Rate-Memory Trade-off for the Two-User Broadcast Caching Network with Correlated Sources

Rate-Memory Trade-off for the Two-User Broadcast Cachng Network wth Correlated Sources Parsa Hassanzadeh, Antona M. Tulno, Jame Llorca, Elza Erkp arxv:705.0466v [cs.it] May 07 Abstract Ths paper studes the fundamental lmts of cachng n a network wth two recevers and two fles generated by a two-component dscrete memoryless source wth arbtrary jont dstrbuton. Each recever s equpped wth a cache of equal capacty, and the requested fles are delvered over a shared errorfree broadcast lnk. Frst, a lower bound on the optmal peak rate-memory trade-off s provded. Then, n order to leverage the correlaton among the lbrary fles to allevate the load over the shared lnk, a two-step correlaton-aware cache-aded coded multcast CACM scheme s proposed. The frst step uses Gray- Wyner source codng to represent the lbrary va one common and two prvate descrptons, such that a second correlatonunaware multple-request CACM step can explot the addtonal coded multcast opportuntes that arse. It s shown that the rate acheved by the proposed two-step scheme matches the lower bound for a sgnfcant memory regme and t s wthn half of the condtonal entropy for all other memory values. I. INTRODUCTION AND SETUP The use of cachng at the wreless network edge has emerged as a promsng approach to effcently ncrease the capacty of wreless access networks. There have been extensve studes characterzng the fundamental rate-memory tradeoff n a broadcast cachng network wth a lbrary composed of ndependent content [] [3]. More recently, [4] [6] have studed the rate-memory trade-off when delverng correlated fles. In [4], the authors consder a sngle-recever multplefle network wth lossy reconstructons and characterze the trade-offs between rate, cache capacty, and reconstructon dstortons. They extend the analyss to a scenaro wth two recevers and one cache, n whch local cachng gans can be explored. The works n [5] and [6] consder a settng wth an arbtrary number of fles and recevers each havng a cache. A practcal correlaton-aware scheme s proposed n [5], n whch content s cached accordng to both the popularty of fles and ther correlaton wth the rest of the lbrary. Then, the requested content s delvered va a multcast codeword composed of compressed versons of the requested fles. Alternatvely, the work n [6] addresses the dependency among content fles by frst compressng the correlated lbrary, whch s then treated as a lbrary of ndependent fles by a conventonal cache-aded coded multcast scheme. Ths work has been supported by NSF grant #699. P. Hassanzadeh and E. Erkp are wth the ECE Department of New York Unversty, Brooklyn, NY. Emal: ph990, elza}@nyu.edu J. Llorca and A. Tulno are wth Bell Labs, Noka, Holmdel, NJ, USA. Emal: jame.llorca, a.tulno}@noka-bell-labs.com A. Tulno s wth the DIETI, Unversty of Naples Federco II, Italy. Emal: antonamara.tulno}@unna.t In ths paper, by focusng on a two-user two-fle settng, we are able to characterze the optmal peak rate-memory tradeoff of the broadcast cachng network wth correlated sources. To ths end, we frst provde a lower bound on the optmal peak rate-memory trade-off, whch s derved usng a cutset argument on the correspondng cache-demand augmented graph [7]. The lower bound mproves the best known bound for correlated sources gven n [8], and when partcularzed to ndependent sources, matches the correspondng best known bound derved n [9]. We then propose a two-step scheme, n whch the source fles are frst encoded based on the Gray- Wyner network [0], and n the second step, they are cached and delvered through