Rate region for a class of delay mitigating codes and P2P networks

Transcription

1 Rate region for a class of delay mitigating codes and P2P networks Steven Weber, Congduan Li, John MacLaren Walsh Drexel University, Dept of ECE, Philadelphia, PA Abstract This paper identifies the relevance of a distributed source coding problem first formulated by Yeung and Zhang in 1999 to two applications in network design: i) the design of delay mitigating codes, and ii) the design of network coded P2P networks When transmitting time-sensitive frames from a source to a destination over a multipath network using a collection of coded packets, the decoding requirements determine which subsets of packets will be sufficient for decoding which frames The rate region of packet sizes consistent with these requirements is shown to be an instance of the aforementioned distributed source coding problem When encoding file chunks into packets in a peer to peer system, the peers wish to receive their chunks as soon as possible while uploading data at as low a rate as possible It is shown that the region of encoded packet sizes consistent with the decoding constraints is another instance of the aforementioned distributed source coding problem These rate regions are placed in the larger context of rate-delay tradeoffs in designing delay mitigating codes and efficient P2P systems I INTRODUCTION This paper identifies the relevance of a distributed source coding problem first formulated by Yeung and Zhang in 1999 [1] to two applications in network design: i) the design of delay mitigating codes, and ii) the design of network coded P2P networks The paper is organized as follows II discusses a broad array of related work in source and network coding (comprising much of the paper), focusing in particular on i) Yeung and Zhang s distributed source coding problem [1] in II-A, ii) multilevel diversity coding [2] in II-B, iii) the multi-source network coding problem (with independent sources) as formulated by Yeung in his book [3] (Chapter 21) in II-C, iv) the Han and Kobayashi multiterminal source coding problem [4] in II-D, v) multi-source network coding with correlated sources (eg, [5]) in II-E, and vi) categorizing these problems in II-F III discusses the design of delay mitigating codes we highlight the inherent rate delay tradeoff, and identify the rate region as an instance of [1] IV discusses the design of network coded P2P networks with decoding constraints we identify key design issues and tradeoffs, and identify the rate region as an instance of [1] There are no new theorems in this paper the contribution is the identification of the relevance of these various source and network coding problems for a structured class of rate delay codes and P2P network coded networks II RELATED WORK A Yeung and Zhang s distributed source coding problem Yeung and Zhang (1999) [1] consider the distributed source coding (YZDSC) problem in Fig 1 Their motivation is satellite communications where encoders E at satellites connect sources S with destinations D The problem consists of: 1) A set of independent sources (with index set S) with each source S j holding an iid sequence of discrete random variables (RVs) {X ji } with specified entropy H(X j ) The RV Y j is associated with source S j in the analysis 2) A set of encoders (with index set E) where encoder E l has access to a subset U l Sof the sources The source to encoder mapping is specified by the relation A S E The RV Z l is associated with encoder E l in the analysis Each encoder operates at rate R l in that H(Z l ) R l, and the rate vector is R =(R l,l E) 3) A set of decoders (with index set D) where decoder D m has access to a subset V m E of the encoded messages The encoder to decoder mapping is specified by the relation B E D Decoder D m is to decode a subset F m Sof the sources A rate vector R is admissible if for arbitrary error probability bound >0there exists a sufficiently large blocklength n such that a code exists with encoder rates n 1 log η l R l + for l S, and decoder Hamming error probabilities m, for m D Here η l is the number of codewords available to encoder l and m is the Hamming error probability for decoder m, cf [1] The rate region is the set of admissibile rate vectors: R = {R : R is admissible} Y j H(X j ) Z l S j S U l A S E R l E l E V m B E D D m D F m S Fig 1 Yeung and Zhang s distributed source coding problem [1] Yeung and Zhang express the rate region R in terms of the region of entropic vectors, Γ N, cf [3] (Chapters 13 16) The entropy vector h R 2N 1 for a finite set of N discrete RVs (W 1,,W N ) (with arbitrary joint distribution) has a component h G = H(W i,i G) for each nonempty G {1,,N} (note there are 2 N 1 such Gs) In turn, an arbitrary vector h R 2N 1 is said to be entropic if there exist a joint distribution on N discrete RVs with h as the corresponding entropy vector

