11,613 research outputs found
Lower Bounds for Multiplication via Network Coding
Multiplication is one of the most fundamental computational problems, yet its true complexity remains elusive. The best known upper bound, very recently proved by Harvey and van der Hoeven (2019), shows that two n-bit numbers can be multiplied via a boolean circuit of size O(n lg n). In this work, we prove that if a central conjecture in the area of network coding is true, then any constant degree boolean circuit for multiplication must have size Omega(n lg n), thus almost completely settling the complexity of multiplication circuits. We additionally revisit classic conjectures in circuit complexity, due to Valiant, and show that the network coding conjecture also implies one of Valiant\u27s conjectures
Hierarchical Coding for Distributed Computing
Coding for distributed computing supports low-latency computation by
relieving the burden of straggling workers. While most existing works assume a
simple master-worker model, we consider a hierarchical computational structure
consisting of groups of workers, motivated by the need to reflect the
architectures of real-world distributed computing systems. In this work, we
propose a hierarchical coding scheme for this model, as well as analyze its
decoding cost and expected computation time. Specifically, we first provide
upper and lower bounds on the expected computing time of the proposed scheme.
We also show that our scheme enables efficient parallel decoding, thus reducing
decoding costs by orders of magnitude over non-hierarchical schemes. When
considering both decoding cost and computing time, the proposed hierarchical
coding is shown to outperform existing schemes in many practical scenarios.Comment: 7 pages, part of the paper is submitted to ISIT201
Latency Analysis of Coded Computation Schemes over Wireless Networks
Large-scale distributed computing systems face two major bottlenecks that
limit their scalability: straggler delay caused by the variability of
computation times at different worker nodes and communication bottlenecks
caused by shuffling data across many nodes in the network. Recently, it has
been shown that codes can provide significant gains in overcoming these
bottlenecks. In particular, optimal coding schemes for minimizing latency in
distributed computation of linear functions and mitigating the effect of
stragglers was proposed for a wired network, where the workers can
simultaneously transmit messages to a master node without interference. In this
paper, we focus on the problem of coded computation over a wireless
master-worker setup with straggling workers, where only one worker can transmit
the result of its local computation back to the master at a time. We consider 3
asymptotic regimes (determined by how the communication and computation times
are scaled with the number of workers) and precisely characterize the total
run-time of the distributed algorithm and optimum coding strategy in each
regime. In particular, for the regime of practical interest where the
computation and communication times of the distributed computing algorithm are
comparable, we show that the total run-time approaches a simple lower bound
that decouples computation and communication, and demonstrate that coded
schemes are times faster than uncoded schemes
Algebraic Methods in the Congested Clique
In this work, we use algebraic methods for studying distance computation and
subgraph detection tasks in the congested clique model. Specifically, we adapt
parallel matrix multiplication implementations to the congested clique,
obtaining an round matrix multiplication algorithm, where
is the exponent of matrix multiplication. In conjunction
with known techniques from centralised algorithmics, this gives significant
improvements over previous best upper bounds in the congested clique model. The
highlight results include:
-- triangle and 4-cycle counting in rounds, improving upon the
triangle detection algorithm of Dolev et al. [DISC 2012],
-- a -approximation of all-pairs shortest paths in
rounds, improving upon the -round -approximation algorithm of Nanongkai [STOC 2014], and
-- computing the girth in rounds, which is the first
non-trivial solution in this model.
In addition, we present a novel constant-round combinatorial algorithm for
detecting 4-cycles.Comment: This is work is a merger of arxiv:1412.2109 and arxiv:1412.266
Coded Computation Against Processing Delays for Virtualized Cloud-Based Channel Decoding
The uplink of a cloud radio access network architecture is studied in which
decoding at the cloud takes place via network function virtualization on
commercial off-the-shelf servers. In order to mitigate the impact of straggling
decoders in this platform, a novel coding strategy is proposed, whereby the
cloud re-encodes the received frames via a linear code before distributing them
to the decoding processors. Transmission of a single frame is considered first,
and upper bounds on the resulting frame unavailability probability as a
function of the decoding latency are derived by assuming a binary symmetric
channel for uplink communications. Then, the analysis is extended to account
for random frame arrival times. In this case, the trade-off between average
decoding latency and the frame error rate is studied for two different queuing
policies, whereby the servers carry out per-frame decoding or continuous
decoding, respectively. Numerical examples demonstrate that the bounds are
useful tools for code design and that coding is instrumental in obtaining a
desirable compromise between decoding latency and reliability.Comment: 11 pages and 12 figures, Submitte
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