a multple-request correlaton-unaware cache-aded coded multcast scheme. In the rest of the paper, we dscuss the optmalty of the proposed two-step scheme by characterzng a lower bound on the rate-memory tradeoff for ths class of schemes, and comparng t wth the lower bound on the optmal trade-off. We dentfy the set of operatng ponts n the Gray-Wyner regon [0], [], for whch a twostep scheme s optmal over a range of cache capactes, and approxmates the optmal rate to wthn half of the condtonal entropy for all cache szes. The paper s organzed as follows. Sec. II presents the nformaton-theoretc problem formulaton. In Sec. III, we ntroduce a class of two-step schemes based on the Gray- Wyner network. The lower bounds for the optmal and twostep schemes are provded n Sec. IV, and later used to establsh the optmalty of an achevable two-step scheme proposed n Sec. V. After analyzng an llustratve example n Sec. V-A, the paper s concluded n Sec. VI. II. NETWORK MODEL AND PROBLEM FORMULATION We consder a broadcast cachng network composed of one sender wth access to a lbrary of two fles generated by a -component dscrete memoryless source -DMS. The -DMS model X X, px, x conssts of two fnte alphabets X, X and a jont pmf px, x over X X. The -DMS generates and..d. random process X, X } wth X, X px, x. For a block length n, the two lbrary fles are represented by sequences X n = X,..., X n and X n = X,..., X n, respectvely, where X n X n and X n X n. The sender communcates wth two recevers r and r over a shared error-free broadcast lnk. Each recever s equpped wth a cache of sze nm bts, where M denotes the normalzed cache capacty. We assume that the system operates n two phases: a cachng phase and a delvery phase. Durng the cachng phase, whch

takes place at off-peak hours when network resources are abundant, the recever caches are flled wth functons of the lbrary fles, such that durng the delvery phase, when recever demands are revealed and resources are lmted, the sender broadcasts the shortest possble codeword that allows each recever to losslessy recover ts requested fle. We refer to the overall scheme as a cache-aded coded multcast scheme CACM. Gven a realzaton of the lbrary, X n, X n }, a CACM scheme conssts of the followng components: Cache Encoder: Durng the cachng phase, the cache encoder desgns the cache content of recever r usng a mappng fr C : X n X n [ : nm. The cache confguraton of recever r s denoted by Z r = fr C X n, X n. Multcast Encoder: Durng the delvery phase, each recever requests a fle from the lbrary. The demand realzaton, denoted by d = d r, d r D, }, where d r, } denotes the ndex of the fle requested by recever r, s revealed to the sender, whch then uses a fxed-to-varable mappng f M : D [ : nm [ : nm X n X n Y to generate and transmt a multcast codeword Y d = f M d, Z r, Z r }, X n, X n } over the shared lnk. The codeword Y d s desgned for each demand realzaton accordng to the cache content, lbrary fles, and jont dstrbuton px, x, to enable almost-lossless reconstructon of the requested fles. Multcast Decoders: Each recever r uses a mappng gr M : D Y [ : nm Xd n r to recover ts requested fle, Xd n r, usng the receved multcast codeword and ts cache content as X n d r = g M r d, Y d, Z dr. The worst-case probablty of error of a CACM scheme s gven by P n e = max d max P Xn dr Xd n r r. In ths paper, we focus on the peak multcast rate, correspondng to the worst-case demand, R n = max d E[LY d ], n where LY denotes the length n bts of the multcast codeword Y, and the expectaton s over the lbrary fles. Defnton : A peak rate-memory par R, M s achevable f there exsts a sequence of