Pub Date : 2009-06-15DOI: 10.1109/NETCOD.2009.5191389
N. Cai, Wangmei Guo
Over acyclic networks, it is well known that the global encoding kernels are uniquely determined by the local encoding kernels. But it is not in the case over cyclic networks. To study this problem, we employ matrix power series to describe the encoding kernels. This arrangement not only makes the physical meaning explicitly, but also makes it easy to obtain the conditions of determining the global encoding kernels from the local encoding kernels. We denote by K0 the constant term of the local encoding kernel matrix. Then the above conditions are characteristic of K0. It is shown that a nilpotent K0 is sufficient to determine F(z). K0 is nilpotent when the encoding topology with respect to K0 is acyclic. This result facilitates convolutional network coding encoder design. Then the equivalent conditions to determine convolutional network coding are deduced, and the inclusion relations among these conditions are further discussed in some examples.
{"title":"The conditions to determine convolutional network coding on matrix representation","authors":"N. Cai, Wangmei Guo","doi":"10.1109/NETCOD.2009.5191389","DOIUrl":"https://doi.org/10.1109/NETCOD.2009.5191389","url":null,"abstract":"Over acyclic networks, it is well known that the global encoding kernels are uniquely determined by the local encoding kernels. But it is not in the case over cyclic networks. To study this problem, we employ matrix power series to describe the encoding kernels. This arrangement not only makes the physical meaning explicitly, but also makes it easy to obtain the conditions of determining the global encoding kernels from the local encoding kernels. We denote by K0 the constant term of the local encoding kernel matrix. Then the above conditions are characteristic of K0. It is shown that a nilpotent K0 is sufficient to determine F(z). K0 is nilpotent when the encoding topology with respect to K0 is acyclic. This result facilitates convolutional network coding encoder design. Then the equivalent conditions to determine convolutional network coding are deduced, and the inclusion relations among these conditions are further discussed in some examples.","PeriodicalId":251096,"journal":{"name":"2009 Workshop on Network Coding, Theory, and Applications","volume":"136 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127376790","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2009-06-01DOI: 10.1109/NETCOD.2009.5191399
Derek Leong, A. Dimakis, T. Ho
We investigate the problem of using several storage nodes to store a data object, subject to an aggregate storage budget or redundancy constraint. It is challenging to find the optimal allocation that maximizes the probability of successful recovery by the data collector because of the large space of possible symmetric and nonsymmetric allocations, and the nonconvexity of the problem. For the special case of probability-1 recovery, we show that the optimal allocation that minimizes the required budget is symmetric. We further explore several storage allocation and access models, and determine the optimal symmetric allocation in the high-probability regime for a case of interest. Based on our experimental investigation, we make a general conjecture about a phase transition on the optimal allocation.
{"title":"Distributed storage allocation problems","authors":"Derek Leong, A. Dimakis, T. Ho","doi":"10.1109/NETCOD.2009.5191399","DOIUrl":"https://doi.org/10.1109/NETCOD.2009.5191399","url":null,"abstract":"We investigate the problem of using several storage nodes to store a data object, subject to an aggregate storage budget or redundancy constraint. It is challenging to find the optimal allocation that maximizes the probability of successful recovery by the data collector because of the large space of possible symmetric and nonsymmetric allocations, and the nonconvexity of the problem. For the special case of probability-1 recovery, we show that the optimal allocation that minimizes the required budget is symmetric. We further explore several storage allocation and access models, and determine the optimal symmetric allocation in the high-probability regime for a case of interest. Based on our experimental investigation, we make a general conjecture about a phase transition on the optimal allocation.","PeriodicalId":251096,"journal":{"name":"2009 Workshop on Network Coding, Theory, and Applications","volume":"164 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131485242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2009-05-18DOI: 10.1109/NETCOD.2009.5191397
Danilo Silva, Weifei Zeng, F. Kschischang
This paper presents a novel approach to network coding for distribution of large files. Instead of the usual approach of splitting packets into disjoint classes (also known as generations) we propose the use of overlapping classes. The overlapping allows the decoder to alternate between Gaussian elimination and back substitution, simultaneously boosting the performance and reducing the decoding complexity. Our approach can be seen as a combination of fountain coding and network coding. Simulation results are presented that demonstrate the promise of our approach.
