used as in memory, in order to perform SIC at a later stage. As an
1. a main ingredient of the coded random access protocols [4]: example, if Y1 = X2+X5 (collision) and Y2 = X5 (single),
0 the users are allowed to transmit packet replicas in multiple then Y1−Y2 =X2, which providesan additionalmessage. In
3 slots of the frame and once a transmission has been resolved, this way the collisions may potentially be resolved, whereby
1 SICisused toremoveitsreplicas,potentially“unlocking”the the corresponding slots contribute to the overall throughput,
v: collisions where these replicas may have occurred. Analogies defined as T = R, where R is the number of recovered
M
i between SIC and iterative belief-propagation (BP) erasure- messagesandM isthenumberofusedslots.Thesystemalso
X
decodingwereestablishedin [5], leadingto the applicationof consistsofafeedbackchannel,throughwhichthereceivercan
r
a the codes-on-graphstheory to the design of framed ALOHA- broadcastinstructions to the transmitters. The performanceof
based random access schemes. In [6], these ideas were ex- the system is defined by the achievable throughput under a
tended by applying the concepts of rateless coding to the given number of feedback messages F.
slotted ALOHA.
In this paper we show how the ideas of coded random III. CODEDSPLITTINGTREEPROTOCOL
accesscanappliedusingsplittingtreeprotocolsasbasis,rather In this section, we describe the proposed scheme. Our
than ALOHA. We propose the coded splitting tree protocol, approach is centered around the tree splitting algorithm first
consisting of a set of partially split trees whose combined introduced in [2]. In tree splitting, collisions are resolved by
leavesconstituteagraphcode,overwhichthereceiverapplies asking the involved users to retransmit their message in one
SIC. The focus of our work is on the split strategy that of the following B slots with equal probabilities. This will
optimizes the distribution of user collisions over the leaves, potentially isolate one or more of the users, whereby their
such that a suitably chosen reward function related to the messages are successfully received. Any remaining collisions
evolution of the SIC is maximized. The proposed scheme is are treated in the same way until all collisions are resolved.
derived for the general case when the number of contending This process can be viewed as a tree, where each node
users is not a priori known. represents a received signal, which can either be an idle (0),
The organization of the rest of the text is as follows. a single (1) or a collision (c). A collision is the parent of B
Section II introduces the system model. Section III presents children, if an attempt to resolve it has been made. When the
1 1) Estimation phase: Perform BTS in each tree, using a
predefined strategy, until a predefined point, where the
2 3
observationsin the tree serve as basis for an estimation
4 5 6 7 of the degree of the remaining collisions.
2) Degreeoptimizationphase:Basedonthepartiallysplit
8 9 10 11
trees and the collision degree estimates from the esti-
mation phase, a strategy for further splitting is chosen,
12 13
which maximizes a reward function on Ω.
1
2
Active users 34 13 1 A. Estimation Phase
5 4 2 4 2 4
6 5 6 5 3 6 5 1 2 6 4 5 In the estimation phase, the goal is to partially split the
Slot index 1 2 3 4 5 6 7 8 9 10 11 12 13 K trees and estimate the degrees of the remaining collisions.
Hence,N isnotexplicitlyestimated,ratherhowthepopulation
Fig.1. Exampleofthetreesplitting algorithm. has been distributed across the transmission slots, which is
more informative and useful in our case. Information for this
estimation comes from idle and single slots, since only these
process is completed, each leaf in the tree has at most degree
have finite entropy in the observation (in fact zero entropy).