CACM schemes for cache capacty M and ncreasng block length n, such that lm n P e n = 0, and lm sup n R n R. Defnton : The peak rate-memory regon, R, s the closure of the set of achevable peak rate-memory pars R, M, and the optmal peak rate-memory functon s R M = nfr : R, M R }. III. GRAY-WYNER CACM SCHEME In ths secton, we descrbe a class of schemes based on a two-step lossless source codng setup, as depcted n Fg.. The frst step nvolves compressng the lbrary va Gray-Wyner We use to ndcate varable length. Source Lbrary Gray-Wyner Source Codng Encoder 0 X # ", X # % W " Encoder Encoder W 0 W % Correlaton- Unaware Multple-Request CACM Recever cache Recever cache Fg. : Gray-Wyner CACM scheme, composed of a frst Gray- Wyner source codng step, and a second correlaton-unaware multple-request CACM step. source codng, and the second step s a correlaton-unaware multple-request CACM scheme. We refer to ths scheme as Gray-Wyner Cache-Aded Coded Multcast -CACM. Gray-Wyner source codng, depcted n Fg., s a dstrbuted lossless source codng setup n whch a -DMS X, X s represented by three descrptons W 0, W, W }, where W 0 [ : nr0, W [ : nr, and W [ : nr. Fle d r can be losslessly recovered from descrptons W 0, W dr, and fle d r can be losslessly recovered from descrptons W 0, W dr, both asymptotcally, as n. As shown n [], the Gray-Wyner rate regon s the closure of the unon over U of S U, where S U denotes the set of rate trplets R 0, R, R such that R 0 IX, X ; U, 3 R HX U, 4 R HX U, 5 gven a condtonal pmf pu x, x wth U X. X +. For a gven U and a rate trplet R 0, R, R S U, a -CACM scheme conssts of: Gray-Wyner Encoder: Gven a lbrary realzaton X n, X n }, the Gray-Wyner encoder at the sender computes three descrptons W 0, W, W } usng a mappng f : X n X n [ : nr0 [ : nr [ : nr. Correlaton-Unaware Cache Encoder: Gven the descrptons W 0, W, W }, the cache encoder at the sender computes the Gray-Wyner based cache contents Z r = f C r W 0, W, W, r, }. Correlaton-Unaware Multcast Encoder: For any demand realzaton d revealed to the sender, the Gray- Wyner based multcast encoder generates and transmts the multcast codeword Y d = f M d, Z r, Z r }, W 0, W, W }. Multcast Decoder: Recever r decodes the descrptons correspondng to ts requested fle as Ŵ0, Ŵd r } = g M r d, Yd, Z. Gray-Wyner Decoder: Recever r decodes ts requested fle usng the descrptons recovered by the multcast

decoder, va a mappng gr : [ : nr0 [ : nr d X n, as Xn d = gr Ŵ0, Ŵd r. Notce that for the class of -CACM schemes, snce R 0, R, R S U and W 0, W dr s a Gray-Wyner descrpton of Xd n r, n order to have lm n P e n = 0, wth P n e as defned n, we only need lm max max P Ŵ0, n d r Ŵd r W 0, W dr = 0. As n, the peak -CACM multcast rate s R n R E[LYd ] 0, R, R = max, 6 d n where we explctly show the dependence on R 0, R, R. In lne wth Defntons and, for a gven U, the peak U-rate-memory regon for the class of - CACM schemes, R U, s defned as the closure of the unon of all the achevable pars R n R 0, R, R, M wth R 0, R, R S U. Analogously, the peak U-ratememory functon of -CACM, R M, U, s defned as R M, U = nfr : R, M R U}. We remark that R M, U s the rate acheved by a - CACM scheme wth the Gray-Wyner encoder operatng at the boundary of the regon S U. Fnally, optmzng over the choce of U, we obtan the peak rate-memory regon, R, and the peak rate-memory functon, R M, as } R = cl R U, R M = nf R M, U, where cls} denotes the closure of S, and the unon and nfmum are over all choces of U wth U X. X +. IV. LOWER BOUNDS In ths secton, we provde lower bounds for R M, the optmal peak rate-memory functon, and R M, U, the peak U-rate-memory functon of -CACM for