## Multicluster interleaving on paths and cycles (2005)

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Venue: | IEEE TRANSACTIONS ON INFORMATION THEORY |

Citations: | 4 - 4 self |

### BibTeX

@ARTICLE{Jiang05multiclusterinterleaving,

author = {Anxiao (Andrew) Jiang and Jehoshua Bruck},

title = {Multicluster interleaving on paths and cycles},

journal = {IEEE TRANSACTIONS ON INFORMATION THEORY},

year = {2005},

volume = {51},

pages = {597--611}

}

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### Abstract

Interleaving codewords is an important method not only for combatting burst-errors, but also for distributed data retrieval. This paper introduces the concept of Multi-Cluster Interleaving (MCI), a generalization of traditional interleaving problems. MCI problems for paths and cycles are studied. The following problem is solved: how to interleave integers on a path or cycle such that any m (m ≥ 2) non-overlapping clusters of order 2 in the path or cycle have at least 3 distinct integers. We then present a scheme using a ‘hierarchical-chain structure’ to solve the following more general problem for paths: how to interleave integers on a path such that any m (m ≥ 2) nonoverlapping clusters of order L (L ≥ 2) in the path have at least L + 1 distinct integers. It is shown that the scheme solves the second interleaving problem for paths that are asymptotically as long as the longest path on which an MCI exists, and clearly, for shorter paths as well.

### Citations

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Citation Context ...nterleaved components of a codeword are transmitted in sequence, and every client can listen to this data flow for a while until enough codeword components are received for recovering the information =-=[6]-=-, [11]. (An example is shown in Fig. 1 (a), where a codeword of 7 components is broadcast repeatedly. We assume that the codeword can tolerate 2 erasures. Therefore every client only needs to receive ... |

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Citation Context ...client can simultaneously listen to multiple channels in order to get data faster, which is equivalent to retrieving data from multiple clusters. Another possible application is data storage on disks =-=[14]-=-, where we assume multiple heads can read different parts of a disk in parallel to accelerate I/O speed. The MCI problem for paths and cycles can be divided into smaller problems based on the values o... |

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Citation Context ...eaved components of a codeword are transmitted in sequence, and every client can listen to this data flow for a while until enough codeword components are received for recovering the information [6], =-=[11]-=-. (An example is shown in Fig. 1 (a), where a codeword of 7 components is broadcast repeatedly. We assume that the codeword can tolerate 2 erasures. Therefore every client only needs to receive 5 diff... |

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Citation Context ...orm ‘1, 2, 3, · · · n, 1, 2, 3, · · · n, · · · · · · ,’ for combatting onedimensional burst-errors of length up to n. This onedimensional interleaving is generalized to higher dimensions in [3], [4], =-=[5]-=- and [7], where integers are used to label the vertices of a two-dimensional or higher-dimensional array in such a way that in every connected subgraph of order t of the array, each integer appears at... |

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Citation Context ...components of the codeword are interleavingly placed on a network, such that every node in the network can retrieve enough distinct codeword components from its proximity for recovering the file [9], =-=[13]-=-. (An example is shown in Fig. 1 (b), where the codeword again has length 7 and can tolerate 2 erasures. We assume that all edges have length 1. Then every network node can retrieve 5 distinct codewor... |

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Citation Context ...2, 3, · · · n, 1, 2, 3, · · · n, · · · · · · ,’ for combatting onedimensional burst-errors of length up to n. This onedimensional interleaving is generalized to higher dimensions in [3], [4], [5] and =-=[7]-=-, where integers are used to label the vertices of a two-dimensional or higher-dimensional array in such a way that in every connected subgraph of order t of the array, each integer appears at most r ... |