2 Collect the RVs for sources and encoders as N = {(Y j,j S) (Z l,l E)}, N = N (1) Define the region R in as the set of rate vectors R such that there exists h Γ N satisfying h Yj,j S = j S h Yj (2) h Zl (Y j,j U l ) = 0, l E (3) h (Yj,j F m) (Z l,l V m) = 0, m D (4) h Yj > H(X j ), j S (5) h Zl R l, l E (6) where expressions involving conditional entropies are understood as the corresponding unconditioned entropy vector inequalities ([3] Chapter 13) These constraints admit natural interpretations: (2) requires the sources be independent; (3) requires each codeword is a function of the available inputs; (4) requires each decoder is able to recover its intended subset of sources; (5) requires each source exceed its rate constraint; and (6) requires each encoder to obey its rate constraint Yeung and Zhang prove R in is an inner bound They also prove an outer bound R out obtained from R in by i) relaxing the strict inequality in (5) to allow for equality (h Yj H(X j ) for j S), and ii) expanding Γ N to include its closure, Γ N Theorem 1 ([1]) R in R R out The outer bound was later shown to be tight for the more general problem discussed in II-C B Multilevel diversity coding (MDC) Multilevel diversity coding (MDC) is the problem of taking several encodings of a set of sources (each encoder with its own rate), and asking each of several decoders (each with access to a specific subset of the various encodings) to decode a specified subset of the sources Applications have been found in distributed storage, network reliability, secret sharing, and satellite communications MDC was first introduced in 1992 in the thesis of Roche [6] Early seminal papers include Yeung (1995) [7], Roche, Yeung, and Hau (1995) [8], [9], and Yeung and Zhang (1999) [2] The connection between MDC and the multiple descriptions problem is explored in [7], Fu and Yeung (2002) [10], as well as in the more recent work of Mohajer, Tian and Diggavi (2008) [11], [12] Fig 2 shows the symmetric MDC (SMDC) problem ([9], [2]) with K co-located iid sources Y 1,,Y K (each Y k representing an iid sequence {X ki }) that are prioritized in that Y k is deemed more important than Y k+1 for each k [K 1] These sources are made available to K encoders E 1,,E K, with rates R 1,,R K There are 2 K 1 decoders one for each nonempty subset of [K] The K k decoders corresponding to subsets of [K] of size k each seek to decode sources Y 1,,Y k, for each k [K] The source prioritization is reflected in the fact that source Y k is recoverable from any decoder of level k or higher Y 1 Y K iid S R 1 E 1 R k E l E K { K decoders at level 1 K k { R K { Y 1 Y 1 decoders at level k Y 1 Y k Y 1 Y k 1 decoder at level K Y 1 Y K Fig 2 Yeung & Zhang s symmetric multilevel diversity coding problem [2] It is shown in [2] that superposition coding suffices to achieve any point in the SMDC rate region Superposition coding in this context means each encoder E k with rate R k separately encodes each of the K sources with rates (rk 1,,rK k ) such that l K rl k = R k [7] ( IIB) The symmetry in SMDC refers to the fact that all decoders at level k recover the same subset of the sources, ie, Y 1,,Y k The optimality of superposition coding is dependent upon the symmetry of the problem A simple counter-example showing the inadequacy of superposition coding for general MDC is shown in [7] ( IIB), while [13], [14] identifies that superposition coding is optimal for 86 of the 100 MDC configurations with three encoders The asymmetric (AMDC) problem of encoding K sources with three encoders and 2 K 1 decoders, where the decoding capability depends upon the particular subset of encodings (ie, not just the cardinality of that set), is studied in [11], [12] They also find superposition coding is in general suboptimal (although linear network coding suffices) 1 The SMDC rate region is given below Theorem 2 ([2]) The rate region R for SMDC with K encoders is the set of R =(R 1,,R K ) such that K R k = rk, l k [K], (7) l=1 for some (rk l 0,l [K],k [K]) satisfying r D k H(X D ), D [K], D = (8) k D The (rk l ) are the superposition coding vectors (7) states the superposition coding vector must cover the rate constraint at each encoder, and (8) states the decoder associated with subset D should receive sufficient rate to decode X D Decoder D is able to decode X k for k< D due to the corresponding constraints for decoders D with D = k This result generalizes the explicit rate region for SMDC with 3 encoders given in [9] An explicit rate region for AMDC with 3 encoders is given in [12] The AMDC problem is an instance of the YZDSC problem in II-A; of course the same is also true of the SMDC problem 1 The problem definitions of AMDC appear to be slightly different in [10] (Fig 8) and [12] (Fig 1)