a gven U. The latter bound can also be used to obtan a lower bound for R M. We then nvestgate condtons on the cache capacty M under whch the lower bounds for R M and R M meet. These condtons are then used n Secton V n order to establsh the optmalty of -CACM, and quantfy the rate gap from the lower bound as a functon of the cache capacty M. A. Lower bound on R M Theorem : For a broadcast cachng network wth two recevers, cache capacty M, and a lbrary composed of two fles wth jont dstrbuton px, x, a lower bound on R M, the optmal peak rate-memory functon, s gven by M = nf R : R HX, X M, R HX, X M, R } } HX, X + max HX, HX M. Proof: Theorem follows from combnng cut-set bounds on the cache-demand-augmented graph, and the tmereplcaton of the cache-demand-augmented graph as descrbed n [7], []. Remark : The outer bound n Theorem mproves the best known bound for correlated sources gven n [8, Theorem ], and when partcularzed to ndependent sources, matches the correspondng best known bound derved n [9]. B. Lower bounds on R M, U and R M Theorem : For a gven U, a lower bound on R M, U, the peak U-rate-memory functon of the -CACM scheme, s gven by R LB M, U = nf R : R IX, X ; U + HX U + HX U M, R IX, X ; U + HX U + HX U M, R IX, X ; U + HX U + HX U M, R IX, X ; U + } HX U + HX U M. Proof: The proof s smlar to that of Theorem ; now appled to Gray-Wyner descrptons at rates R 0, R, R S U. Corollary : A lower bound on R M, the peak ratememory functon of -CACM, s gven by M = nf M, U, where the nfmum s over all choces of U wth U X. X +. C. Where R LB M and RLB M meet By comparng the lower bounds n Theorems and, t M, U, and hence, M R LB M. In the followng, we derve condtons under whch M = R LB M. Theorem 3: Let } M max X U X mn HX U, HX U. s easy to see that M Then, for M [0, M ] [HX, X M, HX, X ], we have R LB M = RLB M. Proof: Theorem 3 follows from comparng M wth R LB M, U for a gven U, over dfferent regons of memory M. It s observed that when M [0, }] mn HX U, HX U [ } ] HX, X mn HX U, HX U, HX, X, M, U RLB M s ndependent from the cache capacty M, and becomes zero when IX, X ; U + HX U + HX U = HX, X. For the choce of U used to obtan M, the regon of memory over whch the two bounds meet s maxmzed.

Remark : The Markov chan X U X n Theorem 3 suggests that for the rate trplet R 0, R, R S U used n -CACM, we requre R 0 + R + R = HX, X. The same Markov chan s also used to defne Wyner s common nformaton []. Whle n Wyner s common nformaton the goal s to mnmze R 0 subject to R 0 +R +R = HX, X, for M n Theorem 3, the goal s to maxmze mnr, R. Corollary : If the fle lbrary X, X s such that } IX ; X, HX X, HX X cl S U, where the unon s over all choces of U wth U X. X +, then M = R LB M. Proof: See [3]. Example : Consder a -DMS whose jont pmf px, x s such that X, X can be represented as X = X, V and X = X, V, where X and X are condtonally ndependent gven V. Takng U = V, the pont R 0 = IX, X ; U = HV = IX ; X, R = HX U = HX V = HX X and R = HX U = HX V = HX X belongs to the Gray-Wyner rate regon. Then, by Theorem, M = M for any M. V. AN ACHIEVABLE SCHEME BASED ON -CACM AND ITS OPTIMALITY In ths secton, we present an achevable -CACM scheme, where the frst step conssts of a Gray-Wyner encoder restrcted to operate on the plane of the Gray-Wyner rate regon wth R = R = ρ, and the second step s a determnstc correlaton-unaware multple-request CACM scheme that combnes deas from conventonal cachng e.g., LFU and uncoded multcastng and correlaton-unaware CACM wth coded placement, as suggested by Tan and Chen n [4], and referred to as TC n the followng. We remark that jontly optmzng these two steps s the key to