Beyond Cut-Set Bounds - The Approximate Capacity of D2D Networks

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1 Beyond Cut-Set Bound - The Approximate Capacity of DD Network Avik Sengupta Hume Center for National Security and Technology Department of Electrical and Computer Engineering Virginia Tech, Blackburg, VA 4060, USA avikg@vt.edu Ravi Tandon Dicovery Analytic Center Department of Computer Science Virginia Tech, Blackburg, VA 4060, USA tandonr@vt.edu Abtract Device-to-Device DD communication i emerging a a viable olution for alleviating the evere capacity crunch in content-centric wirele network. DD encourage backhaulfree communication directly between device with imilar content requirement grouped into cluter. In thi work, a elf-utaining DD network i conidered, where a et of commonly requeted file are completely tored within the collective device memorie in a cluter and file requet from device are erviced by local inter-device multicat tranmiion. For uch a network, new information theoretic convere reult are developed, in the form of a lower bound on the minimum DD multicat rate a a function of the torage per device. The propoed convere i then ued to characterize the approximate tradeoff between the device torage and DD multicat rate to within a contant multiplicative gap of 8. I. INTRODUCTION In recent time, the dynamic of traffic over wirele network ha undergone a paradigm hift to become increaingly content centric with multimedia content ditribution holding precedence. Thi neceitate careful deign of content torage and over-the-air ditribution cheme within the network to efficently ue carce pectral reource to handle heavy traffic load a evidenced by a wealth of recent reearch [] []. Device-to-device DD communication ytem have been propoed a one of the olution that could help in alleviating the evere capacity crunch in wirele network [] [6]. DD ue low-power direct communication within device grouped into cluter to facilitate large cale frequency reue thereby freeing up preciou pectral reource. We conider a DD network with device, and N file denoted by F,..., F N, each of ize B unit, where the torage of each device i M B unit. The bae-tation populate the device with function of file, depending on the number of uer and available device torage. The DD ytem i elf-utaining i.e., all N file mut be recoverable from the collective torage memory of the device, which i equivalent to the condition M N. Once the device reveal their file requet, all the device within the cluter tranmit multicat meage over the hared link to ervice the requet. Figure illutrate the ytem model. The torage/multicat mechanim mut be deigned uch that any device mut be able to decode it requeted file uing it local torage and the multicat tranmiion from other device. Recently, Maddah-Ali and Nieen have invetigated a related caching problem [] [4] for which it ha been hown that, by jointly deigning the content torage and delivery, order-wie improvement in tranmiion rate can be achieved. The main difference between the caching problem tudied in [,] and the DD problem i the ditributed nature of multicat tranmiion. The caching problem allow for centralized delivery, in which the multicat can be any arbitrary function of all N file. In the DD problem however, the outgoing multicat from each device can only depend on the content tored at that device. The focu of thi work i on the fundamental tradeoff between the device torage and the total multicat tranmiion rate referred to a the M, R tradeoff for the DD problem. In [5,6], Ji et.al. preent new torage/delivery mechanim for the DD problem. The reult in [6] how that for a elf-utaining DD network, when the device can ue local inter-device multicat tranmiion to atify the requet of other within the cluter, orderwie rate improvement imilar to [,] can be achieved. The author preent an achievable cheme a well a a cut-et baed argument to obtain upper and lower bound on the M, R tradeoff for the DD problem. In thi work, we develop a new convere lower bound on the M, R tradeoff for the DD problem. The new lower bound addree contraint that are unique to the DD network, i.e., the local dependence of device multicat on it tored content. We how that the new convere i tighter than the exiting one preented in [6, Theorem ] which are primarily baed on the relaxation of the DD problem to the centralized caching problem for all value of problem parameter. Our reult how that the achievable cheme preented in [6] i indeed optimal for the lowet allowable value of torage memory for a elf-utaining DD network i.e., at M=N/. Uing our new convere reult, together with the achievable cheme of [6], we characterize the M, R tradeoff for the DD network to within a maximum contant multiplicative gap of 8 for all value of problem parameter. The paper i organized a follow: Section II outline the ytem model. Section III detail the analyi of the new lower bound on the torage v. rate tradeoff for the DD network, while Section IV illutrate the intuition behind the new bound through an example. Finally, Section V conclude the paper.

