206 IEEE Intenational Symposium on Infomation Theoy Cloud-Aided Wieless Netwos with Edge Caching: Fundamental Latency Tade-Offs in Fog Radio Access Netwos Ravi Tandon Osvaldo Simeone Depatment of ECE Univesity of Aizona, Tucson, AZ tandon@email.aizona.edu CWCSPR, ECE Depatment New Jesey Institute of Technology osvaldo.simeone@njit.edu Key F-RAN Paametes Abstact Fog Radio Access Netwo (F-RAN) is an emeging wieless netwo achitectue that leveages caching capabilities at the wieless edge nodes, as well as edge connectivity to the cloud via fonthaul lins. This pape aims at poviding a latency-centic analysis of the degees of feedom of an F-RAN by accounting fo the total content delivey delay acoss the fonthaul and wieless segments of the netwo. The main goal of the analysis is the identification of optimal caching, fonthaul and edge tansmission policies. The study is based on the intoduction of a novel pefomance metic, efeed to as the Nomalized Delivey Time (NDT), which measues the total delivey latency as compaed to an ideal intefeence-fee system. An infomation-theoetically optimal chaacteization of the tade-off between NDT, on the one hand, and fonthaul and caching esouces, on the othe, is deived fo a class of F-RANs with two edge nodes and two uses. Using these esults, the inteplay between caching and cloud connectivity is highlighted, as well as the impact of both caching and fonthaul esouces on the delivey latency. Index Tems Cloud Radio Access Netwo (C-RAN), caching, 5G, degees of feedom, latency. N : # of popula files M : # of edge nodes (ENs) : stoage capacity L : size of a File K : # of uses CF = log(p ) : fonthaul capacity M =3 Fonthaul Demands (Edge Node) EN g(p CF = lo ) Use Cloud CF = log (P ) EN2 shaed wieless C F = Use 2 channel l og ( P) EN3 Use 3 L Set of popula files N =5 Use 4 () K=4 t= t = 2 Time Fig.. Infomation-theoetic model fo F-RAN. I. I NTRODUCTION Edge pocessing and vitualization ae two of the ey emeging tends in the evolution of 5G netwos [], [2]. Edge pocessing efes to the localization of stoage and computing esouces at the netwo edge. Notably, edge nodes (ENs), such as base stations, can be equipped with local s to stoe popula content, with the aim of educing the delivey latency by limiting the need to communicate with emote content seves []. In a dual manne, vitualization allows the implementation of netwo functionalities at a centalized cloud pocesso. An impotant example is given by the Cloud Radio Access Netwo (C-RAN) achitectue, in which the ENs ae connected to a cloud pocesso by so called fonthaul lins, so as to enable, among othe pefomance gains, enhanced intefeence management capabilities thans to the joint baseband pocessing in the cloud [2], [3]. Bidging the gap between these two complementay tends, the Fog Radio Access Netwo (F-RAN) achitectue has been ecently advocated that combines the benefits of both edge and cloud pocessing (see, e.g., [4]). In this achitectue, as illustated in Fig., ENs may be endowed with caching capabilities, so as to seve local data equests of popula content 978--5090-806-2/6/$3.00 206 IEEE with low latency, while at the same time being contollable fom a cental cloud pocesso, in ode to seve abitay data equests with stonge intefeence management popeties and less stingent delay constaints. The design of F-RANs involves the following intetwined design questions: (a) What to?; (b) How to utilize the limited capacity available on the fonthaul lins?; and (c) How to delive the equested content on the downlin wieless channel? In this wo, we set out to obtain fundamental insights into these questions by means of an infomationtheoetic appoach. The poposed famewo aims at poviding a latency-centic analysis of the degees of feedom of an FRAN by accounting fo the total content delivey delay acoss the fonthaul and wieless segments of the netwo. Related Wo: Questions (a) and (c) wee ecently tacled fom an infomation-theoetic viewpoint by Maddah-Ali and Niesen in [5], [6], fo a -aided scenaio that allows fo edge caching but not fo cloud pocessing. Specifically, fo a scenaio with M = 3 ENs and K = 3 uses, these wos pesent an uppe bound on the invese of the numbe of achievable degees of feedom (DoF). In [7], instead, a lowe 2029
