Table Of Content

User Activity Detection via Group Testing and Coded Computation Matthias Frey, Igor Bjelakovic´ and Sławomir Stanćzak Technische Universita¨t Berlin Abstract—Inspiredbygroup testingalgorithms and thecoded problemoffindingasuitablecomputationalchannelcode,and computationparadigm,weproposeandanalyzeanovelmultiple 2) the problem of using this code to detect the set of active access scheme for detecting active users in large-scale networks. 7 nodesin a mannerthat is efficientin terms of communication Theschemeconsistsof asimplerandomized detectionalgorithm 1 overhead and computational resources used. thatusescomputationcodingasintermediatestepsforcomputing 0 logical disjunction functions over the multiple access channel Computation over multiple-access channels, to the best of 2 (MAC). First we show that given an efficient MAC code for ourknowledge,wasfirstproposedforaspecialcasein[7]and n disjunction computation the algorithm requires O(klog(Nk)) later formalized for a more general setting in [10]. A special a decision steps for detecting k active users out of N +k users. casethathasreceivedmuchattentionisthecomputationofthe J Subsequently we present a simple suboptimal code for a class modulosumofsourcesovermultiple-accesschannelsbecause 3 of MACs with arbitrarily varying sub-gaussian noise that uni- 2 formly requires O(klog(N)max{logk,loglogN}) channel uses it yields a more efficient approach to network coding which for solving the activity detection problem. This shows that even is called Physical Layer Network Coding [11]. For the binary ] in the presence of noise an efficient detection of active users is case,thiscanbeviewedascomputationcodingforthelogical T possible. Our approach reveals that the true crux of the matter Exclusive Or function. We, in contrast, need a computational I lies in constructing efficient codes for computing disjunctions s. over a MAC. channelcode which computeslogical disjunctions(or equiva- lently conjunctions). c [ I. INTRODUCTION The problem of determining the set of active nodes from A. Motivation such disjunctions is remarkably similar to Combinatorial 1 v Novel Machine Type Communication (MTC) applications, Group Testing problems which were first proposed as a way 4 which often involve a massive deployment of MTC devices, to conductsyphilis tests for draftees of the U.S. armed forces 5 during World War II more efficiently by pooling their blood pose some fundamental challenges. One of these is the 3 samples in clever ways [3]. Although this idea was not development of new uncoordinated random access schemes 6 implemented in the end, the approach remains relevant in 0 that incur a small computation and communication overhead. biology and bioinformatics e.g. for drug screening or DNA . An inherent feature of any random access scheme is the 1 sequencing [6], [5]. Existing deterministic and randomized 0 ability to identify the set of active devices that compete for approaches in Combinatorial Group Testing [3], however, 7 access to the channel. This problem is referred to as the user 1 activity detection problem. Unfortunately, conventional pilot- usuallyaimatconstructingpoolingstrategiesthatworkforall : distributionsofagivennumberofactivedevices(or,usingthe v basedapproachestothisproblementailunacceptablesignaling terminologyfromthebiologyapplications,defectivesamples) i overhead in envisioned massive MTC scenarios, because the X asopposedtoconstructingarandomizedstrategywhichworks overheadofsuchschemesscaleslinearlywiththetotalnumber r well with high probability for an unknown, but fixed set of of deployed devices. On the positive side, in most MTC a active devices. There is an approach to pooling samples for applications, the set of active devices at any given point in DNA tests called Random Incidence Designs [5] (a variation timewillbesmallerbyordersofmagnitudethanthesetofall