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Explicit Construction of Linear Sized Tolerant Networks
[chapter]

1988
*
Annals of Discrete Mathematics
*

This settles a problem

doi:10.1016/s0167-5060(08)70766-0
fatcat:kof5ehgpobe65lthp6v3yffqtq
*of*Rosenberg, which was motivated by the study*of*fault*tolerant**linear*arrays. ... For every > 0 and every integer m > 0, we*construct*explicitly graphs with O(m/ ) vertices and maximum degree O(1/ 2 ), such that after removing any (1 − ) portion*of*their vertices or edges, the remaining ... Our*construction*supplies an*explicit*example showing that the*size*Ramsey number for paths is*linear*. ...##
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Explicit construction of linear sized tolerant networks

1988
*
Discrete Mathematics
*

This settles a problem

doi:10.1016/0012-365x(88)90189-6
fatcat:2h6m2f44uzac5fn23qrwi3x4hy
*of*Rosenberg, which was motivated by the study*of*fault*tolerant**linear*arrays. ... For every > 0 and every integer m > 0, we*construct*explicitly graphs with O(m/ ) vertices and maximum degree O(1/ 2 ), such that after removing any (1 − ) portion*of*their vertices or edges, the remaining ... Our*construction*supplies an*explicit*example showing that the*size*Ramsey number for paths is*linear*. ...##
###
Explicit construction of linear sized tolerant networks

2006
*
Discrete Mathematics
*

This settles a problem

doi:10.1016/j.disc.2006.03.025
fatcat:sbp4m3smcbccxkrvvyplfuhfla
*of*Rosenberg, which was motivated by the study*of*fault*tolerant**linear*arrays. ... For every > 0 and every integer m > 0, we*construct*explicitly graphs with O(m/ ) vertices and maximum degree O(1/ 2 ), such that after removing any (1 − ) portion*of*their vertices or edges, the remaining ... Our*construction*supplies an*explicit*example showing that the*size*Ramsey number for paths is*linear*. ...##
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Fault tolerant networks with small degree

2000
*
Proceedings of the twelfth annual ACM symposium on Parallel algorithms and architectures - SPAA '00
*

In this paper, we study the design

doi:10.1145/341800.341809
dblp:conf/spaa/Zhang00
fatcat:gx3jo2kgrbhhrd5w64ivp7nxoe
*of*fault*tolerant**networks*for arrays and meshes by adding redundant nodes and edges. ... The number*of*spare nodes used are O(k log 2 k) and O(k 2 / log k), respectively. Compared to the previous results, the number*of*spare nodes used in our*construction*has one fewer*linear*factor in k. ... Is it possible to*construct**explicit*fault*tolerant*graphs for directed*linear*arrays with O(k log k) spare nodes and constant maximum degree? ...##
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Sparse asymmetric connectors in communication networks

2005
*
Electronic Notes in Discrete Mathematics
*

However, the known

doi:10.1016/j.endm.2005.07.004
fatcat:ll7z2he47ngzrnys3seeuky65i
*explicit**constructions*with n ≤ √ Here we present a simple combinatorial*construction*for (n, N, 2)-connectors*of**size*O(N log 2 n). ... The probabilistic argument in [1] shows the existence*of*(n, N, 2)connectors*of**size*(number*of*edges) O(N ) if n ≤ N 1/2−ε , ε > 0. ... Hwang and Richards [7] gave an*explicit**construction*for (n, N, 2)-connectors*of**size*(1+o(1))N √ n if n ≤ √ N . ...##
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Sparse Asymmetric Connectors in Communication Networks
[chapter]

2006
*
Lecture Notes in Computer Science
*

However, the known

doi:10.1007/11889342_66
fatcat:qsdprszgqrawrgfqszv65qbt6i
*explicit**constructions*with n ≤ √ Here we present a simple combinatorial*construction*for (n, N, 2)-connectors*of**size*O(N log 2 n). ... The probabilistic argument in [1] shows the existence*of*(n, N, 2)connectors*of**size*(number*of*edges) O(N ) if n ≤ N 1/2−ε , ε > 0. ... Hwang and Richards [7] gave an*explicit**construction*for (n, N, 2)-connectors*of**size*(1+o(1))N √ n if n ≤ √ N . ...##
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Contents

2006
*
Discrete Mathematics
*

Chung

doi:10.1016/s0012-365x(06)00296-2
fatcat:36kmqe3r55hgnc277fj36c5g2q
*Explicit**construction**of**linear**sized**tolerant**networks*Discrete Mathematics, Volume 72, Issues 1-3, December 1988, Pages 15-19. ... A proof*of*Andrews' q-Dyson conjecture Discrete Mathematics, Volume 54, Issue 2, April 1985, Pages 201-224. ...##
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Page 765 of Neural Computation Vol. 7, Issue 4
[page]