1. See Fig. 1 for an example of N =6 and B =2. Here the
The strategy for each tree is to perform the BTS algorithm,
nodesarenumberedaccordingtheslotnumbertheyrepresent.
with the modification that only a single child is split, if
Underthetreeitisseenwhichuserstransmitintheindividual
the split of the parent resulted in two collisions. Once an
slots.
idle or a single slot is observed, the algorithm is repeated,
ThisworkfocusesonthesimplestcasewhereB =2,which
starting at a collision node closest to the root. Note that this
is called binary tree splitting (BTS). This approach has an
requires a feedback message, since the users are not aware
asymptotic average throughputof 0.347 for N →∞ [2]. The
of the outcome of a transmission slot. Using the token-based
useofSICenablestoperformabinarysplitusingonlyasingle
representation for tree protocols from [7], it can be seen that
slot. For the example on Fig. 1, after receiving the signal Y
2 any node in the i’th level of the tree covers 1 of the
in slot2,the receivercan locallycreatethe receivedsignal Y3 probability mass, since each user has an a priori2ip−r1obability
usingY1 andY2;thustheslot3 canbesaved.Usingthisidea, of 1 to transmit in the corresponding slot. The described
theaveragethroughputofthesplittingtreealgorithmhasbeen 2i−1
process is iterated until the total fraction of the probability
doubled to 0.693 [3].
mass covered by idle and single slots exceeds a threshold
BTSisakeycomponentoftheproposedscheme.However,
α. Based on the observations, the degree distributions for the
instead of completing the algorithm, the tree is only partially
remaining collisions are calculated, as detailed in section IV.
split, such that a number of collisions remains. The resulting
leaves will follow a certain degree distribution denoted Ω,
B. Degree Optimization Phase
where Ω(d) is the fraction of leaves for which |A| = d.
Note that if BTS was fully completed, such that R = N, Thepurposeof the degreeoptimizationphaseis to perform
then Ω(0) = M−N, Ω(1) = N and Ω(d) = 0 for d > 1, as binary splits in the trees in an order that favors SIC. Hence,
M M
there are no unresolved collisions. For partial BTS, Ω(d) is we are looking for the sequence of splits, referred to as
a random variable for all d ≤ N, where Ω(0) and Ω(1) can the split order, which modifies the trees resulting from the
be observed in the tree. The choice of splitting strategy, i.e. estimation phase, such that the degrees among the leaves
which collisions to resolve, determines the statistics of Ω(d). follow a desirable distribution. The first split, after the esti-
This relationship is subject to analysis in section IV. mation phase, is determined by first calculating the resulting
Themainideaofcodedsplittingtreeprotocol,andthemain degree distribution, Ω1, for each possible node to be split,
conceptualcontributionofthiswork,istogenerateK partially s1 = {1,2,...,C}, where C = M − Ω(0) − Ω(1), is the
splittreesandviewthecombinedsetofleavesasagraphcode number of collisions among the leaves after the estimation
on the N messages from the users. Each user transmits its phase. See details in section IV. The split that maximizes the
messagesuchthatitsmultiplereplicasexistamongtheleaves, scalarproductofΩ1andtherewardfunction,λ,isthenchosen.
whichprovidesanSICpotentialsimilartoin[4].Theresulting Thereward functionassignsa valueto a leaf of anydegree d,
graph code can thus be decoded using SIC (i.e, iterative BP i.e.λ(d)isthevalueofaleafwithdegreed.Thenextsplitsare
decoding), as applied in erasure codes like LDPC codes and choseninthesamemanner,usingΩi−1asthestartingpointfor
LT codes. Hence, existing results in this area can serve as choosingthesplits .NotethateachsplitcausesM toincrease
i
guidelines for the desired Ω and thereby for the splitting by one. Once a split order of the desired length has been
strategy. determined,itisbroadcasttotheusers,whichstarttotransmit
Since N is unknown,it isnotpossibleto design ana priori accordingto the schedule. After each split, belief propagation
splitting strategy a priori to fulfill the requirements to Ω. We decodingis attempted,thusgivingthewholeprotocola flavor
therefore propose a two-phase strategy: of rateless codes, as in [6]. When all collisions have been
1 6 1 6
2 2' 7 7' 2 2' 7 7'
3 3' 5 5' 8 8' 10 10' 3 3' 5 5' 8 8' 10 10'
4 4' 9 9' 11 11' 4 4' 14 14' 9 9' 12 12' 11 11'
2
1 3 13 13'
1 7 3 1 7 5 2 2
2 8 4 2 2 8 6 5 Active users 5 2
Active users 34 190 56 170 35 1 5 27 34 190 14 79 69 190161190 5 171122
5 11 8 11 8 4 8 11 5 11 8 10 1 4 1 10 3
6 12 9 12 9 6 9 3 12106 121211 8 12 8 11 7 3 7 Slot index 1212'1313'1414'
Slot index 1 2 2' 3 3' 4 4' 5 5' 6 7 7' 8 8' 9 9'1010'1111'
Fig.3. Exampleofthedegreeoptimization phase.
