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Hierarchical Coded Caching

Nikhil Karamchandani, Urs Niesen, Mohammad Ali Maddah-Ali, Suhas Diggavi

arXiv:1403.7007v2cs.ITcs.NI

TL;DR

Hierarchical caching raises questions about optimal memory–rate tradeoffs, cross-layer coding, and whether minimizing one link rate worsens another. The paper balances within-layer and across-layer coded multicasting, approximating the feasible rate region within constant additive and multiplicative gaps while simultaneously minimizing both hop rates.

  • Problem

    Hierarchical caching lacks a characterization of memory–rate tradeoffs, the need for cross-layer coding, and whether reducing one layer’s rate increases another’s.

  • Method

    The scheme balances coded multicasting within individual layers against opportunities spanning both cache layers.

  • Results

    The proposed achievable region approximates the feasible two-layer rate region within constant additive and multiplicative gaps.

  • Takeaways & Limitations

    Both hop rates can be simultaneously approximately minimized, so the optimal hierarchical caching scheme has no rate tension up to those gaps.

Abstract

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Caching of popular content during off-peak hours is a strategy to reduce network loads during peak hours. Recent work has shown significant benefits of designing such caching strategies not only to deliver part of the content locally, but also to provide coded multicasting opportunities even among users with different demands. Exploiting both of these gains was shown to be approximately optimal for caching systems with a single layer of caches. Motivated by practical scenarios, we consider in this work a hierarchical content delivery network with two layers of caches. We propose a new caching scheme that combines two basic approaches. The first approach provides coded multicasting opportunities within each layer; the second approach provides coded multicasting opportunities across multiple layers. By striking the right balance between these two approaches, we show that the proposed scheme achieves the optimal communication rates to within a constant multiplicative and additive gap. We further show that there is no tension between the rates in each of the two layers up to the aforementioned gap. Thus, both layers can simultaneously operate at approximately the minimum rate.

I. INTRODUCTION · II. PROBLEM SETTING · III. PRELIMINARIES

The paper studies two-layer hierarchical coded caching, where caches support local and coded multicasting gains across links. It formalizes the feasible rate region and recalls the single-layer coded-caching scheme used in its construction.

  • I. INTRODUCTION: Hierarchical content delivery networks place caches in a tree, with parent caches communicating with children while minimizing transmission rates across layers.The motivating problem is reducing peak network load through content cached during off-peak hours.
  • I. INTRODUCTION: The central design question is whether reducing the rate in one layer necessarily increases the rate in another layer.The paper specifically investigates the memory–rate tradeoff and possible tension between the two layers.
  • I. INTRODUCTION: The proposed two-layer scheme combines single-layer coding within each layer with coding opportunities that operate across multiple layers.The first type operates between a node and its direct children on each layer’s link.
  • II. PROBLEM SETTING: The system has one server, K1 mirrors, K2 users per mirror, mirror memory M1F bits, user memory M2F bits, and K1K2 total users.The server–mirror and mirror–user connections are error-free broadcast links.
  • II. PROBLEM SETTING: Content delivery consists of placement during low traffic followed by delivery, with placement performed without prior knowledge of future user demands.Mirrors and user caches store content related to the N files while satisfying their memory constraints.
  • II. PROBLEM SETTING: A rate tuple is feasible when every user recovers its requested file for all request matrices with probability arbitrarily close to one for sufficiently large F.The feasible rate region consists of all rate pairs associated with feasible tuples for fixed M1 and M2.
  • III. PRELIMINARIES: The preliminaries reduce to a single-layer case with M2 = 0 and K2 = 1, where mirrors are the only caches and each mirror serves one user.In this case, R2 = K2 = 1 is necessary and sufficient, so the objective is minimizing R1.
  • III. PRELIMINARIES: Single-layer coded caching uses local availability and coded multicasting, achieving R1 = r(M/N, K), which is within a constant factor of the minimum rate.For K2 > 1, delivery extends through K2 stages with one unique user per mirror active in each stage.

IV. MAIN RESULTS

The paper constructs a feasible rate region for two-layer hierarchical caching by sharing memory between two coded-caching schemes. This region is within constant additive and multiplicative gaps of the optimal feasible region, while both hop rates can be simultaneously approximately minimized.