3 C Multi-source network coding A key structural property of the YZDSC problem in II-A is the absence of intermediate relay nodes between the encoders and decoders The multi-source network coding (MSNC) problem generalizes the YZDSC problem by allowing for an arbitrary topology of intermediate encoding nodes between the source encoders and decoders Notably, the MSNC problem retains the assumption of independent sources, at least as formulated by Yeung [3] (Chapter 21), whose presentation and notation we follow below An acyclic graph G =(V, E) has two disjoint subsets S, T representing sources and destinations The source RVs (Y s,s S) are independent with entropies H(Y s ) ω s for each s S Each destination t T requests a subset β(t) Sof the sources Each edge e E has an associated RV U e with an entropy H(U e ) R e The following constraints hold: H(Y s ) ω s, s S (9) H(Y S ) = s S H(Y s ), H(U Out(s) Y s ) = 0, s S H(U Out(i) U In(i) ) = 0, i V\(S T) H(U e ) R e, e E H(Y β(t) U In(t) ) = 0, t T These constraints model i) the source rates obey the constraints ω, ii) the sources are independent, iii) the messages sent on edges Out(s) (outgoing from source s) are functions of Y s, iv) the outgoing messages on edges Out(i) are functions of the incoming messages on edges In(i) at each intermediate node i, v) the edge message variables obey the edge rate constraints R, and vi) each destination t is able to recover its subset of source messages β(t) from the messages incoming to t Collect the RVs for the problem as the set N = {(Y s,s S) (U e,e E)} (10) with N = N, and let Γ N be the region of entropic vectors for these variables Yeung [3] (Theorem 215) gives the rate region R for the sources ω =(ω s,s S) as the intersection of Γ N with the constraints in (9) Theorem 3 ([3]) The rate region R for the MSNC problem is the set of ω =(ω s,s S) such that there exist RVs N in (10) in Γ N satisfying (9) Several technical caveats hold, see [3] for a more precise statement of the theorem The rate regions for both the YZDSC and MSNC problems are in terms of the region of entropic vectors Γ N (more precisely, its closure, Γ N ) Although Γ N is known to be a convex cone ([3] Theorem 155), it has not been characterized for N 4 RVs In fact for N 4 it is known that Γ N is curved and hence cannot be described by any finite set of linear inequalities [15] Consequently we must employ computable inner and outer bounds on Γ N The Shannon linear programming outer bound Γ N is the intersection of Shannon s classical information inequalities, which is equivalent to the nonnegativity of conditional mutual information I(X A ; X B X C ) 0, A, B, C N (11) for all (possibly empty and possibly overlapping) subsets A, B, C of N The important fact that the Shannon LP outer bound is loose is due to Zhang and Yeung [16] Ingleton s inequality [17], in its information theoretic form [16] (p 1444), states that for N =4RVs I(X 1 ; X 2 )+I(X 3 ; X 4 X 1 )+I(X 3 ; X 4 X 2 ) I(X 3 ; X 4 ) 0 (12) There are six nonredundant permutations of this inequality Zhang and Yeung prove that the intersection of the Shannon LP outer bound with Ingleton s inequality gives an inner bound on Γ N for N =4variables; see [18] for a discussion Finally, linear network coding is in general suboptimal in that there exist MSNC problems for which the rate region achievable by linear network coding is a loose inner bound [19] Y s s ω s S In(i) i e U e R e Out(i) t β(t) Fig 3 Yeung s formulation of the multi-source network coding problem [3] D Han and Kobayashi multiterminal source coding The YZDSC problem in II-A is similar to the Han and Kobayashi (1980) multiterminal source coding (HKMSC) problem [4], shown in Fig 4 A collection of correlated sources (Y e,e E) is made available to a collection of encoders E each source to exactly one encoder The encoders have rates R = (R e,e E), and each encoder sends its message to a subset of decoders D Each decoder D m receives a subset of encoded messages, and must decode a particular source Y k(m), where k(m) is the index of the encoder of one of the received messages This problem generalizes the YZDSC problem in that it relaxes the assumption of independent sources One may observe two structural differences between YZDSC and HKMSC: i) in YZDSC each encoder receives a subset of the source messages whereas in HKMSC each encoder receives a single source message, and ii) in YZDSC each decoder is responsible for a subset of source messages whereas in HKMSC each decoder must decode a single source message In both cases we can always construct an instance of HKMSC for any instance of YZDSC Namely, i) for every encoder E l in YZDSC, create an encoder E e in HKMSC with the source Y e =(Y j,j U l ) (recall Fig 1), and ii) for every source s F m to be recovered by decoder D m in YZDSC create a decoder (d, s) for Y s in HKMSC with the same set T

