Pliable Index Coding Research Articles

Pliable Index Coding via Conflict-Free Colorings of Hypergraphs

Pliable Index Coding via Conflict-Free Colorings of Hypergraphs

Sparse and Low-Rank Optimization for Pliable Index Coding via Alternating Projection

Pliable index coding (PICOD) has recently been regarded as a promising solution that exploits the coding advantage to improve communication efficiency of content-type systems (e.g., recommendation system), where clients are pliable and are interested in receiving any new message that they do not have. PICOD aims to find an effective coding strategy that satisfies the demands of all clients with the minimum number of transmissions. However, most of the previous works mainly provided theoretical understanding on PICOD in special instances based on greedy algorithms. In contrast, in this paper, we present a flexible sparse and low-rank matrix modeling approach to minimize the number of transmissions for the general PICOD problems. This is achieved by establishing generalized pliable alignment conditions to guarantee the requirements of all clients. As the resulting non-convex problem is highly intractable, we further develop an alternating pursuit framework to detect the rank of the matrix to be recovered by using the rank-increasing strategy. To address the feasibility-detection issues in the existing methods, we propose an alternating projection algorithm, which admits closed-form expressions and avoids excessive sparsity inducing. Moreover, we establish the global convergence of the alternating projection algorithm with random initial points. Simulation results demonstrate that the proposed alternating pursuit algorithm significantly reduces the number of transmissions compared to the state-of-the-art methods.

A Pliable Index Coding Approach to Data Shuffling

A promising research area that has recently emerged, is on how to use index coding to improve the communication efficiency in distributed computing systems, especially for data shuffling in iterative computations. In this paper, we posit that pliable index coding can offer a more efficient framework for data shuffling, as it can better leverage the many possible shuffling choices to reduce the number of transmissions. We theoretically analyze pliable index coding under data shuffling constraints, and design a hierarchical data-shuffling scheme that uses pliable coding as a component. We find benefits up to O(ns/m) over index coding, where ns/m is the average number of workers caching a message, and m, n, and s are the numbers of messages, workers, and cache size, respectively.

Tight Information Theoretic Converse Results for Some Pliable Index Coding Problems

This paper studies the Pliable Index CODing problem (PICOD), which models content-type distribution networks. In the PICOD$(t)$ problem there are $m$ messages, $n$ users and each user has a distinct message side information set, as in the classical Index Coding problem (IC). Differently from IC, where each user has a pre-specified set of messages to decode, in the PICOD$(t)$ a user is "pliable" and is satisfied if it can decode any $t$ messages that are not in its side information set. The goal is to find a code with the shortest length that satisfies all the users. This flexibility in determining the desired message sets makes the PICOD$(t)$ behave quite differently compared to the IC, and its analysis challenging. This paper mainly focuses on the \emph{complete--$S$} PICOD$(t)$ with $m$ messages, where the set $S\subset[m]$ contains the sizes of the side information sets, and the number of users is $n=\sum_{s\in S}\binom{m}{s}$, with no two users having the same side information set. Capacity results are shown for: (i) the \emph{consecutive} complete--$S$ PICOD$(t)$, where $S=[s_{\min}:s_{\max}]$ for some $0 \leq s_{\min} \leq s_{\max} \leq m-t$, and (ii) the \emph{complement-consecutive} complete--$S$ PICOD$(t)$, where $S=[0:m-t]\backslash[s_{\min}:s_{\max}]$, for some $0 < s_{\min} \leq s_{\max} < m-t$. The novel converse proof is inspired by combinatorial design techniques and the key insight is to consider all messages that a user can eventually decode successfully, even those in excess of the $t$ required ones. This allows one to circumvent the need to consider all possible desired message set assignments at the users in order to find the one that leads to the shortest code length. In addition, tight converse results are also shown for those PICOD$(1)$ with circular-arc network topology hypergraph.

A Polynomial-Time Algorithm for Pliable Index Coding

In pliable index coding, we consider a server with $m$ messages and $n$ clients, where each client has as side information a subset of the messages. We seek to minimize the number of broadcast transmissions, so that each client can recover any one unknown message she does not already have. Previous work has shown that the pliable index coding problem is NP-hard and requires at most $\mathcal {O}(\log ^{2}(n))$ broadcast transmissions, which indicates exponential savings over the conventional index coding that requires in the worst case $\mathcal {O}(n)$ transmissions. In this paper, building on a decoding criterion that we propose, we first design a deterministic polynomial-time algorithm that can realize the exponential benefits, by achieving, in the worst case, a performance upper bounded by $\mathcal {O}(\log ^{2}(n))$ broadcast transmissions. We extend our algorithm to the $t$ -requests case, where each client requires $t$ unknown messages that she does not have, and show that our algorithm requires at most $\mathcal {O}(t\log (n)+\log ^{2}(n))$ broadcast transmissions. We construct lower bound instances that require at least $\Omega (\log (n))$ transmissions for linear pliable index coding and at least $\Omega (t+\log (n))$ transmissions for the $t$ -requests case, indicating that both our upper and lower bounds are polynomials of $\log (n)$ and differ within a factor of $\mathcal {O}(\log (n))$ . We provide a probabilistic analysis over random instances and show that the required number of transmissions is almost surely $\Theta (\log (n))$ , as compared with the $\Theta (n/\log (n))$ for index coding. In addition, we show that these upper and lower bounds also hold for vector pliable index coding in the worst case instances and the random graph instances, implying that vector coding does not provide benefits in terms of these bounds. Our numerical experiments show that our algorithm outperforms existing algorithms for pliable index coding by up to 50% less transmissions.

Pliable Index Coding

We formulate a new variant of the index coding problem, where instead of demanding a specific message, clients are pliable, and are interested in receiving any $t$ messages that they do not have. We term this problem pliable index coding or PICOD $(t)$ . We prove that, with this formulation, although some instances of the problem become simple, in general, the problem of finding the optimal linear code remains NP-hard. However, we show that it is possible to construct pliable index codes that are substantially smaller than index codes in many cases. If there are $n$ clients, the server has $m$ messages, and each client has a side information set of cardinality $s\leq m-t$ ; we show that $O(\min \{t\log n, t+\log ^{2} n\})$ broadcast transmissions are sufficient to satisfy all the clients. For $t=1$ , this is an exponential improvement over the $n$ messages required in index coding in the worst case (for $m=n$ ). In addition, for $t\gg \log ^{2} n$ , the number of broadcast transmissions required is only linearly dependent on $t$ . We generalize the results to instances where the side information sets are not necessarily of equal cardinality. When $m = O(n^\delta )$ , for some constant $\delta >0$ , we show that the codes of size $O(\min \{t\log ^{2} n,t\log n+\log ^{3} n\})$ are sufficient in general. We also consider the scenario when the server only knows the cardinality of the side information sets of the clients and each client is interested in receiving any $t$ messages that it does not have. We term this formulation oblivious pliable index coding or OB-PICOD $(t)$ . If the cardinalities of side information sets of all the clients is $s$ (with $s\leq m-t$ ), then we show that $\min \{s+t,m-s\}$ messages are both sufficient and necessary for linear codes. Finally, we develop efficient heuristic approximation algorithms for PICOD $(t)$ and show through simulations on the random instances of PICOD $(t)$ that they perform well in practice.