  • Construction: The proposed region RC(M1, M2) combines two achievable schemes by appropriately sharing memory during placement and file recovery during delivery.The parameters α and β determine the fractions of each file and memory allocated to the two schemes.
  • Optimality: For any memory sizes M1 and M2, RC(M1, M2) approximates the feasible rate region R⋆(M1, M2).Theorem 1 establishes RC(M1, M2) ⊆ R⋆(M1, M2) and R⋆(M1, M2) ⊆ c1 · RC(M1, M2) − c2, with finite positive constants independent of problem parameters.
  • Optimality: The analysis gives constants c1 = 1/60 and c2 = 16 for the additive and multiplicative approximation gaps.Numerical results suggest that the actual constants are much smaller.
  • Rate tradeoff: There is no tension between the first-hop rate R1 and second-hop rate R2 up to a constant additive and multiplicative gap.Some α⋆ and β⋆ simultaneously approximately minimize R1 and R2.
  • Rate tradeoff: In the single-mirror example with M1 = M2 = N/2, α = β = 1/2 achieves (R1, R2) ≈ (1/2, 1), reducing the second-hop rate to one.Compared with α = 1/2 and β = 0, the first-hop rate increases by 1/2 while the second-hop rate decreases from K2/2 to one.

V. CACHING SCHEMES · A. Caching Scheme A

The paper introduces two basic caching schemes for the two-layer hierarchical network, beginning with Scheme A, which independently applies single-layer placement and delivery across the mirror and user-cache layers. Scheme A uses decode-and-forward: mirrors decode attached users’ requested files, then re-encode them for delivery to those users.

  • V. CACHING SCHEMES: The proposed framework constructs two basic two-layer caching schemes from the BasePlacement and BaseDelivery procedures developed for single-layer networks.The schemes are later combined to obtain a near-optimal hierarchical caching scheme.
  • A. Caching Scheme A: Scheme A independently caches content in the mirrors and in the user caches attached to each mirror.Each layer uses the single-layer caching procedure separately during placement.
  • A. Caching Scheme A: During delivery, the server enables each mirror to decode all files requested by its attached users.The server applies BaseDelivery(N, K1, M1) over the server-to-mirror links.
  • A. Caching Scheme A: Each mirror then re-encodes the decoded files for its attached users using the single-layer BaseDelivery procedure.Users recover their requested files from the mirror transmission and their own cached content.
  • A. Caching Scheme A: Scheme A exploits coded multicasting opportunities both between the server and mirrors and between mirrors and their users.Its decode-and-forward operation applies coded caching separately across the two layers.
  • A. Caching Scheme A: The scheme’s performance is characterized by separate rates over the server-to-mirror links and the mirror-to-user links.The paper states these rates using the corresponding single-layer rate expressions.

B. Caching Scheme B · C. Generalized Caching Scheme

Caching scheme B applies single-layer coded caching between the server and users while leaving mirror memories unused, with mirrors forwarding relevant transmissions. The generalized scheme combines schemes A and B over disjoint file and cache-memory subsystems controlled by α and β, and adds their corresponding link rates.

  • B. Caching Scheme B: Scheme B ignores mirror memory and directly applies single-layer coded caching across the K1K2 users.The mirrors only forward server transmissions relevant to their attached users.
  • B. Caching Scheme B: During placement, each user independently stores a random M2F/N-bit subset of every file, while all mirrors remain empty.This uses BasePlacement(N, K1K2, M2) across the K1K2 user caches.
  • B. Caching Scheme B: During delivery, BaseDelivery(N, K1K2, M2) generates sums for all users, and each mirror forwards sums involving at least one file requested by its attached users.Users recover requested files from forwarded transmissions and their own cache contents.
  • B. Caching Scheme B: For N = 4, K1 = 2, K2 = 2, and M2 = 1, scheme B stores a random F/4-bit subset of every file in each user cache.The mirror memory M1 is irrelevant in this example.
  • B. Caching Scheme B: The example’s server transmission uses coded XOR sums of file parts, achieving normalized rate r(1/4, 4) over the server-to-mirrors link.The four users request files A, B, C, and D, respectively.
  • C. Generalized Caching Scheme: The generalized scheme divides the system into disjoint subsystems operated by schemes A and B, splitting each file into αF and (1 −α)F bits.The first subsystem uses all mirror memory and a β fraction of user memory; the second uses the remaining (1 −β) fraction.
  • C. Generalized Caching Scheme: The generalized scheme’s net rate on each transmission link is the sum of the corresponding rates contributed by schemes A and B.Scheme A serves the first file parts, while scheme B serves the second file parts.
  • C. Generalized Caching Scheme: The resulting link rates are R1 = R1(α, β) and R2 = R2(α, β), obtained by combining the rates of the two subsystems.The formal derivation of these rate expressions is provided in Appendix A.