2 For the M, R DD cheme, the probability of error i defined a: P e max max P ˆF Rk F Rk. 5 R,...,R [N] k [] Definition 4 Storage v. Rate Tradeoff. The torage-rate pair M, R i achievable if, for any ϵ > 0, there exit an M, R DD cheme for which P e ϵ. The torage v. rate tradeoff i defined a: RM inf {R : M, R i achievable}. 6 Fig.. Sytem Model for device-to-device DD Network. Notation: Let Y i be a random variable. Y [a:b], where a < b denote the et of random variable {Y i : i = a, a +,..., b, b}. Alo, Y [a,b] denote the et {Y i : i = a, b}. Y [n] denote a et of any n arbitrary random variable Y i uch that Y [n] = n. N + denote the et of natural number; the function x + = max0, x; x denote the ceil function; x denote the floor function. II. SYSTEM MODEL The DD network ha uer, and a total of N file, denoted by F,.., F N, where each file i aumed to be of ize B, for ome B N +. Formally, the file F n are i.i.d and ditributed a : F n Unif{,,..., B }. We next define the key component of the DD problem: Definition Storage. The content torage phae conit of torage function, which map the file F,..., F N into the device torage content Z k ϕ k F,..., F N, for each uer k {,,..., }. The maximum allowable ize of the content of each device torage, Z k, i MB unit. An additional contraint of elf-utainability i enforced i.e., M N, which enure all the file in the ytem are completely tored within the DD cluter. Definition Delivery. In the delivery phae each device k {,,..., } ue one of N encoding function to map it torage content, Z k, to a multicat tranmiion: XR k,...,r ψ R,...,R Zk k {,,..., }. over the hared link in repone to the requet R,..., R {,,..., N }. Each uch tranmiion ha rate RB/ unit. The total multicat rate. when all device tranmit, i RB unit. Definition File Decoding. Once the multicat tranmiion i received, N decoding function map the received ignal over the hared link XR,...,R,..., X R,...,R and the torage content Z k to the etimate ˆF Rk µ R,...,R,k XR,...,R,..., X R,...,R, Z k, 4 of the requeted file F Rk for uer k {,,..., }. III. MAIN RESULTS AND DISCUSSION We next preent our main reult which give a new information theoretic lower bound on the torage v. rate tradeoff for the DD problem. Theorem. In a DD ytem with device and N file, with each device having torage ize M, and M N, the optimal content delivery rate R M i lower bounded by: R M R LB M N M max {,...,}, l {,..., N } where µ = min N l l µ l,, l. N l + Proof. The proof of Theorem i given in Appendix A, 7 The expreion in Theorem ha two parameter -, which i related to the number of device and a well a a new parameter l. Compared to [6, Theorem ], the additional parameter l add further flexibility to the lower bound expreion and better model the file decoding through the interaction of the device torage content and the multicat tranmiion. Conidering the cut-et baed bound preented in [6, Theorem ], the firt term inide the max function i imply the expreion for the centralized caching problem preented in [, Theorem ]. Furthermore, etting l = N/ in Theorem yield { } R N M M max {,...,} N, 8 which i greater than the cut-et expreion in the firt term inide the max function owing to the factor in the denominator. The econd term inide the max function can be obtained by etting =, l = N in Theorem. Thu, the propoed bound i generally tighter than the bound