206 IEEE Intenational Symposium on Infomation Theoy bound is pesented that poves the optimality of the caching and tansmission schemes poposed in [5], [6] fo a given ange of values of the stoage capacity and unde the constaint that no inte-file coding is allowed. Main Contibutions: In contast to the mentioned pio eseach, in this pape, we focus on the F-RAN achitectue in Fig., which allows fo both edge caching and cloud pocessing by means of fonthaul connections. To this end, we fist pesent an infomation-theoetic model of F-RANs that succinctly captues its new design aspects and constaints. We also develop a new pefomance measue, which we efe to as the Nomalized Delivey Time (NDT). The NDT captues the wost-case latency incued ove the fonthaul and wieless access segments of the netwo fo content delivey as compaed to an ideal intefeence-fee system in the high signal-to-noise atio (SNR) egime. Fo the case of a cachingonly system, with no fonthaul lins, the NDT is elated to the invese of the DoF metic studied in [5], [6], as discussed in [7]. Unde the constaint of uncoded inte-file caching, we chaacteize the optimal tade-off between NDT and caching and fonthaul esouces fo an F-RAN with M =2ENs and K =2uses (see Fig. 3). II. INFORMATION THEORETIC MODELING OF F-RAN As illustated in Fig., we conside an F-RAN achitectue with M edge nodes (ENs), which can seve a set of K uses ove a shaed wieless channel. Note that, in Sec. IV, we will specialize the set-up fo ou main esults to the case M = K = 2, but we pesent hee the model in its full geneality in ode to highlight moe geneal eseach questions and open poblems. We assume the pesence of a libay of N files, which epesent the content that may be equested by the uses, whee each file is of size L bits. Time is oganized into tansmission intevals, as shown in Fig. and Fig. 2 and futhe discussed below. The system model, notation and main assumptions ae summaized as follows. The libay of N contents, o files, that may be equested is denoted by F = {F,F 2,,F N }, whee each file is of size L bits. This libay is assumed to be constant acoss many tansmission intevals. At each tansmission inteval t, uses issue a vecto of equests D(t) =(D (t),,d K (t)), whee D (t) 2{,,N} indicates that file F D (t) is equested by use at time t. We mae no assumption on the natue of the time-vaiability of the demands made by the uses. Each EN has a which can stoe bits, whee 2 [0, ] epesents the factional size. The cloud has access to all N files, and each EN is connected to the cloud via a fonthaul lin. The fonthaul capacity is given by C F bits pe symbol fo each EN, whee a symbol efes to a channel use of the downlin wieless channel. The capacity C F is assumed to be fixed, eflecting conventional scenaios in which fonthaul lins coespond to dedicated wied connections (see, e.g., [2], [3]). The collective time-vaying wieless channel state infomation (CSI) at tansmission inteval t is defined as H(t) = {{H m, (t)} K = }M m=, whee H m, (t) denotes the channel coefficients that chaacteize the popagation between the mth EN and the th use. The channel coefficients ae assumed to be dawn in an independent identically distibuted (i.i.d.) manne fom a continuous distibution (as in [5], [6]). Next, we define the caching-fonthaul-tansmission policy. We focus on the case in which, in each tansmission inteval t, the ENs and cloud ae awae of the channel ealization H(t), as well as of the uses demand vecto D(t), but not of any futue channel ealizations and demands H(t 0 ) and D(t 0 ) with t 0 >t. As detailed below, the caching-fonthaul-tansmission policy decides each ENs composition, which is ept fixed fo many tansmission intevals, as well as the duation and content of the tansmissions acoss fonthaul and wieless segments fo each tansmission inteval, as shown in Fig. 2. Definition. (Caching-Fonthaul-Tansmission Policy) A caching-fonthaul-tansmission policy = ( C, F, E ) is chaacteized by the following thee encoding functions. a) Caching policy C : The caching policy is defined by a function F!