259 research outputs found
Interactive Coding Resilient to an Unknown Number of Erasures
We consider distributed computations between two parties carried out over a noisy channel that may erase messages. Following a noise model proposed by Dani et al. (2018), the noise level observed by the parties during the computation in our setting is arbitrary and a priori unknown to the parties.
We develop interactive coding schemes that adapt to the actual level of noise and correctly execute any two-party computation. Namely, in case the channel erases T transmissions, the coding scheme will take N+2T transmissions using an alphabet of size 4 (alternatively, using 2N+4T transmissions over a binary channel) to correctly simulate any binary protocol that takes N transmissions assuming a noiseless channel. We can further reduce the communication to N+T by relaxing the communication model and allowing parties to remain silent rather than forcing them to communicate in every round of the coding scheme.
Our coding schemes are efficient, deterministic, have linear overhead both in their communication and round complexity, and succeed (with probability 1) regardless of the number of erasures T
Adaptive Protocols for Interactive Communication
How much adversarial noise can protocols for interactive communication
tolerate? This question was examined by Braverman and Rao (IEEE Trans. Inf.
Theory, 2014) for the case of "robust" protocols, where each party sends
messages only in fixed and predetermined rounds. We consider a new class of
non-robust protocols for Interactive Communication, which we call adaptive
protocols. Such protocols adapt structurally to the noise induced by the
channel in the sense that both the order of speaking, and the length of the
protocol may vary depending on observed noise.
We define models that capture adaptive protocols and study upper and lower
bounds on the permissible noise rate in these models. When the length of the
protocol may adaptively change according to the noise, we demonstrate a
protocol that tolerates noise rates up to . When the order of speaking may
adaptively change as well, we demonstrate a protocol that tolerates noise rates
up to . Hence, adaptivity circumvents an impossibility result of on
the fraction of tolerable noise (Braverman and Rao, 2014).Comment: Content is similar to previous version yet with an improved
presentatio
Streaming Codes for Channels with Burst and Isolated Erasures
We study low-delay error correction codes for streaming recovery over a class
of packet-erasure channels that introduce both burst-erasures and isolated
erasures. We propose a simple, yet effective class of codes whose parameters
can be tuned to obtain a tradeoff between the capability to correct burst and
isolated erasures. Our construction generalizes previously proposed low-delay
codes which are effective only against burst erasures. We establish an
information theoretic upper bound on the capability of any code to
simultaneously correct burst and isolated erasures and show that our proposed
constructions meet the upper bound in some special cases. We discuss the
operational significance of column-distance and column-span metrics and
establish that the rate 1/2 codes discovered by Martinian and Sundberg [IT
Trans.\, 2004] through a computer search indeed attain the optimal
column-distance and column-span tradeoff. Numerical simulations over a
Gilbert-Elliott channel model and a Fritchman model show significant
performance gains over previously proposed low-delay codes and random linear
codes for certain range of channel parameters
The Optimal Error Resilience of Interactive Communication Over Binary Channels
In interactive coding, Alice and Bob wish to compute some function of their individual private inputs and . They do this by engaging in a non-adaptive (fixed order, fixed length) interactive protocol to jointly compute . The goal is to do this in an error-resilient way, such that even given some fraction of adversarial corruptions to the protocol, both parties still learn .
In this work, we study the optimal error resilience of such a protocol in the face of adversarial bit flip or erasures. While the optimal error resilience of such a protocol over a large alphabet is well understood, the situation over the binary alphabet has remained open. In this work, we resolve this problem of determining the optimal error resilience over binary channels. In particular, we construct protocols achieving error resilience over the binary bit flip channel and error resilience over the binary erasure channel, for both of which matching upper bounds are known. We remark that the communication complexity of our binary bit flip protocol is polynomial in the size of the inputs, and the communication complexity of our binary erasure protocol is linear in the size of the minimal noiseless protocol computing
Synchronization Strings: Codes for Insertions and Deletions Approaching the Singleton Bound
We introduce synchronization strings as a novel way of efficiently dealing
with synchronization errors, i.e., insertions and deletions. Synchronization
errors are strictly more general and much harder to deal with than commonly
considered half-errors, i.e., symbol corruptions and erasures. For every
, synchronization strings allow to index a sequence with an
size alphabet such that one can efficiently transform
synchronization errors into half-errors. This powerful new
technique has many applications. In this paper, we focus on designing insdel
codes, i.e., error correcting block codes (ECCs) for insertion deletion
channels.
While ECCs for both half-errors and synchronization errors have been
intensely studied, the later has largely resisted progress. Indeed, it took
until 1999 for the first insdel codes with constant rate, constant distance,
and constant alphabet size to be constructed by Schulman and Zuckerman. Insdel
codes for asymptotically large or small noise rates were given in 2016 by
Guruswami et al. but these codes are still polynomially far from the optimal
rate-distance tradeoff. This makes the understanding of insdel codes up to this
work equivalent to what was known for regular ECCs after Forney introduced
concatenated codes in his doctoral thesis 50 years ago.
A direct application of our synchronization strings based indexing method
gives a simple black-box construction which transforms any ECC into an equally
efficient insdel code with a slightly larger alphabet size. This instantly
transfers much of the highly developed understanding for regular ECCs over
large constant alphabets into the realm of insdel codes. Most notably, we
obtain efficient insdel codes which get arbitrarily close to the optimal
rate-distance tradeoff given by the Singleton bound for the complete noise
spectrum
Sublinear-Time Computation in the Presence of Online Erasures
We initiate the study of sublinear-time algorithms that access their input
via an online adversarial erasure oracle. After answering each query to the
input object, such an oracle can erase input values. Our goal is to
understand the complexity of basic computational tasks in extremely adversarial
situations, where the algorithm's access to data is blocked during the
execution of the algorithm in response to its actions. Specifically, we focus
on property testing in the model with online erasures. We show that two
fundamental properties of functions, linearity and quadraticity, can be tested
for constant with asymptotically the same complexity as in the standard
property testing model. For linearity testing, we prove tight bounds in terms
of , showing that the query complexity is . In contrast to
linearity and quadraticity, some other properties, including sortedness and the
Lipschitz property of sequences, cannot be tested at all, even for . Our
investigation leads to a deeper understanding of the structure of violations of
linearity and other widely studied properties. We also consider implications of
our results for algorithms that are resilient to online adversarial corruptions
instead of erasures
Efficient and Effective Schemes for Streaming Media Delivery
The rapid expansion of the Internet and the increasingly wide deployment of wireless networks provide opportunities to deliver streaming media content to users at anywhere, anytime. To ensure good user experience, it is important to battle adversary effects, such as delay, loss and jitter. In this thesis, we first study efficient loss recovery schemes, which require pure XOR operations. In particular, we propose a novel scheme capable of recovering up to 3 packet losses, and it has the lowest complexity among all known schemes. We also propose an efficient algorithm for array codes decoding, which achieves significant throughput gain and energy savings over conventional codes. We believe these schemes are applicable to streaming applications, especially in wireless environments. We then study quality adaptation schemes for client buffer management. Our control-theoretic approach results in an efficient online rate control algorithm with analytically tractable performance. Extensive experimental results show that three goals are achieved: fast startup, continuous playback in the face of severe congestion, and maximal quality and smoothness over the entire streaming session. The scheme is later extended to streaming with limited quality levels, which is then directly applicable to existing systems
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