Fig.2. Exampleoftheestimation phase.
IV. ANALYSIS
resolved,eitherthrougha splitorSIC, a terminatingfeedback
Two elements in the proposed scheme are subject to anal-
message is broadcast.
ysis. First, the leaf degree probability distribution is derived
C. Example on the basis of an observation of the tree. This is the essence
of the estimation phase. Second, the expressions necessary to
Fig. 2 and Fig. 3 show an exampleof the proposedscheme
optimize the split order for the degree optimization phase are
with K = 2, N = 12 and α = 0.2. The tree nodes are
derived.
labeled with the slot they represent, which makes it possible
to follow the evolution of the trees. Underneath the trees, it A. Leaf Degree Probability Distribution
is illustrated which users are active in the individualslots and We introduce D = {D1,D2,...,DM}, which is a vector
thereby also the degrees of the nodes. Note that since SIC is
of random variables, where D is the degree of the ℓ-th leaf,
ℓ
utilized,onlyasingleslotisnecessaryforeachsplit,suchthat ℓ = 1,2,...,M. Note that given a population N, D follows
the complementarynodeof the noderepresentingthe i-thslot
a multinomial distribution, denoted M, with parameters N
isdenotedby i′. In otherwords,node i′ canbe obtainedfrom and probability vector P = {P1,P2,...,PM}, where Pℓ is
i and the parent node, and thus a slot corresponding to node
the fraction of probability mass covered by the ℓ-th leaf, ℓ=
i′ iIsnnFoitg.n2eetdheed.estimation phase is illustrated. In the left tree 1p,o2ss,i.b..l,eMre.alMizoatrieoonvsero,fwDe ianstrcoodnuscteraiDnNeˆd, bwyhitchhe iosbstherevasetitonosf
the first single or idle slot is observed in slot 4′, after three in the tree and an estimate of the population N =Nˆ. Hence,
s1pli=ts0fr.o1m25thoef trhoeotp,rwobhaicbhilimtyemanasssth.eSisnlocet cαov=er0s.a2firsaacptipolnieodf, for all D∈DNˆ, we have that ℓDℓ =Nˆ. Also, Dℓ =|Aℓ|,
23 if |Aℓ|<2; otherwise Dℓ is unPknown (i.e, unobserved).
the estimation phase continues, starting from the node closest
For the unobserved leaf degrees, we find the probability
to the root, 2′. After only a single split, anothersingle or idle
distribution, Pr(D = d) for d ≥ 2, by summing the
ℓ
sthloetnisodoebsceorvveerds,t2h12is=tim0.e2i5nothfeththeirpdrolbevaebliloitfythmeatsrse,eb.rHinegnicneg, wpreobhaabvielitnieosporfioarlldirsetarliibzuattiioonnsoninNˆD,NˆwwehmeurestDalℓso=sudm. Soinvceer
thetotalmasscoveredbysingleandidleslotsto 0.375,which Nˆ =1,2,...,∞, weighting each Nˆ equally:
exceeds α and the estimation phase of this tree is terminated.