{S,S 2,,S M }, which maps the set F of files into the content S m of the mth EN fo m =,,M, which in tun cannot exceed bits. Fo the scope of this pape, we focus on the pactically elevant class of caching policies that do not allow fo inte-file coding but do include abitay inta-file coding. Within this class, the content S m of the mth EN can be patitioned into N independent sub-s, i.e., S m =(S mf,s mf2,,s mfn ), whee S mfn can be any abitay function of file F n, fo n =,,N. We emphasize that the caching policy is fixed fo many tansmission intevals and is only a function of the libay F of files. b) Fonthaul policy F : The fonthaul policy is defined by a function {F,D,H}! {T F,U T F,U T F 2,,U T F M }, which maps the set of files F, instantaneous demands and channels to the duation T F of the fonthaul tansmission (see Fig. 2) and to the message U T F m, of duation T F, sent on the mth lin. In eeping with the definition of the fonthaul capacity C F, all time intevals, including T F, ae nomalized to the symbol tansmission time on the downlin wieless channel. Accodingly, the fonthaul message cannot exceed T F C F bits. We emphasize that the fonthaul policy, as well as the tansmission policy discussed below, adapts to the instantaneous demands and to the CSI in each tansmission inteval, unlie the caching policy. c) Edge tansmission policy E : The edge tansmission policy, o tansmission policy fo shot, is defined by the collection of functions E =( E0, E, E2,, EM ), which chaacteize the tansmission on the wieless downlin channel as a function of the cuent demands and CSI and of the fonthaul messages. Specifically, the function E0 : {F,D,H,U T F,,U T F M }! T E selects the tansmission duation, in numbe of symbols, on the downlin wieless channel fo all the ENs (see Fig. 2). Instead, the M local tansmission functions Em : {S m,d,h,t E,U T F m }!X T E m, one fo each EN, detemine the codewod X T E m, of duation symbols, sent on the wieless channel by the mth EN, T E 2030
206 IEEE Intenational Symposium on Infomation Theoy unde an aveage powe constaint given by the paamete P. Fig. 2. Illustation of tansmission intevals and of the definition of latency. Upon the tansmission by the ENs at a tansmission inteval t, the th use eceives the signal Y T E = NX m= H m, X T E m + N T E, () on each channel use of the wieless downlin channel, whee H m, is the channel coefficient between the th use and the mth EN; N T E denotes the additive noise at the th use, which is assumed to be complex Gaussian with unitay powe, i.i.d. ove time and uses and also independent of the channel coefficients. Each use maps its eceived signal Y T E to an estimate ˆF D of the demanded file (i.e., F D ), incuing a pobability of eo P( ˆF D 6= F D ). The pobability of eo of the policy is then defined fo the wost-case equest vecto as P e = max D max P( ˆF D 6= F D ). (2) 2{,,K} A policy is said to be feasible if, fo almost all ealizations H of the channel, i.e., with pobability, we have P e! 0 when L!. III. LATENCY-CENTRIC DOF ANALYSIS: NORMALIZED DELIVERY TIME (NDT) We now define the poposed delivey metic by fist intoducing the notion of delivey time pe bit. Definition 2. (Delivey time pe bit) Fo a given factional size, fonthaul capacity C F, and an EN powe constaint P, conside a sequence of feasible policies indexed by the file size L. Denote as T F and T E the duations of the fonthaul tansmission and edge tansmission as in Definition fo a file size L. The aveage achievable delivey time pe bit fo a given sequence of feasible policies is then defined as (, C F,P) = max D lim L! L (T F + T E ) (3) The delivey time pe bit (3) captues the latency within each tansmission inteval, which is depicted in Fig. 2, as evaluated fo the wost-case uses equests and on aveage ove the channel distibution. It is noted that, in ode to obtain vanishing pobabilities of eo, as equied by Definition 2, the latencies T F and T E need to scale with L, and it is this scaling that is measued by (3). We also obseve that the definition of delivey time pe bit in (3) is ain to the completion time studied in [8], [9] fo standad channel models, such as boadcast and multiple access. The optimal latency pefomance is in pinciple obtained by minimizing the delivey time pe bit (3) ove all possible policies. This optimization is geneally pohibitive and it is also dependent on all paametes (, C F,P). With the aim of obtaining analytical insights, we next popose a novel tactable metic that etains the ey dependence of latency on size and fonthaul capacity while adopting a high-snr appoximation in the vein of the by now standad DoF analysis of