178,226 research outputs found
Efficient Computation of the Characteristic Polynomial
This article deals with the computation of the characteristic polynomial of
dense matrices over small finite fields and over the integers. We first present
two algorithms for the finite fields: one is based on Krylov iterates and
Gaussian elimination. We compare it to an improvement of the second algorithm
of Keller-Gehrig. Then we show that a generalization of Keller-Gehrig's third
algorithm could improve both complexity and computational time. We use these
results as a basis for the computation of the characteristic polynomial of
integer matrices. We first use early termination and Chinese remaindering for
dense matrices. Then a probabilistic approach, based on integer minimal
polynomial and Hensel factorization, is particularly well suited to sparse
and/or structured matrices
On the Hardness of the Lee Syndrome Decoding Problem
In this paper we study the hardness of the syndrome decoding problem over
finite rings endowed with the Lee metric. We first prove that the decisional
version of the problem is NP-complete, by a reduction from the 3-dimensional
matching problem. Then, we study the actual complexity of solving the problem,
by translating the best known solvers in the Hamming metric over finite fields
to the Lee metric over finite rings, as well as proposing some novel solutions.
For the analyzed algorithms, we assess the computational complexity in both the
finite and asymptotic regimes.Comment: Part of this work appeared as preliminary results in arXiv:2001.0842
Universal and Robust Distributed Network Codes
Random linear network codes can be designed and implemented in a distributed
manner, with low computational complexity. However, these codes are classically
implemented over finite fields whose size depends on some global network
parameters (size of the network, the number of sinks) that may not be known
prior to code design. Also, if new nodes join the entire network code may have
to be redesigned.
In this work, we present the first universal and robust distributed linear
network coding schemes. Our schemes are universal since they are independent of
all network parameters. They are robust since if nodes join or leave, the
remaining nodes do not need to change their coding operations and the receivers
can still decode. They are distributed since nodes need only have topological
information about the part of the network upstream of them, which can be
naturally streamed as part of the communication protocol.
We present both probabilistic and deterministic schemes that are all
asymptotically rate-optimal in the coding block-length, and have guarantees of
correctness. Our probabilistic designs are computationally efficient, with
order-optimal complexity. Our deterministic designs guarantee zero error
decoding, albeit via codes with high computational complexity in general. Our
coding schemes are based on network codes over ``scalable fields". Instead of
choosing coding coefficients from one field at every node, each node uses
linear coding operations over an ``effective field-size" that depends on the
node's distance from the source node. The analysis of our schemes requires
technical tools that may be of independent interest. In particular, we
generalize the Schwartz-Zippel lemma by proving a non-uniform version, wherein
variables are chosen from sets of possibly different sizes. We also provide a
novel robust distributed algorithm to assign unique IDs to network nodes.Comment: 12 pages, 7 figures, 1 table, under submission to INFOCOM 201
Distributed matrix multiplication with straggler tolerance using algebraic function fields
The problem of straggler mitigation in distributed matrix multiplication
(DMM) is considered for a large number of worker nodes and a fixed small finite
field. Polynomial codes and matdot codes are generalized by making use of
algebraic function fields (i.e., algebraic functions over an algebraic curve)
over a finite field. The construction of optimal solutions is translated to a
combinatorial problem on the Weierstrass semigroups of the corresponding
algebraic curves. Optimal or almost optimal solutions are provided. These have
the same computational complexity per worker as classical polynomial and matdot
codes, and their recovery thresholds are almost optimal in the asymptotic
regime (growing number of workers and a fixed finite field)
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