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Explicit Construction of Linear Sized Tolerant Networks [chapter]

N. Alon, F.R.K. Chung
1988 Annals of Discrete Mathematics  
This settles a problem 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.  ... 
doi:10.1016/s0167-5060(08)70766-0 fatcat:kof5ehgpobe65lthp6v3yffqtq

Explicit construction of linear sized tolerant networks

N. Alon, F.R.K. Chung
1988 Discrete Mathematics  
This settles a problem 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.  ... 
doi:10.1016/0012-365x(88)90189-6 fatcat:2h6m2f44uzac5fn23qrwi3x4hy

Explicit construction of linear sized tolerant networks

N. Alon, F.R.K. Chung
2006 Discrete Mathematics  
This settles a problem 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.  ... 
doi:10.1016/j.disc.2006.03.025 fatcat:sbp4m3smcbccxkrvvyplfuhfla

Fault tolerant networks with small degree

Li Zhang
2000 Proceedings of the twelfth annual ACM symposium on Parallel algorithms and architectures - SPAA '00  
In this paper, we study the design 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?  ... 
doi:10.1145/341800.341809 dblp:conf/spaa/Zhang00 fatcat:gx3jo2kgrbhhrd5w64ivp7nxoe

Sparse asymmetric connectors in communication networks

R. Ahlswede, H. Aydinian
2005 Electronic Notes in Discrete Mathematics  
However, the known 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 .  ... 
doi:10.1016/j.endm.2005.07.004 fatcat:ll7z2he47ngzrnys3seeuky65i

Sparse Asymmetric Connectors in Communication Networks [chapter]

R. Ahlswede, H. Aydinian
2006 Lecture Notes in Computer Science  
However, the known 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 .  ... 
doi:10.1007/11889342_66 fatcat:qsdprszgqrawrgfqszv65qbt6i

Contents

2006 Discrete Mathematics  
Chung 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.  ... 
doi:10.1016/s0012-365x(06)00296-2 fatcat:36kmqe3r55hgnc277fj36c5g2q

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

S. Jaggi, P. Sanders, P.A. Chou, M. Effros, S. Egner, K. Jain, L.M.G.M. Tolhuizen
2005 IEEE Transactions on Information Theory  
We give deterministic polynomial time algorithms and even faster randomized algorithms for designing 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.  ... 
doi:10.1109/tit.2005.847712 fatcat:peg6v5ovwrg6zb3vaazlqysdjm

Self -Routing Superconcentrators

Nicholas Pippenger
1996 Journal of computer and system sciences (Print)  
Specifically, we can adapt the 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.  ... 
doi:10.1006/jcss.1996.0005 fatcat:d26ddwr6dngepb7gple3et6ib4

XORing Elephants: Novel Erasure Codes for Big Data [article]

Maheswaran Sathiamoorthy, Megasthenis Asteris, Dimitris Papailiopoulos, Alexandros G. Dimakis, Ramkumar Vadali, Scott Chen, Dhruba Borthakur
2013 arXiv   pre-print
Our modified HDFS implementation shows a reduction 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.  ... 
arXiv:1301.3791v1 fatcat:7u5w67cebbbm3eumjdxm2vhpje

Erasure coding for cloud storage systems: A survey

Jun Li, Baochun Li
2013 Tsinghua Science and Technology  
Though it is easy to store replicated data to 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.  ... 
doi:10.1109/tst.2013.6522585 fatcat:4a5uadseg5bg7fg5yu2uojs764

GRAND: Graph Neural Diffusion [article]

Benjamin Paul Chamberlain, James Rowbottom, Maria Gorinova, Stefan Webb, Emanuele Rossi, Michael M. Bronstein
2021 arXiv   pre-print
We develop 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.  ... 
arXiv:2106.10934v2 fatcat:ownoxv6vzjgurdnllqxllkwwgu

XORing elephants

Maheswaran Sathiamoorthy, Megasthenis Asteris, Dimitris Papailiopoulos, Alexandros G. Dimakis, Ramkumar Vadali, Scott Chen, Dhruba Borthakur
2013 Proceedings of the VLDB Endowment  
Our modified HDFS implementation shows a reduction 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.  ... 
doi:10.14778/2535573.2488339 fatcat:ncei5mlrfjhjvgndoj463tqv6e

Locally repairable codes with multiple repair alternatives

Lluis Pamies-Juarez, Henk D.L. Hollmann, Frederique Oggier
2013 2013 IEEE International Symposium on Information Theory  
Finally, we present pg-BLRC, an 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  ... 
doi:10.1109/isit.2013.6620355 dblp:conf/isit/Pamies-JuarezHO13 fatcat:5pqynarqz5hqrgos2ktdeatg3y
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