intefeence netwos [0]. To this end, we let the fonthaul capacity scale with the SNR paamete P as C F = log(p ), whee is a paamete that measues the multiplexing gain of the fonthaul lins. Definition 3. (NDT) Fo any achievable (, C F,P), with C F = log(p ), the nomalized delivey time (NDT) (, ) = lim P! (, log(p ),P) / log(p ) is said to be achievable. Fo a given pai (, ) the minimum NDT is defined as (4) (, ) =inf{ (, ) : (, ) is achievable}. (5) In (5), the delivey time pe bit (3) is nomalized by the tem / log(p ), which measues fo the delivey time pe bit, at high SNR, of a baseline system with no intefeence and unlimited caching, in which each use can be seved by a dedicated EN that has all files. As such, an NDT of indicates that the wost-case time equied to seve any possible equest is times lage than the time that would be needed by the baseline system. Based on the definitions above, ou goal is to chaacteize the novel metic NDT, (, ) that captues the inteplay between the nomalized stoage and the fonthaul multiplexing gain. We note that the minimum NDT (5) geneally depends on the numbe N of files, although we do not mae this dependence explicit to simplify the notation. We close this section with a ey popety of NDT that lends futhe evidence to its suitability as a pefomance metic fo the analysis of F-RANs. Rema (Time Shaing between Policies and Convexity of Minimum NDT). Conside two (sequences of) policies and 2, equiing caching and fonthaul esouces (,) and ( 2,) and achieving NDTs (,) and ( 2,), espectively. An F-RAN is now given with caching and fonthaul esouces chaacteized as (, ) = ( +( ) 2,) fo some paamete 2 [0, ]. On this F-RAN, one could then opeate with policy on a faction of the stoage, fonthaul capacity and spectal esouces (i.e., time o fequency), and with policy 2 on the emaining pats. It can be eadily shown, based on additivity aguments, that this file splitting and -shaing stategy achieves an NDT equal to the convex combination (,)+( ) ( 2,), which is lowe bounded by (, ). The above agument demonstates that the NDT pefomance measue (, ) is convex in fo any value of. We note that a simila obsevation was made in [5], [6] fo a system with caching only (i.e., =0). 203
206 IEEE Intenational Symposium on Infomation Theoy NDT This obsevation motivated the authos of [5], [6] to study the invese of the DoF metic, instead of the DoF itself, as the pefomance citeion of inteest, since the DoF is shown not to have the same convexity popeties (see [7, Rema 2]). Rema 2 (Pipelined NDT) We note hee that the NDT in Definition 3 is based on the assumption of seial opeation of the fonthaul and edge segments. This definition can be extended to delivey stategies in which fonthaul and wieless channels ae opeated in a pipelined (paallel) manne, so that fonthaul and wieless tansmissions can tae place at the same time. This vaiation, along with moe geneal esults on NDT chaacteization, can be found in []. CF = log(p ) fonthaul capacity (, ) + cloud + (a) 3 2 NDT 0 A. Achievability of Minimum NDT Hee we descibe the specific policies that achieve the NDT tade-off cuve in Fig. 3. Without loss of geneality, we focus on the cone points in both low-fonthaul and high-fonthaul capacity egimes. This is because all the emaining points on the tade-off cuve can be achieved by time- and memoyshaing between the policies coesponding to the cone points as pe Rema. Achieving (, ) = fo = : With =, each EN can all files. This enables full coopeation between the ENs, since each EN can the entie libay. Thus, the set of ENs foms a vitual multiple-input single-output (MISO) boadcast channel, and zeo-focing beamfoming yields paallel intefeence-fee channels to the two uses. Theefoe, the latency equals the time needed by the mentioned efeence intefeence-fee channel with full caching, esulting in the achievable NDT =. Achieving (, ) = 3/2 fo = /2: With = /2, each EN can only at most half of each file. The caching policy is to split each file Fi into two equal sized sub-files, i.e., Fi = (Fi, Fi2 ) fo i =,..., N. EN stoes the fist cloud + In this section, we pesent the optimum NDT as intoduced in Definition 3 fo the special case of M = 2 and K = 2. As illustated in Fig. 3, the NDT tade-off analysis in Theoem identifies two distinct egimes in tems of the fonthaul capacity, namely a low-fonthaul capacity egime with and a high-fonthaul capacity egime with >. In the latte case, the use of both fonthaul and caching esouces is necessay in ode to obtain the optimal NDT pefomance, while, in the fome, if the capacity is sufficiently lage, namely if /2, it is sufficient to leveage the caching esouces to achieve the optimal pefomance. We next discuss the caching-fonthaul-tansmission policies that achieve the NDT tade-off in Theoem. Fo a setch of the convese, we efe to Appendix A (see [] fo a detailed poof). 