in [6, Theorem ]. It i to be noted that the bound in [6] i tight only for large value of device torage ize M. A hown in the equel, the new bound in Theorem i tighter for maller value of M and yield the exiting bound a a pecial cae for large value of M. Next, we preent our econd main reult, which etablihe the optimal torage v. rate tradeoff for the DD network to within a contant multiplicative factor. In [6], the author propoe a DD content ditribution cheme which achieve a rate given by [6, Theorem ]. Uing the reult in 7, we how a contant multiplicative gap exit between the new lower bound and the achievable rate in [6, Theorem ]:

3 Theorem. Let R UB M be the achievable rate of the DD cheme given in [6, Theorem ] and let R LB M be the lower bound on the optimal rate given in 7. For any N + device, N N + file, and device torage in the range N M N, we Gap = R UBM R LB M = M = N N < M 6 < M. 9 8 M N Proof. The proof of Theorem i omitted for brevity. IV. INTUITION BEHIND PROOF OF THEOREM We next preent a detailed example to illutrate the intuition behind the propoed convere. Example. Conider a DD ytem with = device, each with a torage memory of M, and N = file A, B, C each of unit ize. For thi problem, the lower bound [6, Theorem ] are given by: R + M 0 R + M. In contrat, the propoed bound in Theorem give following tighter bound for different choice of and l: R + 6M 8, =, l = 8R + 6M 5, =, l = R + M, =, l =, 4 where 4 i the bound in. Next, we detail the derivation of the new lower bound and. Firt, we conider the requet vector R, R, R = A, B, C and R, R, R = B, C, A. The firt = device torage content Z, Z along with two tranmiion X ABC = {XABC }, X BCA = {XBCA } from the third device correponding to the two requet vector are able to decode all file. Here each tranmiion ha the rate of R. We upper bound the entropy of l = tranmiion with thi rate, while uing the other tranmiion decoding capability in conjunction with the device torage content to derive a tighter bound. We HZ [,], X ABC, X BCA 5 HZ [,] + H X ABC, X BCA Z [,] 6 M + H X ABC + H X BCA Z [,], X ABC 7 + H X BCA Z [,], X ABC, A, B 8 + H X BCA, Z Z [,], X ABC, A, B 9 + HZ Z [,], X ABC, A, B + H X BCA Z [:], X ABC, A, B 0 + HZ Z [,], A, B + H X BCA Z [:], X ABC, A, B, C + HZ Z [,], A, B, where follow from the fact that H X BCA Z [:], X ABC, A, B, C = 0 ince X BCA i a function of the file A, B, C which are preent inide the conditioning. Conidering the term HZ Z [,], A, B, we HZ Z [,], A, B = HZ [:] A, B HZ [,] A, B. Uing in, we + HZ [:] A, B HZ [,] A, B. 4 Now conidering all poible ubet of Z [:] having cardinality, in the RHS of 4, we have the following: + HZ [:] A, B HZ [,] A, B. 5 + HZ [:] A, B HZ [,] A, B. 6 Symmetrizing over the inequalitie in 4, 5 and 6, we + HZ [:] A, B i,j=,i j HZ [i,j] A, B. 7 Han Inequality: We next tate Han Inequality [7, Theorem 7.6.] on ubet of random variable, which we ue throughout the paper for deriving the new lower bound. Let {X, X,..., X n } denote a et of random variable. Further, let X [] {X, X,..., X n } denote a ubet of cardinality. Then given two ubet X [r], X [m] where r m, the tatement of Han Inequality can be tated a: n r X [r] : X [r] =r H X [r] r n m X [m] : X [m] =m H X [m] m, 8 where the um are over all poible ubet of ize r, m repectively. Next, conider Z [:] a the et of random variable and the ubet Z [i,j] Z [:] : i j, i, j =,, of cardinality. Applying Han Inequality uing the conditional entropy of the et, we H Z [:] A, B H Z [i,j] A, B 9 i= H Z [:] A, B i,j= i j i,j=,i j H Z [i,j] A, B. 0 Subtituting 0 into 4, we + HZ [:] A, B HZ [:] A, B + HZ [:] A, B + HZ [:], C A, B + HC A, B + HZ [:] A, B, C 4 + R + 6M 8. 5