4 of incoming edges V m as D m 2 Thus we see YZDSC is in fact a generalization of HKMSC on account of allowing for correlation among the sources The main result in [4] is an achievable region (the Berger- Tung inner bound [20], [21]) that is tight for all known sub-cases for which the rate region is known (Theorem 2) Further, it is worth emphasizing that the achievable region is defined as the union of a collection of half-spaces, where each half-space is defined over all collections of message variables (the (U e,e E) in [4]) obeying certain distribution constraints Taking a union over all possible collections of random variables is equivalent to requiring membership in Γ N, the region of entropic vectors Y e R e E e E Y D k(m) m Fig 4 The Han and Kobayashi multiterminal source coding problem [4] E Multi-source network coding with correlated sources Finally, we briefly mention more recent work on multisource network coding with correlated sources (MSNCCS) Key references include Ho, Médard, Effros, Koetter (2004) [22], Ho et al (2006) [5], Barros and Servetto (2006) [23], Ramamoorthy, Jain, Chou and Effros (2006) [24], Han (2010) [25], and Ramamoorthy (2011) [26] As a representative significant result we mention [5] (cf [26]) Consider a capacitated directed acyclic graph G with a collection of correlated sources (Y s,s S) (with rates R =(R s,s S)) and assume a set of terminals T such that each terminal wishes to receive all sources This last assumption is critical for what follows Ignoring the network for a minute, the rate region R is the contra-polymatroid Slepian Wolf region [27], ie, those R obeying the following 2 S 1 linear inequalities: s S R s H(Y S Y c S ), S S (13) The capacity region is determined by the cutsets in G between each subset of sources and each terminal Namely, it is the set of R obeying the following 2 S 1 linear inequalities: R s min c G(S,T t ), S S, (14) t T s S where c G (S,T t ) is the min cut in G between sources S and terminal T t Theorem 4 ([5]) If there exists a rate vector R satisfying both the Slepian-Wolf constraints (13) and the cutset constraints 2 Some care must be taken in this construction, the details are omitted D (14) then random linear network coding suffices for the terminals to recover the sources F Classification of related work Here is a coarse classification of these problems Let intermediate denote that there are intermediate nodes that re-encode encoded messages, while no intermediate means encoded source messages are sent directly to decoders Let prioritized ind mean the sources are independent and prioritized into levels; independent refers to independent sources, and correlated refers to general correlated sources no intermediate intermediate prioritized ind SMDC ( II-B) independent YZDSC ( II-A) MSNC ( II-C) correlated HKMSC ( II-D) MSNCCS ( II-E) Note the rows are increasingly general from top to bottom and the columns are increasingly general from left to right Thus SMDC is an instance of YZDSC; YZDSC is an instance of both MSNC and HKMSC, and both MSNC and HKMSC are instances of MSNCCS Our focus in the remainder of this paper is on the YZDSC problem, highlighting structural reasons when restrictions to SMDC instances are of interest III DESIGN OF DELAY MITIGATING CODES AND RATE DELAY TRADEOFFS Consider the problem faced by a source wishing to transmit time-sensitive data to a destination over a multipath network The source consists of K temporally ordered independent B- bit frames (F k,k [K]), which are to be encoded into N packets (C n,n [N]) Each packet is to be sent over one of P paths between the source and destination peer At the sink, the desired frame playback schedule and the time at which frames may actually be decoded is quantified with a delay measure The key decision variables under control of the source are: the time t n R + at which the nth encoded packet C n is sent Collect these into a vector t the path p n P along which the nth encoded packet is sent Collect these into a vector p the subsets of received packets from which