{"title":"Sparse network coding with overlapping classes","authors":"Danilo Silva, Weifei Zeng, F. Kschischang","doi":"10.1109/NETCOD.2009.5191397","DOIUrl":"https://doi.org/10.1109/NETCOD.2009.5191397","url":null,"abstract":"This paper presents a novel approach to network coding for distribution of large files. Instead of the usual approach of splitting packets into disjoint classes (also known as generations) we propose the use of overlapping classes. The overlapping allows the decoder to alternate between Gaussian elimination and back substitution, simultaneously boosting the performance and reducing the decoding complexity. Our approach can be seen as a combination of fountain coding and network coding. Simulation results are presented that demonstrate the promise of our approach.","PeriodicalId":251096,"journal":{"name":"2009 Workshop on Network Coding, Theory, and Applications","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124307370","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2009-03-25DOI: 10.1109/NETCOD.2009.5191395
D. Lucani, M. Médard, M. Stojanovic
We study random linear network coding for broadcasting in time division duplexing channels. We assume a packet erasure channel with nodes that cannot transmit and receive information simultaneously. The sender transmits coded data packets back-to-back before stopping to wait for the receivers to acknowledge the number of degrees of freedom, if any, that are required to decode correctly the information. We study the mean time to complete the transmission of a block of packets to all receivers. We also present a bound on the number of stops to wait for acknowledgement in order to complete transmission with probability at least 1 − ε, for any ε ≫ 0. We present analysis and numerical results showing that our scheme outperforms optimal scheduling policies for broadcast, in terms of the mean completion time. We provide a simple heuristic to compute the number of coded packets to be sent before stopping that achieves close to optimal performance with the advantage of a considerable reduction in the search time.
{"title":"Broadcasting in time-division duplexing: A random linear network coding approach","authors":"D. Lucani, M. Médard, M. Stojanovic","doi":"10.1109/NETCOD.2009.5191395","DOIUrl":"https://doi.org/10.1109/NETCOD.2009.5191395","url":null,"abstract":"We study random linear network coding for broadcasting in time division duplexing channels. We assume a packet erasure channel with nodes that cannot transmit and receive information simultaneously. The sender transmits coded data packets back-to-back before stopping to wait for the receivers to acknowledge the number of degrees of freedom, if any, that are required to decode correctly the information. We study the mean time to complete the transmission of a block of packets to all receivers. We also present a bound on the number of stops to wait for acknowledgement in order to complete transmission with probability at least 1 − ε, for any ε ≫ 0. We present analysis and numerical results showing that our scheme outperforms optimal scheduling policies for broadcast, in terms of the mean completion time. We provide a simple heuristic to compute the number of coded packets to be sent before stopping that achieves close to optimal performance with the advantage of a considerable reduction in the search time.","PeriodicalId":251096,"journal":{"name":"2009 Workshop on Network Coding, Theory, and Applications","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133337123","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2008-11-03DOI: 10.1109/NETCOD.2009.5191400
B. K. Rai, B. Dey, A. Karandikar
We consider the problem of communicating the sum of m sources to n terminals in a directed acyclic network. Recently, it was shown that for a network of unit capacity links with either m = 2 or n = 2, the sum of the sources can be communicated to the terminals using scalar/vector linear network coding if and only if every source-terminal pair is connected in the network. We show in this paper that for any finite set of primes, there exists a network where the sum of the sources can be communicated to the terminals only over finite fields of characteristic belonging to that set. As a corollary, this gives networks where the sum can not be communicated over any finite field using vector linear network coding even though every source is connected to every terminal.
{"title":"Some results on communicating the sum of sources over a network","authors":"B. K. Rai, B. Dey, A. Karandikar","doi":"10.1109/NETCOD.2009.5191400","DOIUrl":"https://doi.org/10.1109/NETCOD.2009.5191400","url":null,"abstract":"We consider the problem of communicating the sum of m sources to n terminals in a directed acyclic network. Recently, it was shown that for a network of unit capacity links with either m = 2 or n = 2, the sum of the sources can be communicated to the terminals using scalar/vector linear network coding if and only if every source-terminal pair is connected in the network. We show in this paper that for any finite set of primes, there exists a network where the sum of the sources can be communicated to the terminals only over finite fields of characteristic belonging to that set. As a corollary, this gives networks where the sum can not be communicated over any finite field using vector linear network coding even though every source is connected to every terminal.","PeriodicalId":251096,"journal":{"name":"2009 Workshop on Network Coding, Theory, and Applications","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131165143","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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