D. Choice of α⋆and β⋆ · E. Achievable rates R1(α⋆, β⋆), R2(α⋆, β⋆)

The paper selects regime-dependent parameters α⋆ and β⋆ for the generalized caching scheme, then evaluates the resulting rates R1(α⋆, β⋆) and R2(α⋆, β⋆). These rates are shown to be within a constant multiplicative and additive gap of the corresponding minimum feasible rates.

  • D. Choice of α⋆and β⋆: The parameter choice (α⋆, β⋆) is selected separately for three memory regimes defined by M1 and M2.The regimes are I: M1 + M2K2 ≥ N and 0 ≤ M1 ≤ N/4; II: M1 + M2K2 < N; and III: M1 + M2K2 ≥ N and N/4 < M1 ≤ N.
  • D. Choice of α⋆and β⋆: In regime I, the scheme balances optimization of both link rates, choosing α⋆ and β⋆ to make the second-link rate approximately optimal.Setting α = β yields R2(α, β) = r(M2/N, K2), which is approximately optimal for that link.
  • D. Choice of α⋆and β⋆: In regime II, the optimizing parameters are (α⋆, β⋆) = (M1/(M1 + M2K2), 0), prioritizing the first-link rate because R2 is necessarily on the order of K2.With β⋆ = 0, no user cache memory is assigned to caching scheme A.
  • D. Choice of α⋆and β⋆: In regime III, β is thresholded at 1/4 to limit its effect on R1 while still reducing R2 sufficiently.The resulting scheme exploits coded multicasting within each mirror-to-user group and across the server-to-user connection.
  • E. Achievable rates R1(α⋆, β⋆), R2(α⋆, β⋆): The achievable rates R1(α⋆, β⋆) and R2(α⋆, β⋆) are calculated from the generalized scheme’s rate expressions involving r(·, ·).The evaluation proceeds separately because (α⋆, β⋆) takes different values in the three memory regimes.
  • E. Achievable rates R1(α⋆, β⋆), R2(α⋆, β⋆): In regime I, (α⋆, β⋆) = (M1/N, M1/N), and the resulting R1(α⋆, β⋆) and R2(α⋆, β⋆) are upper bounded using the scheme’s rate formulas.Regime I satisfies M1 + M2K2 ≥ N and 0 ≤ M1 ≤ N/4.
  • E. Achievable rates R1(α⋆, β⋆), R2(α⋆, β⋆): In regime II, (α⋆, β⋆) = (M1/(M1 + M2K2), 0), and both achievable rates receive upper bounds under M1 + M2K2 < N.The derivation uses M1 < α⋆N, M2 < (1 − α⋆)N, and M2K2 < N.
  • E. Achievable rates R1(α⋆, β⋆), R2(α⋆, β⋆): In regime III, (α⋆, β⋆) = (M1/N, 1/4), and combining the regime-specific bounds yields upper bounds for both achievable rates.These bounds, including the stated regime-III term 3NM2, are used to establish the final approximation guarantee.

VI. PROOF OF THEOREM 1 A. Proof of RC(M1, M2) ⊆R⋆(M1, M2) · B. Proof of R⋆(M1, M2) ⊆c1 · RC(M1, M2) −c2

The proof establishes both inclusions needed for Theorem 1: every achievable rate pair in RC(M1, M2) lies in R⋆(M1, M2), while every feasible pair is within constant multiplicative and additive gaps of RC(M1, M2). It further shows that both hop rates can be simultaneously approximately minimized, with no tension between them up to these gaps.