2 Factional Cache Size (, ) + IV. O PTIMUM NDT T RADE -O FF FOR M = K = 2 Theoem. The optimal NDT tade-off fo the M = 2-EN, K = 2-use F-RAN with numbe of files N 2 is given as ( max + + 2, 2 fo 0 (6) (, ) = + fo >. > (b) 0 Factional Cache Size Fig. 3. Optimal NDT tade-off fo the M = 2-EN, K = 2-use F-RAN as a function of (factional size pe EN) and fo fonthaul capacity CF = log(p ). The tade-off has distinct egimes of opeations: (a) lowfonthaul capacity egime, with ; (b) high-fonthaul capacity egime, with >. sub-file, Fi, i =,..., N, and EN2 stoes the second subfile Fi2, fo all files i =,..., N. Fo any equest of distinct files, say Fi by use, and Fj by use 2, the tansmission poblem on the wieless channel is then equivalent to an Xnetwo with fou vitual messages, namely Fi, Fj available at EN, and Fi2, Fj2 available at EN2. Thus, we can use the intefeence alignment scheme poposed in [2], which achieves an equal ate of 2/3 log(p ) towads each of the two uses, ignoing o(log(p )) tems. The delivey time is, neglecting o(log(p )) tems, given by the edge tansmission time TE = 3L/2 log(p ), and the delivey time pe bit in Definition 2 is then appoximately = 3/(2 log(p )), yielding an achievable NDT of = 3/2. Inteestingly, as shown in Theoem, in the low-fonthaul egime when, if /2, the usage of the fonthaul esouce cannot futhe educe the NDT and this scheme achieves the minimum NDT. Achieving (, ) = + / fo = 0: The case = 0 coesponds to the setting in which the ENs have no caching capability. A finite NDT can hence only be achieved by using the fonthaul esouces. The fonthaul lins can be utilized in two distinct ways, efeed to hee as had- and soft-tansfe modes. With the had-tansfe mode, the cloud can diectly tansmit both equested files to each EN, and then the ENs can use the same fully coopeative zeo-focing appoach adopted above fo =. Since the fonthaul lins have capacities CF = log(p ) each, the fonthaul delivey time is TF = 2L/( log(p )), while the edge tansmission time, following the same aguments as fo the fist cone point, is appoximately TE = L/ log(p ). This yields an appoximate total delivey time pe bit of = ( + 2/)/(log(P )), and hence the achievable NDT = +2/. Howeve, had tansfe 2032
206 IEEE Intenational Symposium on Infomation Theoy tuns out to be suboptimal in this scenaio. The optimal NDT is in fact achieved though a soft-tansfe mode appoach typical of C-RAN (see, e.g., [3]): the cloud implements zeo-focing beamfoming and quantizes the esulting baseband signals [3]. Using a esolution of log(p ) bits pe downlin baseband sample, it can be shown that the effective SNR in the downlin scales popotionally to the powe P (see [3, Eq. (5)]). As a esult, this scheme entails a fonthaul latency T F that equals the edge latency T E of the zeo-focing beamfoming scheme, namely T E = L/(log(P )), multiplied by the time needed to cay each baseband sample on the fonthaul lin, namely log(p )/( log(p )), yielding the NDT =+/ (we efe the eade to [] fo details). V. CONCLUSIONS In this pape, we pesented an infomation-theoetic analysis of Fog Radio Access Netwos (F-RANs), an emeging wieless achitectue that encompasses both edge caching and cloud pocessing. The study aims at poviding a latency-centic undestanding of the degees of feedom, in the high-snr egime, of F-RAN netwos by accounting fo the available limited esouces in tems of fonthaul capacity, stoage sizes, as well as powe and bandwidth on the wieless channel. We detailed a geneal model and a novel pefomance metic, efeed to as Nomalized Delivey Time (NDT), which captues the wost-case delivey latency with espect to an ideal intefeence-fee system. Fo the special case of M =2edge nodes and K =2uses, we fully chaacteized the tade-off between the NDT and the fonthaul and caching