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Multi-Server Coded Caching

Seyed Pooya Shariatpanahi, Seyed Abolfazl Motahari, Babak Hossein Khalaj

arXiv:1503.00265v1cs.IT

TL;DR

The paper asks how topology affects coded caching when multiple servers connect to clients through an intermediate network. It develops topology-aware strategies for three network classes and analyzes their coding delays against information-theoretic bounds. For flexible networks, the scheme is order-optimal, with the optimal memory-delay curve achieved in certain regimes.

  • Problem

    The paper investigates how intermediate-network topology affects the optimal placement and delivery strategy, beyond single-server broadcast and simple tree networks.

  • Method

    The paper defines dedicated, flexible, and linear network classes and develops topology-aware coding strategies with coding-delay analyses and lower-bound comparisons.

  • Results

    For flexible networks, the proposed scheme is order-optimal in coding delay, and certain regimes achieve the optimal memory-delay curve.

  • Takeaways & Limitations

    Multi-server coded caching can exploit intermediate-network topology to reduce service delay.

  • Takeaways & Limitations

    The delay analysis assumes a routing strategy enabling parallel, interference-free packet transmission, although the paper establishes such a strategy for flexible networks.

Abstract

from arXiv · show

In this paper, we consider multiple cache-enabled clients connected to multiple servers through an intermediate network. We design several topology-aware coding strategies for such networks. Based on topology richness of the intermediate network, and types of coding operations at internal nodes, we define three classes of networks, namely, dedicated, flexible, and linear networks. For each class, we propose an achievable coding scheme, analyze its coding delay, and also, compare it with an information theoretic lower bound. For flexible networks, we show that our scheme is order-optimal in terms of coding delay and, interestingly, the optimal memory-delay curve is achieved in certain regimes. In general, our results suggest that, in case of networks with multiple servers, type of network topology can be exploited to reduce service delay.

I. INTRODUCTION

The paper extends coded caching from single-server and tree networks to multi-server networks, studying how intermediate-network topology and internal operations affect coding design and service delay.

  • Topology-aware strategies can outperform the super-server approach and improve coding-delay constants beyond tree-network solutions.
  • Multi-server coded caching is studied over an error-free, delay-free intermediate network connecting servers and clients.
  • The paper distinguishes network delay from coding delay, focusing on the transmission block length required to satisfy all client demands.Total service delay is defined as T = TN + TC.
  • Three topology classes—dedicated, flexible, and linear—are characterized by internal connectivity and available coding operations.The classes are introduced according to the richness of internal connections and server-client contact structure.
  • Flexible networks allow server-to-client assignments to adapt during delivery, unlike dedicated networks with fixed assignments.

II. MODEL AND ASSUMPTIONS

The model contains cache-enabled users, multiple servers, and a memoryless intermediate network whose topology and node operations define dedicated, flexible, or linear operation.

  • The network is a DAG with unit-capacity error-free, delay-free links, and users receive memoryless functions of the servers’ transmitted symbols.
  • A library of N files, each of F bits, is available to all L servers, while each user has cache size MF bits.
  • During placement, users cache data without knowing demands; during delivery, servers transmit over TC channel uses so users decode requested files.
  • Dedicated Networks: Dedicated networks route each server’s packets to a fixed, non-overlapping user subset, enabling concurrent interference-free service.
  • Flexible Networks: Flexible networks can realize every partition of users among servers during delivery, making them richer than dedicated networks.
  • Linear Networks: Linear networks use random linear combinations at internal nodes without requiring topology knowledge, producing a random network transfer matrix.The model assumes sufficiently large q = 2^m so the matrix has full-rank properties with high probability.

III. MAIN RESULTS: REVIEW AND DISCUSSION

The paper compares topology-aware multi-server coded-caching strategies across dedicated, flexible, and linear networks. Its results show that exploiting parallelism and flexible assignments can reduce coding delay, with optimality guarantees in important flexible-network regimes.

  • Naive strategies: The super-server transformation and user partitioning provide two simple multi-server baselines, with parallel sub-networks reducing delay when servers serve disjoint user groups.The partitioning approach assigns approximately K/L users to each server, while parallel operation reduces the resulting delay.
  • Dedicated networks: A topology-aware dedicated-network scheme improves the pre-constant factor over the tree-based super-server strategy.The paper gives a piecewise-linear coding-delay upper bound for dedicated networks.
  • Flexible networks: Flexible networks permit server-user assignments to change across transmissions, enabling parallel interference-free delivery and lower coding delay than the single-server case.In the example, six single-server transmissions can be paired across two servers, yielding an achievable pair (M, T_C) = (1, 3).
  • Flexible networks: The flexible-network achievable region is described by coding-delay bounds parameterized by Q and server partitions, and these bounds upper-bound the optimum delay.The construction requires p_1 + ... + p_L = K − Q with p_i >= 2; K' is the smallest number at least K divisible by L.
  • Flexible networks: For flexible networks with K divisible by L, the achievable upper bound is optimal within a multiplicative constant gap.In a specific example, the flexible strategy also gives a scaling improvement over the super-server tree strategy and achieves the optimal delay.
  • Linear networks: Linear networks use topology-oblivious random linear coding at internal nodes, trading possible optimality for practical robustness, while an example achieves multiplexing gain 4 for L = 4.For L in {2, 3}, interference management is required to achieve gain min(4, L + M) in the denominator.

IV. FLEXIBLE NETWORKS: DETAILS

The flexible-network scheme combines cache placement, file partitioning, and coordinated delivery across servers. Its achievable delay is compared with a general lower bound, yielding optimality in specified regimes.