each frame may be recovered: i k 1, A = Fk can be determined from packets A 0, else (15) Collect these into a vector i =(i k A,k [K], A [N]) the sizes s =(s n,n [N]) of the encoded packets The sizes must lie in S(i), the region of packet sizes compatible with the decoding requirements i Given t, p, s, the network determines the series of random packet arrival times a = (a n,n [N]) according to the probability distribution p(a t, p, s) From a and i one can determine the first times d =(d k,k [K]) at which each source frame k and all frames prior to it are decodable at the destination We can compare these times with a desired playback schedule b =(b k,k [K]) of times the destination would like to decode these ordered frames While alternative

5 schedules may be interesting, in the framework of rate delay tradeoffs we will select b k = k R where R is the rate of playback in source frames per second One then selects a delay metric based on the difference between these desired playback times and when the associated frames are first available (d k b k ) + Viable options include the i) mean sum delay and ii) outage sum delay: D EΣ = E (d k b k ) + (16) k [K] D Σ = min D : P (d k b k ) + >D (17) k [K] and the iii) mean worst delay (DÊ), and iv) outage worst delay (Dˆ ), defined by replacing the sum with the max in (16) and (17), respectively These depend on decision variables t, p, i, s and frame rate R The rate delay tradeoff 3 in this context is the tradeoff between the frame rate R in frames per second (equivalently the data rate RB in bits per second) and the minimum delay: D o (R) = min D o(r, t, p, i, s), o {EΣ, Ê,Σ, ˆ} (18) t,p,i,s Here the minimum is take over all viable values of the control variables: t R N +, p [P ] N, i {0, 1} K2N, s S(i) When set up this way, this problem breaks up naturally into the following sub-problems Information theory part: find region S(i) for each i Coding part: given an i and s S(i), build a code with a low complexity encoder and decoder achieving the decoding requirements in i with packet lengths s Modeling and estimation part: determine a family of statistical models p(a t, p, s) appropriate for the networks of interest and provide estimators of their parameters Optimization part: given region S(i) from the information theory part, and model p(a t, p, s) from the modeling and estimation part, perform the optimization in (18) We now make a useful observation about S(i) Observation 1 The region S(i) is the rate region of a particular instance of the YZDSC problem in II-A In the specific case where K = N and i is such that frames 1,,k are recoverable from any subset of packets of size k, then the region S(i) is that of the SMDC problem in II-B Given i, we construct the corresponding instance of the YZDSC problem as follows There are K sources S 1,,S K and N encoders E 1,,E N All source variables Y 1,,Y K are available to all encoders, ie, A in Fig 1 is the complete bipartite graph connecting S to E Equivalently, there is a single source S with variables Y 1,,Y K connected to N encoders, as in Fig 2 There are 2 N 1 decoders, one for each nonempty subset of [N], and B in Fig 1 is configured accordingly Each decoder D m (with associated subset D m [N]) 3 A special issue on rate delay tradeoffs has recently appeared [28] seeks to recover the source variables (frames): {Y k : i k A =1, A D m } (19) The particular case where the rate region S(i) is that of the SMDC problem was addressed in our earlier work [29] We observe each i is an instance of YZDSC with a specific structure that conceivably may yield a more explicit representation of S(i) than that of Theorem 1 This is the subject of our current work It does not seem likely that superposition coding will achieve the entire rate region S(i) based on the examples demonstrating the suboptimality of superposition coding for non-symmetric MDC IV DESIGN OF NETWORK CODED P2P NETWORKS WITH DECODING CONSTRAINTS A peer to peer (P2P) file sharing system must be designed with several important metrics and constraints in mind, including: i) peers wish to obtain their desired chunks/files in a minimal amount of time, and ii) peers wish to minimize their required upload rate These two desires are in tension with one another Hence it is of interest to design P2P file sharing systems to optimally tradeoff between these two metrics [30] We consider the problem of optimally trading rate for delay in a class of P2P file sharing systems utilizing a restricted form of network coding The problem is formulated from a centralized omniscient standpoint, and it is shown that the overall problem of trading rate for delay decomposes