esouces of the system. This esult eveals optimal caching-fonthaultansmission policies as a function of the system esouces. Ongoing wo focuses on extending the NDT tade-off to the moe geneal setting intoduced hee. REFERENCES [] E. Bastug, M. Bennis, and M. Debbah, Living on the edge: The ole of poactive caching in 5G wieless netwos, IEEE Communications Magazine, vol. 52, no. 8, pp. 82 89, Aug. 204. [2] A. Checo, H. L. Chistiansen, Y. Yan, L. Scolai, G. Kadaas, M. S. Bege, and L. Dittmann, Cloud RAN fo mobile netwos: a technology oveview, IEEE Communications Suveys & Tutoials, vol. 7, no., pp. 405 426, Fist Quate 204. [3] O. Simeone, A. Maede, M. Peng, O. Sahin, and W. Yu, Cloud Radio Access Netwo: Vitualizing Wieless Access fo Dense Heteogeneous Systems, AXiv e-pints, Dec. 205. [Online]. Available: http://axiv.og/abs/52.07743 [4] M. Peng, S. Yan, K. Zhang, and C. Wang, Fog Computing based Radio Access Netwos: Issues and Challenges, AXiv e-pints, Jun. 205. [Online]. Available: http://axiv.og/abs/506.04233 [5] M. A. Maddah-Ali and U. Niesen, Cache-aided intefeence channels, in Poc. IEEE Intenational Symposium on Infomation Theoy (ISIT), Hong Kong, June 205, pp. 809 83. [6] M. A. Maddah-Ali and U. Niesen, Cache-aided intefeence channels, AXiv e-pints, Oct. 205. [Online]. Available: http://axiv.og/abs/50.062 [7] A. Sengupta, R. Tandon, and O. Simeone, Cache Aided Wieless Netwos: Tadeoffs between Stoage and Latency, AXiv e-pints, Dec. 205. [Online]. Available: http://axiv.og/abs/52.07856 [8] Y. Liu and E. Eip, Completion time in multi-access channel: An infomation theoetic pespective, in Poc. IEEE Infomation Theoy Woshop, Paaty, Bazil, 20, pp. 708 72. [9], Completion time in boadcast channel and intefeence channel, in Poc. 49th Annual Alleton Confeence on Communication, Contol and Computing, Monticello, IL, 20, pp. 694 70. [0] S. A. Jafa, Intefeence alignment a new loo at signal dimensions in a communication netwo, Foundations and Tends in Communications and Infomation Theoy, vol. 7, no., pp. 34, 200. [Online]. Available: http://dx.doi.og/0.56/000000047 [] A. Sengupta, R. Tandon, and O. Simeone, Cloud and -aided wieless netwos: Fundamental latency tadeoffs, CoRR, vol. abs/605.0690, 206. [Online]. Available: http://axiv.og/abs/605.0690 [2] M. A. Maddah-Ali, A. S. Motahai, and A. K. Khandani, Communication ove MIMO X channels: Intefeence alignment, decomposition, and pefomance analysis, IEEE Tansactions on Infomation Theoy, vol. 54, no. 8, pp. 3457 3470, Aug. 2008. [3] O. Simeone, O. Someh, H. V. Poo, and S. Shamai, Downlin multicell pocessing with limited-bachaul capacity, EURASIP Jounal on Advances in Signal Pocessing, vol. 2009, p. 3, Feb. 2009. VI. APPENDIX A: LOWER BOUNDS ON NDT (CONVERSE) In this Appendix, we pesent the main idea behind the convese which povides a matching lowe bound on the achievable NDT, and leads to the optimal NDT tade-off in Theoem. The convese is based on the idea of consideing subsets of infomation esouces acoss the caching, fonthaul and wieless segments of the F-RAN, with the popety that the infomation in each subset is sufficient to decode one o moe of the equested files fo any sequence of feasible policies in the high-snr egime. In paticula, the fist subset encompasses caching, fonthaul and wieless esouces and yields a lowe bound on a linea combination of (T E,T F ) as a function of and. The second subset concens the caching and fonthaul esouces of F-RAN, yielding a lowe bound on T F as a function of and. The thid subset petains only to the wieless segment of F-RAN and yields a lowe bound on T E. As shown in [], we aive at the following inequalities coesponding to the thee mentioned infomation subsets: Inequality : (T E + T F ) log(p )+(T E + T F ) P (2 )L L L ; (7) Inequality 2: Inequality 3: T F log(p ) ( 2)L L L ; (8) T E log(p ) L L L. (9) Hee, L is a function that vanishes fo L!, and P is a function such that P / log(p )! 0 as P!(i.e., P is any o(log(p )) function). Recognizing that the NDT is defined by (T (, ) =lim P! lim E +T F )log(p) L! L, the inequalities in (7), (8), and (9) ae used to fist obtain lowe bounds on (T F + T E ) log(p )/L fo vaious fonthaul and size egimes. Taing the limits L!and then P!ove these bounds leads to the lowe bound on NDT pesented in Theoem (see [] fo full details). 2033