maxmzng the overall performance. We then refer to the overall scheme as -LFU-TC, whch works as follows: Gray-Wyner Encoder: generates three lbrary descrptons W 0, W, W } usng a condtonal pmf pu x, x such that px u = px u wth R 0, ρ, ρ S U. Cache Encoder: populates the recever caches as: If M [0, ρ, the common descrpton W 0 s not cached at ether recever, and descrptons W, W } are cached accordng to the cachng phase of TC. If M [ρ, R 0 + ρ, the frst nm ρ bts of descrpton W 0 are cached at both recevers as per LFU cachng, and descrptons W, W } are cached accordng to TC over the remanng cache capacty ρ. If M [R 0 +ρ, R 0 +ρ], the common descrpton W 0 s fully cached at both recevers, and descrptons W, W } are cached accordng to TC over the remanng cache capacty M R 0. LFU s a local cachng polcy that, n the settng of ths paper, leads to all recevers cachng the same part of the fle. Multcast Encoder: transmts the descrptons W, W } accordng to conventonal coded multcast schemes [] [3], [4], whle the porton of W 0 mssng at each recever cache s transmtted va uncoded nave multcast. Remark 3: Dfferently from the sngle-cache settng analyzed n [4], where cachng the common descrpton frst s always optmal, n our case, when the cache capacty s smaller than the prvate descrpton sze, ρ, t s optmal to frst cache the prvate descrptons. Theorem 4: Gven a condtonal pmf pu x, x such that px u = px u, a cache capacty M, and a rate trplet R 0, ρ, ρ S U, the peak U-rate acheved by -LFU- TC s gven by M, U = nf R ach R 0, ρ, 7 where the nfmum s over all rate trplets R 0, ρ, ρ S U, and R ach R 0, ρ s R 0 + ρ M, M [0, ρ R ach R 0, ρ = R 0 + 3 ρ M, M [ ρ, R 0 + ρ R 0 + ρ M, M [R 0 + ρ, R 0 + ρ]. Furthermore, optmzng over U, the peak rate acheved by -LFU-TC s gven by M = nf M, U, where the nfmum s over all choces of U wth U X. X +. Proof: See [3]. A. Optmalty of -LFU-TC In order to prove the optmalty of the -LFU-TC scheme, we frst state the followng theorem: Theorem 5: For any U and M, M, U = R M, U. Proof: Smlar to the proof of Theorem 3, R UB M, U s compared to the lower bound R LB M, U n each memory regon, and for any U formng a Markov chan, X U X. Example : Assumng the same settng as n Example, snce M = R LB M for any M, t follows from Theorem 5 that the -LFU-TC scheme s optmal for all values of memory M. The followng theorem characterzes the performance of the -LFU-TC scheme for dfferent regons of M, and delneates the cache capacty regon for whch the scheme s optmal or near optmal. Theorem 6: Let M max X U X mn HX U, HX U }, where the max s over all choces of U such that px u = px u. Then, for M [0, M ] [HX, X M, HX, X ], the -LFU-TC scheme s optmal.e., M = R M.

Peak Rate In addton, for M M, HX, X M, we have R UB M R M } HX mn X, HX X M. Proof: See [3] B. Illustraton of Results: Doubly Symmetrc Bnary Source Consder, as a -DMS, a doubly symmetrc bnary source DSBS wth jont pmf px, x = p 0δ x,x + p 0 δ x,x, x, x 0, }, and parameter p 0 [0, ]. Then, HX = HX =, HX X = HX X = hp 0, HX, X = + hp 0, where hp = p logp p log p s the bnary entropy functon. As derved n [], an achevable Gray-Wyner rate regon of a DSBS restrcted to the plane R 0, R, R : R = R = ρ}, s descrbed by the set of rate trplets R 0, ρ, ρ wth R 0 gven by + hp 0 ρ, 0 ρ < hp R 0 fρ hp ρ, 8 where p = p 0, fρ + hp 0 + h ρ p 0 p0 p 0 log + h ρ p 0 log log h ρ p 0 + h ρ p 0 and h ρ s the nverse of the bnary entropy functon. We compare the performance of the proposed -LFU-TC scheme wth respect to: LFU cachng wth uncoded multcastng LFU-UM, the determnstc correlaton-unaware CACM n [4], referred to as TC, 3 the lower bound on the -CACM peak rate-memory