naturally into several decoupled parts The solution of one of these parts is linked to an important result in distributed source coding As stated, the form of network coding we exploit for the P2P system is restricted in that we assume an overlay network directly connecting peers, ie, the encoded packets traversing the multi-hop paths underyling the overlay network are not encoded by intermediate routers, as in network coding Rather than being done for theoretical convenience, this model restriction is done with practical several system level issues in mind, in particular: Security: allowing for coded files/chunks to be combined/reencoded at intermediate network nodes, as in network coding, increases the risk of malicious or accidental file corruption Incentives: peers have a stronger incentive to obtain and then share chunks/files they can decode than to obtain and share chunks/files they cannot decode In particular, let us consider a model for a P2P file sharing system utilized for the dissemination of a certain collection of source file chunks F 1,,F K (these chunks may be chunks from several files) Peer p seeks file chunks with indices D(p) [K] by communicating with other peers over time Time is slotted, and at the beginning of any given time slot t, peer p has a subset of chunks H(p, t) D(p) it has decoded thus far During the time slot, it can encode the chunks it has into messages sent to other peers In particular, let the set of peers that peer p uploads messages to during time slot t be P(p) and the associated size of the message that p sends to p P(p, t) be R p,p (t) Naturally, the

6 uploaded amount of information between peer p and p can not exceed the amount of free capacity between these peers C p,p (t) during this time slot, ie R p,p (t) C p,p (t) The messages received during time slot t are then decoded, and peer p then can start the next time slot with a new, possible larger, set of decoded chunks H(p, t + 1) The delay p that a peer experiences is naturally the minimum t such that H(p, t) =D(p) The aggregate upload rate from peer p in time slot t is naturally U p (t) = p P(p,t) R p,p(t) The simplest model reflecting the desire to share the load of uploading files and minimizing the upload rate would choose the maximum (R = max p,t U p (t)) or total (R = p,t U p(t)) peer upload rate as the metric The delay experienced by different peers can similarly be aggregated into either a maximum or average delay One view of designing efficient P2P systems is then optimally trading these two aggregrate metrics R and against one another The centralized problem of trading rate for delay in peer to peer networks utilizing this class of coding schemes can be broken up into several parts: System monitoring part: monitor the network to determine D(p) the desired chunks for peer p and conservative estimates of C p,p (t) the free capacities between peers p, p in time t Information theory part: find the region R(t, {P(p, t), H(p, t), H(p, t + 1)}) of rates {R p,p (t)} such that peers have chunks {H(p, t)} and {H(p, t + 1)} at the beginning and end of the t time slot, respectively Coding part: given rates [R p,p (t)] design low complexity codes with these rates capable of satisfying the requirements reflected by {H(p, t)} and {H(p, t + 1)} and P Optimization part: given R(t, {P(p, t), H(p, t), H(p, t + 1)}) from the information theory part, and D(p) and C p,p (t) from the network monitoring part, select {P(p, t), H(p, t), H(p, t + 1)} and R p,p (t) Rto minimize the rate metric R subject to a bound on the delay metric We make an observation about the information theory part Observation 2 The region R(t, {P(p, t), H(p, t), H(p, t + 1)}) is an instance of the YZDSC problem ( II-A) In particular, i) add a source variable Y k for each chunk F k, ii) add an encoder for each peer p transmitting in t, iii) add an edge from each chunk to each peer consistent with H(p, t), iv) add a decoder for each peer p receiving in t, v) connect encoders with decoders specified by P(p, t), vi) require each decoder to recover source variables consistent with H(p, t+1) REFERENCES [1] R Yeung and Z Zhang, Distributed source coding for satellite communications, IEEE Transactions on Information Theory, vol 45, no 4, pp , May 1999 [2], On symmetrical multilevel diversity coding, IEEE Transactions on Information Theory, vol 45, no 2, pp , March 1999 [3] R W Yeung, Information Theory and Network Coding Springer, 2008 [4] T S Han and K Kobayashi, A unified achievable rate region for a general class of multiterminal source coding systems, IEEE Transactions on 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