  • VI. PROOF OF THEOREM 1 A. Proof of RC(M1, M2) ⊆R⋆(M1, M2): Any rate pair in RC(M1, M2) is achievable using the generalized caching scheme, proving RC(M1, M2) ⊆ R⋆(M1, M2).This follows from the achievable-rate result in Section V-C.
  • B. Proof of R⋆(M1, M2) ⊆c1 · RC(M1, M2) −c2: The converse first derives lower bounds Rlb 1 (M1, M2) and Rlb 2 (M1, M2) on the feasible rates.The lower-bound derivation is provided in Appendix B.
  • B. Proof of R⋆(M1, M2) ⊆c1 · RC(M1, M2) −c2: It then bounds the gaps between achievable rates R1(α⋆, β⋆), R2(α⋆, β⋆) and the corresponding lower bounds.The gap analysis covers several regimes of M1 and M2 and is deferred to Appendices C and D.
  • B. Proof of R⋆(M1, M2) ⊆c1 · RC(M1, M2) −c2: The constants c1 and c2 are finite, positive, and independent of all problem parameters.These constants quantify the multiplicative and additive gap between the achievable region and the lower bounds.
  • B. Proof of R⋆(M1, M2) ⊆c1 · RC(M1, M2) −c2: Combining the lower bounds and gap estimates shows that every feasible rate pair lies in c1 · RC(M1, M2) −c2.The argument applies for any M1, M2 and every (R1, R2) ∈ R⋆(M1, M2).
  • B. Proof of R⋆(M1, M2) ⊆c1 · RC(M1, M2) −c2: Because RC(M1, M2) consists precisely of rate tuples (R1(α, β), R2(α, β)) for α, β ∈ [0, 1], the inclusion completes the converse proof.This identifies the achievable region used in the final containment argument.
  • B. Proof of R⋆(M1, M2) ⊆c1 · RC(M1, M2) −c2: For any M1 and M2, parameters α⋆ and β⋆ make both R1(α⋆, β⋆) and R2(α⋆, β⋆) simultaneously approximately close to their minimum values.This is stronger than the theorem statement.
  • B. Proof of R⋆(M1, M2) ⊆c1 · RC(M1, M2) −c2: Up to a constant additive and multiplicative gap, the optimal caching scheme has no tension between rates over the first and second network hops.Both layers can therefore operate near their respective minimum rates under the stated gap.

APPENDIX A RATES FOR THE GENERALIZED CACHING SCHEME · APPENDIX B LOWER BOUNDS · A. Rate R2

Appendix A derives the two subsystem rate expressions for the generalized caching scheme, while Appendix B establishes cut-set lower bounds on R1 and R2 through carefully chosen request matrices and cache-memory accounting.

  • APPENDIX A RATES FOR THE GENERALIZED CACHING SCHEME: The generalized caching scheme’s rates are derived separately for two subsystems, yielding expressions (11) and (12).The first subsystem uses caching scheme A, and the second uses caching scheme B.
  • APPENDIX A RATES FOR THE GENERALIZED CACHING SCHEME: The first subsystem caches and delivers the first α fraction of each file, using all mirror memory and the first β fraction of each user cache.Its equivalent file size, mirror memory, and user-cache memory are normalized by the equivalent file size as α.
  • APPENDIX A RATES FOR THE GENERALIZED CACHING SCHEME: The second subsystem caches and delivers the remaining 1 −α fraction of each file, using only the second 1 −β fraction of each user cache.Its rates are normalized by the file size F.
  • APPENDIX B LOWER BOUNDS: Appendix B seeks lower bounds on R1 and R2 for any feasible tuple (M1, M2, R1, R2), using an argument similar to a single-layer caching bound.The construction considers arbitrary choices of s1 and s2 and users indexed across the two cache layers.
  • APPENDIX B LOWER BOUNDS: Two distinct request matrices show that users can recover different requested files from the same cache contents and server transmission rate R1.The cache contents remain fixed across request matrices, although the server transmission may depend on the request matrix.
  • APPENDIX B LOWER BOUNDS: The resulting cut-set accounting combines server transmissions, mirror memories, and user memories against the distinct files reconstructed across repeated request matrices.The bound uses ⌊N/(s1s2)⌋ request matrices and can be rewritten into inequalities that hold for every allowed s1 and s2.
  • A. Rate R2: For rate R2, the lower-bound argument restricts attention to t users attached to mirror one and repeats two request patterns across ⌈N/t⌉ request matrices.Each user recovers its requested file from mirror-one transmission of rate R2 and its cache of size M2, with cache contents fixed across request matrices.
  • A. Rate R2: Because the resulting inequality holds for every t ∈{1, 2, . . . K2}, it yields a lower bound on R2 for any feasible tuple (M1, M2, R1, R2).The appendix states this lower bound as equation (26).