ofwhichappearedfirstinapurelymathematicalcontextin[4]) deployeddevices.Thiscanbeexploitedtoreducetheoverhead thatissimilartowhatwepropose;however,theauthorsof[5] significantly so that it scales logarithmically with the total donotgiveacomparableanalysisaimedatdetectinganumber number of deployed devices. of active devices (or in their language:positive clones) which B. Related Work is much smaller than the number of total devices. We point The problem of user activity detection in large-scale net- out that our results in Theorem 1 and Corollary 1 may also works has been already addressed in the context of compres- be of interest as schemes for Combinatorial Group Testing. sive sensing in [8], [9] where the authors show the existence A result similar to our Corollary 1 appearedin an indepen- of a non-adaptive optimal activity detection strategy. The dent work [13] using a slightly differentscheme; we do point approach in [8], [9] relies heavily on methods from high out, however,that our result is more general in the sense that dimensionalprobabilityandonmethodsforestimatingcertain we do not restrict the growth of the number of active devices partial sums affected by noise. depending on the number of total devices, while the result In contrast to this, we propose to decompose the user in [13], on the other hand, offers a slightly better constant activity detection problem into two sub-problems: 1) The ((log2)−2 2.1 where we have e 2.7). ≈ ≈ C. Outline An (n,ℓ,δ)-disjunction code for consists of encoding C functions for r 1,...,R In the following section, we state the underlying problem, ∈{ } while Section III introduces the frameworkfor activity detec- E : true,false ℓ n tion. In Section IV, we provide an analysis of the proposed r { } →Xr and a decoding function scheme. Section V contains simulation results evaluating the impact of true common randomness versus using pseudo- D : n true,false ℓ, random numbers for the execution of the scheme. In Sec- Y →{ } tion VI, we give a simple suboptimalcode enabling detection such that of active users simultaneously and uniformly for a class of P D(Y ,...,Y )=dis(x ,...,x ) 1 n 1 R MACs with additive sub-gaussian noise satisfying an upper 6 r(X ,...,X )=E (x ) δ boundonthesub-gaussiannormofthenoiserandomvariables. (cid:0) ∀ r,1 r,n r r(cid:12)(cid:12) ≤ for every x ,...,x true,false ℓ, where (cid:1) 1 R D. Notation ∈{ } x(1) x(1) x(1) x(1) We use e to denote Euler’s number and log to denote the 1 R 1 ∨···∨ R . . . logarithm with base e. dis: .. ,..., ..   ..  7→ x(ℓ) x(ℓ) x(ℓ) x(ℓ) II. PROBLEMSTATEMENTANDOBJECTIVES  1   R   1 ∨···∨ R        calculates logical disjunctions component-wise. Weconsideramultipleaccesschannelwithasinglereceiver Throughout Sections III and IV, we make the following and N + k transmitters (also called nodes). The nodes are assumption: indexed by numbers 1,...,k + N in some arbitrary order, which is known both to the receiver and to the nodes. Of Assumption 1. There exists a family of (n,ℓ,δ)-disjunction these N +k nodes, N are considered to be inactive, while codes for arbitrarily small δ >0 and sufficiently large ℓ N the remaining k are active. Each node knows only its own such that n/ℓ is bounded. ∈ state which is either active or inactive. The notion of being In Section VI, we present crude codes for a class of active could mean that a node wishes to make a subsequent additivenoisechannelswithunboundedn/ℓthatdo,however, transmission;but for the sake of this problem,being active or achieve practically relevant bounds on the number of channel inactive is just a labeling dividing the nodes into two classes. uses needed to guarantee low overall errors for our scheme. The problem of interest in this paper is the following: Classifying channels for which good disjunction codes in the Problem1. Givenatotalnumbern