∞
The same procedure is performed in the right tree, where
Pr(D =d)= M(D,Nˆ,P). (1)
single or idle slots are observed in 9′, 11 and 11′, who cover ℓ
atotalof 1 + 1 + 1 =0.375oftheprobabilitymass.Based NXˆ=1D∈DXNˆ:Dℓ=d
23 23 23
ontheobservationsinthetwotrees,thedegreedistributionsof The sum in (1) converges whenever at least one
the remainingcollisionsare calculated as describedin section idle or single slot is observed, since in that case
IV. M(D,Nˆ,P)→0forNˆ →∞.Forthatreason,
D∈DNˆ:Dℓ=d
The degree optimization phase is illustrated in Fig. 3. This Pit is in practice enough to sum until a sufficiently high Nˆ,
phase continues on the trees resulting from the estimation such that convergence is achieved. The overall leaf degree
phase. It is assumed for this example that splitting node 10 probability distribution is now found as:
will maximize the scalar product of Ω1 and λ, which means
M
this node is scheduled to be split first. Following this, a split
Ω(d)= Pr(D =d) for d>0. (2)
of node 12 is assumed to maximize the scalar product, and ℓ
Xℓ=1
finallynode5.Hence,thedeterminedsplitorderis 10,12and
B. Split Order Selection
5,whichis broadcasttothe users,whoflip coinsandtransmit
accordingly. The splits performed in the degree optimization The degree probability distribution of the individual leaves
phase are marked with gray in Fig. 3. After each split, SIC is and the resulting Ω(d) constitute the starting point of the
attempted. degree optimization phase. For this phase we are interested
in the optimal split order, across all K trees, according to 0.8
N =32
a reward function λ, for the remainder of the transmission. N =64
We will add a superscript i to the derived distributions, 0.78 NN ==122586
which denotes the number of performed splits in the degree
0.76
optimization phase. Moreover, we concatenate the random
variables from the individual trees in a single vector, D0, ut
p 0.74
representing all leaves. Hence, D0 = {D1,...,Dk,...,DK}, ugh
where Dk,ℓ is the ℓ-th leaf of the k-th tree. hro 0.72
The result of a split is that one leaf is replaced by two T
children. Hence, if e.g. the ℓ-th leaf is chosen for the first 0.7
split, then D0 will be replaced by two new random variables,
ℓ
denoted D1 and D1 . For the i-th split, we are interested 0.68
ℓ ℓ+1
in finding the leaf that maximizes the scalar product of the
0.66
resulting Ωi and λ. Hence, if si denotes the index of the leaf 0.08 0.1 0.12 0.14 α 0.16 0.18 0.2 0.22
forthei-thsplit,thenthefollowingoptimizationisperformed:
∞ Fig.4. ThroughputasafunctionofαforK=3.
maximize Ωi(d)λ(d), (3)
si Xd=0
For the coded splitting tree protocol, the reward function,
where the dependency on s lies in Ωi(d), since the choice λ, is chosen to be:
i
of split determines how Ωi−1(d) maps into Ωi(d). As in (2),
0.5 for d=2,3,
we find the overall leaf degree distribution by summing the λ(d)= (7)
(cid:26) 0 elsewhere.
contributions from the individual leaves:
The motivation behind this choice is the dominance of
M
degreestwo and three in well performingdegree distributions
Ωi(d)= Pr(Di =d) for d>0. (4)
ℓ for LT codes [8,9]. Experiments show that the performance
Xℓ=1
of the proposed scheme is limited by the amount of leaves
Leaves not chosen for the split are unaffected. However, the with these degrees. Hence, maximizing the amount of leaves
indexof the leavesfollowingthe chosen leaf in the vectorDi with degreestwo and three seems reasonable.Note that using
isincreasedbyone,sinceasplitcausesoneleaftobereplaced such a reward function will not provide trees containing only
by two new leaves. Hence, leaves with degrees two and three, due to the randomness
of the scheme. For this reason, leaves with degree one will
Pr(Dℓi =d)=Pr(Dℓi−1 =d) for ℓ<si, occur, as is necessary in order to initiate SIC (i.e., iterative
Pr(Di =d)=Pr(Di−1 =d) for ℓ>s +1. (5) BP erasure-decoding). Further investigation in the choice of
ℓ ℓ−1 i
reward function is subject to future work.