APPENDIX C GAP BETWEEN ACHIEVABLE RATE R1(α⋆, β⋆) AND LOWER BOUND Rlb

The generalized caching scheme’s first-hop achievable rate R1(α⋆, β⋆) is within a constant additive and multiplicative gap of the lower bound Rlb_1(M1, M2) across regimes I and II.

  • Regime I: In regime I, R1(α⋆, β⋆) and Rlb_1(M1, M2) are within a constant multiplicative and additive gap.The proof covers all cases through subregimes I.A, I.B, and I.C.
  • Regime II: Regime II is defined by M1 + M2K2 < N, and its subregimes consider M2 < N/4.The analysis covers the resulting cases using separate lower-bound parameter choices.
  • Regime II: In regime II, R1(α⋆, β⋆) and Rlb_1(M1, M2) are within a constant multiplicative and additive gap.The proof combines bounds from all cases in Sections II.A–II.G.

Regime III: M1 + M2K2 ≥N, N

In Regime III, the achievable rate R1(α⋆, β⋆) is upper bounded, and the regime is divided into two subregimes to characterize its gap from lower bounds.

  • The achievable rate R1(α⋆, β⋆) is upper bounded in Regime III.
  • The regime condition is M1 + M2K2 ≥ N, N.
  • Regime III is divided into two subregimes to characterize the gap between the achievable rate and the lower bounds.

III.A) N … III.B) N

The analysis covers subregimes within regime III and combines their bounds to show that the achievable rate is within constant multiplicative and additive gaps of the lower bound. Since regimes I–III exhaust all cache-size pairs, the generalized caching scheme is approximately optimal for every feasible configuration.

  • III.B) N: The proof analyzes the subregimes one by one within regime III.
  • III.A) N: The III.A) N analysis selects parameters satisfying the lower bound and substitutes them into the bound.
  • III.A) N: Using (46), N ≥K1K2, and K1 ≥4, the III.A) N analysis combines the resulting inequality with (45).
  • III.B) N: For the considered III.B) N case, the proof combines (45) with the condition (N −M1)/M2 ≤2.
  • III.B) N: The two sections cover all cases in regime III, yielding constant multiplicative and additive gaps between R1(α⋆, β⋆) and Rlb 1 (M1, M2).
  • III.B) N: Regimes I, II, and III cover all possible values of (M1, M2), so the generalized caching scheme’s R1 is within constant additive and multiplicative gaps of the minimum feasible rate.

APPENDIX D GAP BETWEEN ACHIEVABLE RATE R2(α⋆, β⋆) AND LOWER BOUND Rlb

Appendix D establishes that the generalized caching scheme’s rate R2(α⋆, β⋆) is within a constant additive and multiplicative gap of the lower bound Rlb2(M1, M2) for all memory sizes. The proof compares achievable and lower-bound expressions by considering two cases, focusing on K1, K2 ≥ 4.

  • Assumptions: The proof focuses on K1, K2 ≥4, while cases with K1 < 4 or K2 < 4 are handled using results from.The stated restriction applies because the smaller-parameter cases are considered readily analyzable from prior results.
  • Case analysis: The gap analysis studies two different cases that together cover all memory sizes M1 and M2.The appendix analyzes the cases one by one before combining their resulting bounds.
  • Gap characterization: R2(α⋆, β⋆) and Rlb2(M1, M2) are within a constant multiplicative and additive gap for all M1, M2.This conclusion follows by combining the bounds established for the two cases.
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