functon R LB, and 4 the lower bound on the optmal peak rate-memory functon. Fg. dsplays the rate-memory trade-offs for p 0 = 0.. In lne wth Theorems 3 and 6, Fg. shows that the lower bound on the Gray-Wyner rate-memory functon R LB concdes wth the lower bound on the optmal rate-memory functon for M M = 0.5 and M HX, X M =., and -LFU-TC s optmal n ths regon, whle correlaton-unaware schemes, LFU-UM and TC, fall short. Furthermore, the gap between the rate acheved wth -LFU-TC and the optmal peak rate-memory functon s less than 0., whch s less than half of the condtonal entropy, 0.36. Fnally, n lne wth Theorem 5, -LFU-TC acheves for any M. VI. CONCLUDING REMARKS In ths paper, we have studed the fundamental lmts of cache-aded communcaton systems under the assumpton of correlated content for a two-user two-fle network. We have derved a lower bound on the peak rate-memory functon for such systems and proposed a class of schemes based on a twostep source codng approach. Fles are frst compressed usng Gray-Wyner source codng, and then cached and delvered,.5 0.5 0 0 0.5.5 Memory Fg. : Rate-memory trade-off for a DSBS wth p 0 = 0.. usng a combnaton of exstng correlaton-unaware cached aded coded multcast schemes. We have fully characterzed the rate-memory trade-off of such class of schemes, proposed an achevable two-step scheme, and proved ts optmalty for dfferent memory regmes. Fnally, n [3], we provde an extended analyss that ncludes the characterzaton of both peak and average rate-memory trade-offs n more general userfle settngs. ACKNOWLEDGEMENT The authors would lke to thank M. Wgger and D. Gündüz for ther useful dscussons on the Gray-Wyner network. REFERENCES [] M. A. Maddah-Al and U. Nesen, Fundamental lmts of cachng, IEEE Transactons on Informaton Theory, vol. 60, no. 5. [] M. J, A. Tulno, J. Llorca, and G. Care, Order-optmal rate of cachng and coded multcastng wth random demands, arxv:50.034, 05. [3], Cachng-aded coded multcastng wth multple random requests, n Proc. IEEE Informaton Theory Workshop ITW, 05. [4] R. Tmo, S. S. Bdokht, M. Wgger, and B. C. Geger, A rate-dstorton approach to cachng, arxv preprnt arxv:60.07304, 06. [5] P. Hassanzadeh, A. Tulno, J. Llorca, and E. Erkp, Cache-aded coded multcast for correlated sources, Proc. IEEE Internatonal Symposum on Turbo Codes and Iteratve Informaton Processng ISTC, 06. [6], Correlaton-aware dstrbuted cachng and coded delvery, Proc. IEEE Informaton Theory Workshop ITW, 06. [7] J. Llorca, A. M. Tulno, K. Guan, and D. Klper, Network-coded cachng-aded multcast for effcent content delvery, n Proc. IEEE Internatonal Conference on Communcatons ICC. [8] S. H. Lm, C.-Y. Wang, and M. Gastpar, Informaton theoretc cachng: The mult-user case, n Proc. IEEE Internatonal Symposum on Informaton Theory ISIT, 06, pp. 55 59. [9] H. Ghasem and A. Ramamoorthy, Improved lower bounds for coded cachng, n Proc. IEEE Internatonal Symposum on Informaton Theory ISIT, 05, pp. 696 700. [0] R. Gray and A. Wyner, Source codng for a smple network, Bell System Techncal Journal, 974. [] A. Wyner, The common nformaton of two dependent random varables, IEEE Transactons on Informaton Theory, 975. [] J. Llorca and A. M. Tulno, Mnmum cost cachng-aded multcast under arbtrary demand, n Proc. IEEE Aslomar Conference on Sgnals, Systems and Computers, 03, pp. 36 37. [3] P. Hassanzadeh, A. Tulno, J. Llorca, and E. Erkp, Rate-memory tradeoff for the broadcast cachng network wth correlated sources, preprnt at http://engneerng.nyu.edu/eelab/people/parsa. [4] C. Tan and J. Chen, Cachng and delvery va nterference elmnaton, n Proc. IEEE Internatonal Symposum on Informaton Theory ISIT, 06.