1995
*
Neural Computation
*

mesh

*size*. ... Use*of*such devices in place*of**explicit*, prior knowledge about the dynamic structure thus intro- ...##
###
Polynomial Time Algorithms for Multicast Network Code Construction

2005
*
IEEE Transactions on Information Theory
*

We give deterministic polynomial time algorithms and even faster randomized algorithms for designing

doi:10.1109/tit.2005.847712
fatcat:peg6v5ovwrg6zb3vaazlqysdjm
*linear*codes for directed acyclic graphs with edges*of*unit capacity. ... We demonstrate examples*of**networks*where the achievable rates obtained by coding at intermediate nodes are arbitrarily larger than if coding is not allowed. ... FASTER*CONSTRUCTION*We now outline an alternative algorithm for*constructing**linear**network*coding schemes. ...##
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Self -Routing Superconcentrators

1996
*
Journal of computer and system sciences (Print)
*

Specifically, we can adapt the

doi:10.1006/jcss.1996.0005
fatcat:d26ddwr6dngepb7gple3et6ib4
*constructions**of*Leighton and Maggs [11] for fault-*tolerant*packet-routing*networks*and*of*Arora, Leighton, and Maggs [6] for self -routing non-blocking*networks*to use ... No*explicit**constructions*have yet been found for expanders*of*grade 3. Our original*construction*for self -routing superconcentrators required grade-3 expanders and thus was not*explicit*. ...##
###
XORing Elephants: Novel Erasure Codes for Big Data
[article]

2013
*
arXiv
*
pre-print

Our modified HDFS implementation shows a reduction

arXiv:1301.3791v1
fatcat:7u5w67cebbbm3eumjdxm2vhpje
*of*approximately 2x on the repair disk I/O and repair*network*traffic. ... Recently, erasure codes have been used to reduce the large storage overhead*of*three-replicated systems. ... Instead*of*the RLNC scheme, we could use the deterministic*construction*algorithm*of*Jaggi et al. [18] to*construct**explicit*capacity achieving*linear*codes for multicast*networks*. ...##
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Erasure coding for cloud storage systems: A survey

2013
*
Tsinghua Science and Technology
*

Though it is easy to store replicated data to

doi:10.1109/tst.2013.6522585
fatcat:4a5uadseg5bg7fg5yu2uojs764
*tolerate*a certain amount*of*data losses, it suffers from a very low storage efficiency. ... Conventional erasure coding techniques, such as Reed-Solomon codes, are able to achieve a much lower storage cost with the same level*of**tolerance*against disk failures. ... Wang and Zhang [29] showed that as for all possible values*of*.n; k; d; t/, there exist*explicit**constructions**of*exact MBCR on a field*of**size*at least n. ...##
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GRAND: Graph Neural Diffusion
[article]

2021
*
arXiv
*
pre-print

We develop

arXiv:2106.10934v2
fatcat:ownoxv6vzjgurdnllqxllkwwgu
*linear*and nonlinear versions*of*GRAND, which achieve competitive results on many standard graph benchmarks. ... Key to the success*of*our models are stability with respect to perturbations in the data and this is addressed for both implicit and*explicit*discretisation schemes. ... Adaptive step*size*ODE solvers require two*tolerance*parameters; the relative*tolerance*rtol and the absolute atol. ...##
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XORing elephants

2013
*
Proceedings of the VLDB Endowment
*

Our modified HDFS implementation shows a reduction

doi:10.14778/2535573.2488339
fatcat:ncei5mlrfjhjvgndoj463tqv6e
*of*approximately 2× on the repair disk I/O and repair*network*traffic. ... Recently, erasure codes have been used to reduce the large storage overhead*of*threereplicated systems. ... Instead*of*the RLNC scheme, we could use the deterministic*construction*algorithm*of*Jaggi et al. [18] to*construct**explicit*codes. ...##
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Locally repairable codes with multiple repair alternatives

2013
*
2013 IEEE International Symposium on Information Theory
*

Finally, we present pg-BLRC, an

doi:10.1109/isit.2013.6620355
dblp:conf/isit/Pamies-JuarezHO13
fatcat:5pqynarqz5hqrgos2ktdeatg3y
*explicit**construction**of*locally repairable codes with multiple repair alternatives,*constructed*from partial geometries, in particular from Generalized Quadrangles. ... Distributed storage systems need to store data redundantly in order to provide some fault-*tolerance*and guarantee system reliability. ... BALANCED LOCALLY REPAIRABLE CODES FROM PARTIAL GEOMETRIES We present a way to generate*explicit*BLRC*constructions*from partial geometries and analytically evaluate the new codes in terms*of*repair*tolerance*...
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