Nofchanneluses,find sense of Assumption 1 exist and finding such codes remain ∈ a scheme that enables the nodes to encode the information open problems for future research. whether or not they are active as n channel inputs in such a way that the receiver can determine with a sufficiently small III. SCHEME probability of error which of the k+N nodes are active. The scheme consists of ℓ slots in each of which one disjunction needs to be computed over the channel. We use an Our objective is therefore to design a multiple access (n,ℓ,δ)-disjunction code as in Assumption 1 to convey the communication scheme that enables the receiver to recover resulting ℓ disjunctions. the indices of the active nodes from n uses of the multiple accesschannel,whilemeetingagivenrequirementontheerror Remark 1. There will be no dependence between different probability(fortheexactdefinitionoftheerrorprobability,we slots, so a block coding scheme as in Definition 1 that calls refer the reader to Theorem 1). At this point, it is important for simultaneous encoding of the input bits and simultaneous toemphasizethatourgoalisnotto designan optimalscheme decoding of the output bits can be used. for a given channel in the sense of minimizing the number Thereceivermaintainsa setofpotentiallyactivenodes.All of channel uses for some given error probability among all active nodes are contained in this set at all times. By we i possible multiple access schemes. In fact, in this paper, we N denotethesetofpotentiallyactivenodesaftersloti.Wedefine introduceacomputationcodeasalayerofabstractionbetween the number of potentially but not actually active nodes after our proposed detection scheme and the channel. slot i as N := k. The scheme works as follows: i i |N |− Definition 1. Let be a memoryless multiple-access channel 1) At the beginning, all N + k nodes are defined to be C defined by input alphabets ,..., , output alphabet potentially active, i.e. consists of all N +k nodes. 1 R 0 X X Y N (the alphabets need not be discrete) and stochastic kernels We start the scheme with slot i:=1. (W ) that define the transition from channel inputs 2) At the beginning of slot i, a set of chosen nodes is t t∈{1,...} (X ,...,X ) toa channeloutputY determined randomly. We assume common randomness 1,t R,t 1 R t ∈X ×···×X ∈Y at time instant t (in the case of discrete input and output atthe receiverandall the nodes,i.e. everynodeandthe alphabets,thestochastickernelscanbethoughtofasmatrices receiver know which nodes are chosen. Each node is of conditional probabilities). chosen independently with probability p. We often use q := 1 p to denote the probability that a given node Corollary1. Thereexists p (0,1)suchthatP(N >0) ε ℓ − ∈ ≤ is not chosen. By , we denote the set of nodes which provided that i U are chosen but not active. 1 3) Each node uses the following rule to determine the ℓ e(k+1) logN +log . (2) ≥ ε truth value that it contributes to the logical disjunction (cid:18) (cid:19) conveyed over the channel in slot i: Thus, for the cost of a slightly worse bound with regard to a) It sends the message true if it is both active and dependency on k, we can determine the exact set of active chosen. nodes without having to employ another scheme to make a b) It sends the message false otherwise. final determination. In order to prove Theorem 1, we will bound the expecta- 4) Upondecodingthelogicaldisjunctionofthetransmitted messages, the receiver applies the following rule: tion of Nℓ in the following lemmas and then use Markov’s inequality to bound the probability. a) If the receiver decodes true, it can only conclude that at least one of the chosen nodes is active. It Lemma 1. EN Ckε if ℓ ≤ discards this information and sets := . Ni Ni−1 1 N b) If the receiver decodesfalse, it concludesthat none ℓ log . (3) ≥ −log(1 pqk) Ckε of the chosen nodes are active and removes all (cid:18) − (cid:19)(cid:18) (cid:19) chosen nodes from the set of potentially active A similar result already appeared in the context of pooling nodes by setting i := i−1 i. DNA tests in [5, Corollary 3.4], but for the sake of complete- N N \U 5) If i = ℓ, the scheme terminates with the set as ness, we give here an easier and more self-contained proof. ℓ N output. Otherwise, we increment i by 1 and continue Proof: We define random variables (A ) by i i∈{1,...,ℓ} from step 2. 