  • Optimality: The proposed delivery scheme is achievable and is analyzed against a general-network lower bound to establish optimality in some parameter ranges.The lower bound supports the corollary that all Theorem 2 pairs with Q = 0 are optimal.
  • Scheme parameters: The scheme begins by selecting integer parameters Q and p1,...,pL+1 subject to the theorem’s constraints.The virtual server uses pL+1 = Q, while real servers satisfy pi ⩾ 2.
  • Content delivery: During delivery, transmit slots correspond to partitions of the user set, and each server sends a common message to its assigned subset.Flexible connectivity permits simultaneous, interference-free transmissions by all servers.
  • Cache placement: Each file is split into L server-assigned sub-files, which are further divided into mini-files and pico-files for placement and delivery.The virtual server receives no file portion, with xL+1 = 0.
  • Cache placement: Users cache pico-files indexed by user subsets, following the single-server coded-caching placement pattern for each server-assigned sub-file.The required cache contents are constructed from subset-indexed pieces such as W i,j n,τi.
  • Optimality: For the first cache-size regime, the optimal curve is achieved; more generally, when K is divisible by L, the Theorem 2 upper bound is optimal within a multiplicative constant gap.The paper separates the proof into regimes above and below a threshold M*.

V. LINEAR NETWORKS: DETAILS

The linear-network strategy combines cache placement with random linear network coding across multiple servers, enabling users to decode requested files while exploiting caching and multiplexing gains.

  • Strategy: Linear networks use random linear combinations transmitted by multiple servers, with internal nodes performing random linear network coding.The network is modeled through a random transfer matrix, and transmitted blocks are constructed from linear combinations of sub-files.
  • Decoding: Users subtract cached interference and solve independent linear equations to recover missing sub-files and their requested files.In the example, user 1 recovers independent combinations of A2 and A3, then combines them with cached A1.
  • Example: 2-by-2F/3m transmit blocks allow all users in the example to decode their requested files.The decoding argument holds with high probability for sufficiently large field size q.
  • Gains: The strategy is designed to exploit both global caching gain and multiplexing gain simultaneously.The paper contrasts this with single-server caching and two-server delivery without cache memory, each of which provides only one of these gains.
  • Cache placement: The cache placement follows the single-server strategy by partitioning each file into sub-files indexed by user subsets and storing each sub-file at users in its index.The placement uses t = MK/N and stores Wn,τ in user k’s cache whenever k ∈ τ.
  • Vector design: For each (t + L)-subset of users, coding vectors are chosen to satisfy orthogonality constraints, with sufficiently large fields ensuring the constraints hold with high probability.The construction assigns L-dimensional vectors to (t + 1)-subsets and uses a union-bound argument over non-orthogonality constraints.
  • Guarantees: The formal strategy guarantees that users can decode the required sub-files, and its resulting coding delay is derived from the transmitted blocks for each subset.Lemmas 3 and 4 establish successful decoding for the relevant sub-files, while the delivery construction yields the stated delay.

VI. CONCLUSIONS

The paper studies coded caching in multi-server networks and shows that topology and internal-node operations shape coding design, while all considered network types benefit from caching and multiplexing gains.

  • VI. CONCLUSIONS: The paper defines dedicated, flexible, and linear networks according to network topology and internal-node coding operations.Dedicated and flexible networks use topology-aware routing, whereas linear networks use random linear network coding without topology knowledge at internal nodes.
  • VI. CONCLUSIONS: All three network types benefit from both caching and multiplexing gains.In dedicated and linear networks the gains appear additively, while in flexible networks they appear multiplicatively.
  • VI. CONCLUSIONS: Flexible networks achieve an order-optimal coding-delay solution because their caching and multiplexing gains combine multiplicatively.The conclusion identifies this multiplicative combination as distinct from the additive form in dedicated and linear networks.

APPENDIX A: CONVERSE PROOF

The appendix develops converse bounds and explicit achievable schemes for a two-server, four-user example across cache sizes, showing how cache placement and server activation determine coding delay.

  • Converse proof: The converse proof groups users into batches and defines transmissions that enable each batch to decode its requested files using cached contents.The proof maximizes over the free parameter s to obtain the tightest bound.
  • M = 0: At M = 0, six transmitted 2-by-F/(3m) blocks deliver all requested files without user cache contents.The construction divides each file into three parts and uses transmissions that allow every user to recover its requested file.
  • M = 1: At M = 1, four transmitted 2-by-F/m blocks let users recover their requested files after eliminating cached interference.The first user obtains three independent combinations of A4, A5, and A6, then combines them with cached A1, A2, and A3.
  • M = 3: At M = 3, activating one server achieves coding delay TC = 1/m.The single-server coded-caching strategy already provides useful information to all four users in one transmission.
  • M = 4: At M = 4, storing all files in every user cache gives zero delivery delay, TC = 0.No second-phase transmission is required when each user already stores the complete library.

APPENDIX C: DETAILS OF EXAMPLE 4 (L = 3, N = 4, K = 4)

The three-server example extends the linear-network construction across cache sizes, using vector transmissions to recover requested files and activating fewer servers when convenient.

  • M = 0: At M = 0, three-dimensional transmit vectors enable users to decode their requested files without cache memory.The construction uses three servers and produces four transmitted 3-by-F/m blocks.
  • M = 1: At M = 1, users combine cached sub-files with three independent linear combinations to recover all missing parts of their requested files.For user 1, the transmissions recover A2, A3, and A4 after cached interference is removed, while A1 is already cached.
  • M = 2: At M = 2, activating two servers reduces the problem to the two-server, four-user case with coding delay TC = 1/m.The construction therefore reuses the previously achieved two-server result.
  • M = 3: At M = 3, activating one server reduces the problem to the single-server setting with coding delay TC = 1/m.The other servers are not activated in this memory regime.
  • M = 4: At M = 4, all requested files are already cached and the delivery delay is TC = 0.No transmission is needed in the second phase.
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