4 Rate R New Lower Bound Theorem 0. Lower Bound in [6] Upper Bound in [6] Storage per Device M a Rate R New Lower Bound Theorem Lower Bound in [6] Upper Bound in [6] Storage per Device M Fig.. M v. R tradeoff for a DD cluter with a N = = and b N = = 0. b Next, we conider = device torage content, Z, and three requet vector R, R, R = A, B, C, R, R, R = B, C, A and R, R, R = C, A, B along with tranmiion XABC = {XABC, X ABC }, X BCA = {XBCA, X BCA }, X CAB = {XCAB, X CAB } which are capable of decoding all file. Each compoite tranmiion i of rate R. We upper bound the entropy of l = tranmiion with their rate. Thu we HZ, X ABC, X BCA, X CAB 6 HZ + H X ABC, X BCA, X CAB Z 7 M + H X ABC, X BCA Z + H X CAB Z, X ABC, X BCA 8 M + R + H X CAB, Z Z, X ABC, X BCA 9 + HZ Z, X ABC, X BCA, A, B + H X CAB Z [,], X ABC, X BCA, A, B, C 40 + HZ Z, A, B 4 + HZ [,] A, B HZ A, B, 4 where, 40 follow from the fact that H X CAB Z [,], X ABC, X BCA, A, B, C = 0 ince X CAB i a function of the file A, B, C. Conidering all poible ubet of Z [,] of cardinality and uing a imilar ymmetrization argument and Han Inequality, we get: + HZ [,] A, B HZ [,] A, B + HC A, B + HZ [,] A, B, C R + 6M Finally, conidering again, = device torage content, Z, and three requet vector R, R, R = A, B, C, R, R, R = B, C, A and R, R, R = C, A, B along with three tranmiion XABC = {XABC, X ABC }, X BCA = {XBCA, X BCA }, X CAB = {XCAB, X CAB } which are capable of decoding all file. Each tranmiion ha rate R /. We upper bound the entropy of l = tranmiion by their rate. We HZ, X ABC, X BCA, X CAB 47 HZ + H X ABC + H X BCA + H X CAB 48 M + R 49 R + M. 50 Finally, combining, and 4, give the propoed lower bound on the optimal tranmiion rate a hown in Figure a. The lat bound on the rate given in 50 i tight only in the cae when the tranmiion are almot independent of each other. The propoed convere i tight at the point M = N where we how that the achievable point propoed in [6, Theorem ] i optimal. For increaing N, we can ee from Figure b that the propoed convere ignificantly improve on the exiting bound from [6, Theorem ]. V. CONCLUSION In thi paper, we preented a new information theoretic lower bound for a elf-utaining DD network. We have leveraged Han Inequality to better model the interaction of hared content tored in the device memorie and file decoding capability of multicat tranmiion to derive new lower bound which are generally tighter than the exiting cut-et baed bound. Uing the new lower bound, we characterized the device torage v. tranmiion rate tradeoff of the DD network to within a maximum contant multiplicative factor of 8 for all poible value of problem parameter. APPENDIX A PROOF OF THEOREM Let there be a library of N N + file F [:N] each of unit ize without lo of generality, and N + device in the DD cluter, with torage content Z [:]. The elf utainability condition of the DD cluter, we have M N. Let be an integer uch that {,,..., }. Conider