0, only inactive nodes are chosen in slot i It is shown in Section IV that for any ε > 0, C > 0 and A := i sufficiently large ℓ, the cardinality of the output is less (1, otherwise. ℓ N than k(C+1) with probability1 ε. Also, the outputclearly − Note that E(A )=1 qk and E(1 A )=qk. contains all active nodes. i − − i Denote with B the number of potentially but not actually i Remark 2. Since ℓ has a size linear in k, narrowing active nodes which are not chosen in slot i. Clearly, for any N down this set to the exact set of active nodes can be done n and i under the condition N = n, B is binomially i−1 i in time linear in k; one possibility would be to count the distributed with n trials and success probability q. Clearly, number of active nodes in certain subsets of or to at ℓ least obtain parity information aboutthese numbNers and then Ni =AiNi−1+(1 Ai)Bi. − solve a linear equation system to determine which nodes We observe that by the law of total expectation for all i are active. An additional coding technique would have to be used that can count or obtain this parity information in an EB = P(N =n)E(B N =n) i i−1 i i−1 efficient way, for which the schemes employed in Physical | n X Layer Network Coding [11] may be promising candidates. =q nP(N =n)=qEN i−1 i−1 This issue is however not addressed in this paper. n X Remark 3. A major advantage of this approach is that the and further that because nodes are chosen or not chosen algorithmusedatthenodesandthereceiverissimpleinterms independently in different slots, for all i, A is independent i of both conceptual complexity and computational resources of Ni′ for i′ < i and because different nodes are chosen or used. The amount of memory needed in total as well as the not chosen independently even within the same slot, A is i computational effort for the computation during each slot is independent of Bi′ for i′ i (although, of course, for other ≤ constantatthetransmittersandlinearinN+katthereceiver. values of i′ there may be dependences). We can now use these observations to calculate IV. ANALYSIS In this section, we analyze the performance of the scheme ENi =EAiENi−1+E(1 Ai)EBi − described in Section III. We use the definitions introduced =(1 qk)EN +qk qEN i−1 i−1 before. − · =EN (1 pqk). i−1 − Theorem 1. There exists p (0,1) such that P(N Ck) ∈ ℓ ≥ ≤ From this recursion and the definition N =N, we conclude ε provided that 0 N 1 1 ENi =N 1 pqk i. ℓ e(k+1) log +log +log . (1) − ≥ k ε C (cid:18) (cid:19) In particular, for i=ℓ, we ha(cid:0)ve (cid:1) By taking C =k−1, we obtain the following corollary: EN =N 1 pqk ℓ Ckε. ℓ − ≤ (cid:0) (cid:1) Takingthelogarithmonbothsidesandsolvingforℓyields(3), cy 100 n which completets the proof. ue (104,20) Nextweshowhowtochoosepsoastominimizethefactor req (105,20) ( log(1 pqk))−1 in (3), which is equivalentto maximizing f p−qk. The−derivative has zeros at p = 1 and p = (k +1)−1. ror 10−1 (104,30) r Since pqk =0 for p=0 and p=1, the maximummust be at E / p=(k+1)−1. ε nd 10−2 Lemma 2. If p=(k+1)−1, then u o b 1 y −log(1 pqk) <e(k+1). bilit 10−3 − a b o Proof: Using the fact that log(1 x) x for x 1, r p − − ≥ ≤ and substituting 1 p for q as well as (k+1)−1 for p, we r o obtain − rr 10−4 E 400 600 800 1,000 1,200 1,400 1,600 1 1 k+1 k Number of slots ℓ =(k+1) −log(1 pqk) ≤ pqk k − (cid:18) (cid:19) Fig.1. Plotsofobservederrorfrequencies inthesimulations