Conditioned on the degree of the chosen leaf, Dsi−i1 = dp, Thelengthofthesplitorder,appliedinthedegreeoptimiza-
the new leaves will both follow the binomial distribution, tion phase, is determined by the point, where an increase in
denoted B, with parameters dp and 0.5. The probability the scalar product is no longer possible. From this point, the
distribution of the degree of the chosen leaf is found using leaf with the highest expected degree is chosen for the next
(1). We thus have: split, until decoding succeeds.
Fig.4andFig.5showtheachievedthroughputasafunction
Pr(Dsii =d)=Pr(Dsii+1 =d) ofαforK ={3,4}andN ={32,64,128,256}.Thenumber
∞
ofnecessaryfeedbackmessagesisplottedin Fig.6andFig.7
= Pr(Di−1 =d )B(d,d ,0.5) , (6)
si p p forthe same choicesof parameters.It is seen that the optimal
dXp=d(cid:0) (cid:1) choice of α dependson K but not on N. Moreoverit is seen
that K =3 providesthe better results, which shows that only
which concludes the analysis.
very few trees are necessary in order to have a SIC potential.
A comparison with BTS is shown in Fig. 8 for K =3 and
V. NUMERICALRESULTS
N = {32,64,128,256}, where optimized α has been used
The proposed scheme has been evaluated and compared to forthe codedsplittingtree scheme,accordingto the resultsin
the BTS algorithm. BTS is applied on a single tree only and Fig. 4. It is seen that the proposed scheme outperforms BTS
at one level of the tree at a time. Hence, initially the root with respect to throughputat all N and that the improvement
is split, resulting in a maximum of two remaining collisions. increases with N. This is in line with the efficiency increase
A feedbackmessage reports which leaves are in collision and as a function of the message length seen for erasure codes
mustbefurthersplit.Thisprocesscontinuesuntilallcollisions such as LT codes. Fig. 8 also shows that the performance
have been resolved. This constitutes the reference scheme. improvementcomes at the price of increased feedback.
0.8 0.5
N =32 N =32
N =64 N =64
0.78 NN ==122586 FN 0.45 NN ==122586
0.76 k
c
ba 0.4
0.74 d
hput 0.72 offee 0.35
ug nt
Thro 0.7 mou 0.3
0.68 a
ve 0.25
ti
0.66 a
el
R 0.2
0.64
0.62
0.08 0.1 0.12 0.14 α 0.16 0.18 0.2 0.22 0.08 0.1 0.12 0.14 α 0.16 0.18 0.2 0.22
Fig.5. ThroughputasafunctionofαforK=4. Fig.7. Therelative amountoffeedback asafunction ofαforK=4.
0.8 0.4
0.45
N =32
N =64
FN 0.4 NN ==122586 FckN
eedback 0.35 put offeedba
off 0.3 ugh0.7 0.2 unt
mount 0.25 Thro veamo
a ti
tive 0.2 CSTP throughput Rela
Rela 0.15 CBSTTSP t hfreoeudgbhapcukt
BTS feedback
0.6 0
00 5500 110000 115500 220000 225500 330000
0.1 N
0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22
α
Fig. 8. Comparison of BTS and coded splitting tree protocol (CSTP) for
Fig.6. Therelative amountoffeedback asafunctionofαforK=3. K=3andoptimized α.
VI. CONCLUSIONS [3] Y.YuandG.B.Giannakis,“Sicta:a0.693contentiontreealgorithmusing
successive interference cancellation,” in INFOCOM 2005. 24th Annual
A novel approach to multiple access control called coded
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