5 the firt device torage content, Z [:], and a requet vector R, R,..., R, R +,..., R =,,...,, ϕ,..., ϕ i.e., the firt requet are unique file while the remaining can be for arbitrary file. To ervice thi requet conider the compoite tranmiion vector: X = X +,,...,,ϕ,...,ϕ X +,,...,,ϕ,...,ϕ. X +,,...,,ϕ,...,ϕ, 5 where X +,,...,,ϕ,...,ϕ = ψf [:], F [ ] denote the multicat tranmiion from the + th device to ervice the requet of the device. Note that the file in the et F [ ], denoting the requet of the other device, are conidered arbitrary. For achieving optimal tranmiion rate R M, each device in the DD cluter multicat with a rate of R M/. Thu the rate of each compoite tranmiion vector i given by R M. Next, we note that the compoite tranmiion vector X, compoed of ubmulticat tranmiion, along with the content of device memorie i capable of decoding file i.e., F [:]. Similarly, conider a requet vector R, R,..., R, R +,..., R = +, +,...,, ϕ,..., ϕ where the firt requet are again for unique file and ret are arbitrary. A compoite tranmiion vector X = X + +,+,...,,ϕ,...,ϕ X + +,+,...,,ϕ,...,ϕ. X + +,+,...,,ϕ,...,ϕ, 5 along with device torage content Z [:], can decode another file F [+:]. Conidering N/ compoite tranmiion vector X [: N/ ] and device torage content Z [:], the library of file F [:N] can be completely decoded. Thu, we N H Z [:], X [: N/ ] 5 H Z [:] + H X[: N/ ] Z [:] 54 M + H X[: N/ ] Z [:] 55 M + H X[:l] Z [:] + H X[l+: N/ ] Z [:], X [:l] M + l R M + H X[l+: N/ ] Z [:], X [:l], F [:l] M + l R M + H X[l+: N/ ], Z [+:] Z [:], X [:l], F [:l] 58 M + l R M + H Z [+:] Z [:], X [:l], F [:l] }{{} δ + H X[l+: N/ ] Z [:], X [:l], F [:lµ+l], 59 }{{} λ where 57 follow from the fact that each compoite tranmiion vector ha rate R M and that the device torage content, Z [:], along with the compoite tranmiion vector X[:l] are capable of decoding the file F [:l]. In 58, µ = min { } N l l, i the number of additional device memorie which, along with the tranmiion X [:l] can decode all N file. For =, we H X[: N/ ] Z [:] = 0, 60 and 55 yield the elf-utainability condition M N. Upper Bound on δ: We conider the factor δ, from 59: δ = H Z [+:] Z [:], X [:l], F [:l] 6 H Z [+:] Z [:], F [:l] 6 = H Z [:] F [:l] H Z[:] F [:l]. 6 Conidering all poible ubet of Z [:] having cardinality, i.e., all poible combination of device torage content in 5, and all poible combination of file in the et F [ ] for each tranmiion X [:l] in 57, we can obtain different inequalitie of the form of 6. Symmetrizing over all the inequalitie, we δ H Z [:] F [:l] i= H Z i [] F [:l], 64 where, Z i [] i the i-th ize- ubet of Z [:]. Next, conider Z [:] a a et of random variable and ubet Z i [] Z [:] i =,...,. Applying Han Inequality from 8, we H Z [:] F [:l] + µ i= H Z i [] F [:l] i= + µ H Z [:] F [:l] i= 65 H Z i [] F [:l]. Subtituting 66 into 64, we δ H Z [:] F [:l] + µ H Z [:] F [:l] = µ + µ H Z [:] F [:l] µ + µ H Z [:], F [l+:n] F [:l] = µ + µ H F [l+:n] F [:l] + H Z[:] F [:N] 70 }{{} =0 µ + µ N l+, 7 where 70 follow from the fact that the device torage content are function of all the N file in the library. Upper Bound on λ: We conider the factor λ, from 59. We have two cae: Cae : N l + lµ: We conider the cae that all N file