asthicksolid 1 k linesandtheoretical errorboundsaccordingtoCorollary1inthinnerdashed =(k+1) 1+ <(k+1)e, linesforvarious pairs (N,k). k (cid:18) (cid:19) whereinthelaststepwehaveusedthefactthate>(1+1/x)x for all x>0. VI. CODING DISJUNCTIONS OVERCHANNELS WITH Proof of Theorem 1: By Lemma 2, (1) implies (3) if we SUB-GAUSSIAN NOISE choosep=(k+1)−1.So,byLemma1,wehaveENℓ Ckε. Inthissection,weconstructacodefortheclassofchannels ≤ Hence, by Markov’s inequality we obtain witharbitrarilyvaryingsub-gaussiannoise(asdefinedbelow). This code does not satisfy Assumption 1, but is close to it as EN Ckε P(N Ck) ℓ =ε. n/ℓ growsonly logarithmicallywith 1/δ. In conjunctionwith ℓ ≥ ≤ Ck ≤ Ck our scheme from Section III, it leads to good bounds on the total number of channel uses necessary to solve Problem 1. Definition 2. A random variable X is called sub-gaussian if V. SIMULATIONS for some α R, we have ∈ Simulations of the proposed scheme were conducted for n 1 (E X n)1/n α√n. (4) three different value pairs of (N,k). For each value pair and ∀ ≥ | | ≤ each of 1.2 105 simulation runs, the scheme was carried out If X is sub-gaussian, the infimum over all α that satisfy (4) · untilonlyactuallyactivenodeswerepotentiallyactiveandthe exists and is called the sub-gaussian norm of X and denoted resulting values of ℓ were recorded. In Figure 1, we plotted ℓ by X . k kΨ2 versus the observed frequencies of simulation runs in which Definition 3. A channelas in Definition1 is called a channel theexactsetofactivenodeswasnotdeterminedatthereceiver witharbitrarilyvaryingsub-gaussiannoiseofnormatmostK in fewer than ℓ slots. For comparison, we also plotted the and power constraint P if corresponding theoretical error bound given by Corollary 1. For the generation of random numbers called for by the = = = √P,√P , =R 1 R scheme, we used the standard Mersenne Twister pseudo- X ··· X − Y h i random number generator shipped with Python 3.5 [2] which and the transition kernels are such that there exists an was seeded before each simulation run with the hardware independent sequence (Z ) of sub-gaussian random t t∈{1,...} random number generator provided by Linux 4.4 [1]. variables (called the noise) with mean 0 and sub-gaussian From the simulation results, we conclude that the bound norm bounded by K (in particular, this covers independent providedforℓbyCorollary1isrelativelytightforsmallerrors and identically distributed Gaussian noise) such that for any and that it can be achieved using a standard pseudo-random time instant t, number generator and thus the assumption that receiver and R nodes can observe a common source of randomness might Yt = Xn,t+Zt. (5) be relaxed in practice e.g. by having the receiver broadcast nX=1 a random seed to all nodes which then use identical and Transmitters and receiver are not assumed to know anything identically seeded random number generatorsto carry out the about the noise random variables except for their indepen- scheme. denceandthe boundK ontheirsub-gaussiannorm.Thusthe employedcodingschemeshouldworksimultaneouslyandwith Proof: We first choose ℓ to satisfy (2), and thus, by uniform error bound for any such selection of noise random Corollary 1, if we conduct the scheme with an error free variables. channel code, we can bound the error probability by ε. Next, we choose m according to (6) with δ = ε/ℓ and use the Theorem 2. For every K >0,P >0,δ >0,ℓ N and ∈ channelcode given by Theorem2 to conveyour disjunctions. K2 log1 +1 Then, by the union bound, the overall error probability (i.e. m δ , (6) ≥ P · c the probability of the event that one of our ℓ disjunctions is where c > 0 is an absolute constant, there is a code decodedincorrectlyor the applicationof the scheme does not whichis an (mℓ,ℓ,δ)-disjunctioncodeforevery channelwith result in exact recovery of the set of active nodes) does not arbitrarilyvaryingsub-gaussiannoiseofnormatmostK and exceed ε+ℓ ε/ℓ = 2ε. We can do this as long as we are · power constraint P. allowed an overall number of channel uses K2 1 ℓ Proof: For each r, we define a repetition encoder n ℓm= ℓ log +1 , ≥ P · c ε E :x =(x ,...,x ) (X ) (cid:18) (cid:19) r r r,1 r,ℓ r,t t∈{1,...,ℓm} 7→ and substituting (2) we get requirement (8). by insisting that for each slot i 1,...,ℓ ∈{ } ACKNOWLEDGEMENT √P, x =true X = =X = r,i We would like to thank Peter Jung and Richard Ku¨ng for a r,(i−1)m+1 r,im ··· (0, xr,i =false. pleasantdiscussionabouttheirwork[8],[9]aswellasSteffen Limmerforahelpfuldiscussiononpossibleconnectionsofour As an intermediate step, the decoder computes for each i 1,...,ℓ a value Y˜ as ∈ work to discrete valued compressive sensing. This work was i { } supportedbythe GermanMinistryofResearchandEducation im 1 (BMBF)undergrant16KIS0605andbytheGermanResearch Y˜ := Y . i m t Foundation (DFG) under grant STA 864/8-1. t=(i−X1)m+1 We denote the total averaged channel noise in slot i by REFERENCES [1] LinuxProgrammer’sManual:random,urandom-kernel randomnum- im Z˜ = Zt bersourcedevices. i m [2] The Python Standard Library: random — Generate pseudo-random t=(i−X1)m+1 numbers. https://docs.python.org/3.5/library/random.html. [3] Ding-Zhu Du and F. K. Hwang. Combinatorial Group Testing and and note that if the messages x are false for all r in slot r,i Its Applications, volume 12 of Series on Applied Mathematics. World i, then Y˜ = Z˜ , whereas if for at least one r in slot i the Scientific, 2ndedition, Dec1999. i i message x is true, then Y˜ √P +Z˜ . [4] P.A. Erdo˝s and A. Reńyi. On two problems of information theory. r,i i ≥ i Magyar Tud.Akad.Mat.Kutato´ Int.Ko˝zl.,8:229–243, 1963. So define the decoderD :(Y ,...,Y ) (y ,...,y ) by 1 ℓm 1 ℓ [5] F.K.HwangandY.C.Liu. Ageneralapproachtocomputetheproba- 7→ bilities ofunresolved clones inrandom poolingdesigns. Probability in y := true, Y˜i > 21√P theEngineeringandInformationalSciences,18(2):161–183,Apr2004. i (false, otherwise. [6] R. M. Kainkaryam and P. J. Woolf. Pooling in high-throughput drug screening. CurrOpinDrugDiscovDevel.,12(3):339–350, May2009. [7] J. Ko¨rner and K. Marton. How to encode the modulo-two sum of Clearly,yi willbecorrectlydecodedaslongas Z˜i <√P/2, binary sources (corresp.). IEEE Transactions on Information Theory, and the Hoeffding-type inequality for sequence(cid:12)s o(cid:12)f indepen- 25(2):219–221, Mar1979. (cid:12) (cid:12) [8] R. Ku¨ng and P. Jung. Robust nonnegative sparse recovery and 0/1- dent centered sub-gaussian random variables [1(cid:12)2, P(cid:12)roposition bernoulli measurements. In 2016 IEEE Information Theory Workshop 5.10] tells us that (independently of i) (ITW),pages 260–264,Sept2016. [9] R. Ku¨ng and P. Jung. Robust nonnegative sparse recovery and the P Z˜i ≥ 21√P ≤e1−c4′KPm2 , (7) [10] nBu.llNspaazceer apnrodpMert.yGoafst0p/a1r.mCeoamsupreumtaetinotns.oavreXrimv:u1l6t0ip3l.e0-7a9cc9e7s,s2c0h1a6n.nels. (cid:18)(cid:12) (cid:12) (cid:19) IEEETransactionsonInformationTheory,53(10):3498–3516,Oct2007. where c′ > 0 is a(cid:12)n a(cid:12)bsolute constant. By insisting that the [11] B. Nazer and M. Gastpar. Reliable physical layer network coding. (cid:12) (cid:12) righthandsidein(7)belessthanorequalδ,settingc:=c′/4 Proceedings oftheIEEE,99(3):438–460, Mar2011. [12] Roman Vershynin. Introduction to the non-asymptotic analysis of and solving for m, we obtain (6). randommatrices. arXiv:1011.3027, 2010. [13] Wenyi Zhang and Lingyan Huang. On or many-access channels. Corollary 2. For every K >0,P >0,ε>0 and arXiv:1701.03620, 2017. K2 1 1 n e(k+1) logN +log ≥ P · c ε · (cid:18) (cid:19) (8) N 1 2+log(k+1)+loglog +log , ε ε (cid:18) (cid:19) where c>0 isanabsoluteconstant,Problem1canbesolved over a channelwith arbitrarily varying sub-gaussiannoise of normatmostK andpowerconstraintP witherrorbound2ε.