6 can be decoded with µ additional device torage content and tranmiion X [:l], within the conditioning in the factor λ in 59i.e., l + lµ N. We λ = H X[l+: N/ ] Z [:], X [:l], F [:N] H X[l+: N/ ] Z [:], F [:N] = 0, 7 which follow from the fact that the tranmiion are function of all N file. Cae : N > l + lµ: We conider the cae where µ = additional device torage content along with the tranmiion X [:l], cannot decode all N file. We λ = H X[l+: N/ ] Z [:], X [:l], F [:l] H X[l+: N/ ] Z [:] = 0, 7 which follow from the fact that M N i.e., all file are tored within the collective device memorie in the DD cluter and hence all tranmiion are function of all the device torage content. Thu combining 7 and 7 we λ = Subtituting 7 and 74 into 59, we l N M + R M + µ N l+ N M R M µ + µ N l l Optimizing over all parameter value of, l, µ, we R M R LB M N M max {,...,}, l {,..., N } µ l which complete the proof of Theorem. REFERENCES N l +, 76 [] M. A. Maddah-Ali and U. Nieen, Fundamental Limit of Caching, IEEE Tranaction on Information Theory, vol. 60, no. 5, pp , May 04. [] M. Maddah-Ali and U. Nieen, Decentralized Coded Caching Attain Order-Optimal Memory-Rate Tradeoff, IEEE/ACM Tranaction on Networking, April 04. [] U. Nieen and M. A. Maddah-Ali, Coded Caching with Nonuniform Demand, in IEEE Conference on Computer Communication Workhop INFOCOM WSHPS, April 04, pp. 6. [4] R. Pedarani, M. Maddah-Ali, and U. Nieen, Online Coded Caching, in IEEE International Conference on Communication ICC, June 04, pp [5] A. Sengupta, R. Tandon, and T. C. Clancy, Fundamental Limit of Caching with Secure Delivery, Wirele Phyical Layer Security Workhop - IEEE International Conference on Communication ICC, pp , June 04. [6], Decentralized Caching with Secure Delivery, IEEE International Sympoium on Information Theory ISIT, pp. 4 45, July 04. [7], Secure Caching with Non-Uniform Demand, IEEE Global Wirele Summit GWS, pp. 5, May 04. [8], Fundamental Limit of Caching with Secure Delivery, IEEE Tranaction on Information Forenic and Security, vol. 0, pp , Feb 05. [9] A. Sengupta, S. Amuru, R. Tandon, R. M. Buehrer, and T. C. Clancy, Learning Ditributed Caching Strategie in Small Cell Network, The Eleventh International Sympoium on Wirele Communication Sytem ISWCS, pp. 97 9, Aug 04. [0] M. Ji, G. Caire, and A. F. Molich, Optimal Throughput-Outage Tradeoff in Wirele One-Hop Caching Network, in IEEE International Sympoium on Information Theory ISIT, July 0, pp [] M. Ji, A. M. Tulino, J. Llorca, and G. Caire, Order Optimal Coded Caching-Aided Multicat under Zipf Demand Ditribution, in The Eleventh International Sympoium on Wirele Communication Sytem ISWCS, 04. [] N. Golrezaei,. Shanmugam, A. Dimaki, A. Molich, and G. Caire, FemtoCaching: Wirele Video Content Delivery through Ditributed Caching Helper, IEEE Tranaction on Information Theory, vol. 59, pp , Dec. 0. [] M. Ji, G. Caire, and A. F. Molich, Wirele Device-to-Device Caching Network: Baic Principle and Sytem Performance, arxiv : 05.56, May 0. [4] N. Golrezaei, A. G. Dimaki, and A. F. Molich, Wirele Device-to- Device Communication with Ditributed Caching, in IEEE International Sympoium on Information Theory Proceeding ISIT, July 0, pp [5] M. Ji, G. Caire, and A. F. Molich, Fundamental Limit of Ditributed Caching in DD Wirele Network, in IEEE Information Theory Workhop ITW 0, Sept 0, pp. 5. [6], Fundamental limit of caching in wirele DD network, arxiv:405.56, 04. [Online]. Available: 56 [7] T. M. Cover and J. A. Thoma, Element of Information Theory, nd Edition. Hoboken, NJ, USA: Wiley-Intercience, John Wiley and Son.