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Citation Context ...m of =-=[1, 3, 1, 4, 1, 5, 1, 2, 3, 2, 5, 2, 4, 3, 4, 5, 3, 5]-=-. B6 is shown in Figure 11 (e). Then Algorithm 2 inserts vertices into B6 and gets a new path A6, whose assigned integers are in the form of [6, 1, 3, 6, 1, 4, 6, 1, 5, 6, 1, 2, 6, 3, 2, 6, 5, 2, 6, 4, 3, 6, 4, 5, 6, 3, 5]. A6 is shown in Figure 11 (f). Finally, Algorithm 2 combines A6, A5 and A4 such that the last L − 1 = 2 vertices of A6 overlap the first 2 vertices of A5, and the last L − 1 = 2 vertices of A5 overla... |

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Citation Context ...Bruck is with the Department of Electrical Engineering, California Institute of Technology, MC 136-93, Pasadena, CA 91125, USA (e-mail: bruck@paradise.caltech.edu). interleaving scheme includes [10], =-=[12]-=-, [15], [16] and [17], where the underlying graphs on which integers are interleaved include tori, arrays and circulant graphs. In [1], [2] and [3], codewords are interleaved on arrays to correct burs... |

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Citation Context ... bruck@paradise.caltech.edu). interleaving scheme includes [10], [12], [15], [16] and [17], where the underlying graphs on which integers are interleaved include tori, arrays and circulant graphs. In =-=[1]-=-, [2] and [3], codewords are interleaved on arrays to correct burst-errors of rectangular shapes, circular shapes, or arbitrary connected shapes. Applications of interleaving in distributed data retri... |

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Citation Context ... has the form ‘1, 2, 3, · · · n, 1, 2, 3, · · · n, · · · · · · ,’ for combatting onedimensional burst-errors of length up to n. This onedimensional interleaving is generalized to higher dimensions in =-=[3]-=-, [4], [5] and [7], where integers are used to label the vertices of a two-dimensional or higher-dimensional array in such a way that in every connected subgraph of order t of the array, each integer ... |

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Citation Context ... and components of the codeword are interleavingly placed on a network, such that every node in the network can retrieve enough distinct codeword components from its proximity for recovering the file =-=[9]-=-, [13]. (An example is shown in Fig. 1 (b), where the codeword again has length 7 and can tolerate 2 erasures. We assume that all edges have length 1. Then every network node can retrieve 5 distinct c... |

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Citation Context ...). J. Bruck is with the Department of Electrical Engineering, California Institute of Technology, MC 136-93, Pasadena, CA 91125, USA (e-mail: bruck@paradise.caltech.edu). interleaving scheme includes =-=[10]-=-, [12], [15], [16] and [17], where the underlying graphs on which integers are interleaved include tori, arrays and circulant graphs. In [1], [2] and [3], codewords are interleaved on arrays to correc... |

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Citation Context ...leaving is an MCI, every client can retrieve enough data for decoding the codeword. Multi-Cluster Interleaving on paths and cycles appears to have natural applications in data-streaming and broadcast =-=[8]-=-. Imagine that the components of a codeword interleaved the same way are transmitted asynchronously in several channels. Then a client can simultaneously listen to multiple channels in order to get da... |

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Citation Context ...is with the Department of Electrical Engineering, California Institute of Technology, MC 136-93, Pasadena, CA 91125, USA (e-mail: bruck@paradise.caltech.edu). interleaving scheme includes [10], [12], =-=[15]-=-, [16] and [17], where the underlying graphs on which integers are interleaved include tori, arrays and circulant graphs. In [1], [2] and [3], codewords are interleaved on arrays to correct burst-erro... |

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Citation Context ...h the Department of Electrical Engineering, California Institute of Technology, MC 136-93, Pasadena, CA 91125, USA (e-mail: bruck@paradise.caltech.edu). interleaving scheme includes [10], [12], [15], =-=[16]-=- and [17], where the underlying graphs on which integers are interleaved include tori, arrays and circulant graphs. In [1], [2] and [3], codewords are interleaved on arrays to correct burst-errors of ... |

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Citation Context ...artment of Electrical Engineering, California Institute of Technology, MC 136-93, Pasadena, CA 91125, USA (e-mail: bruck@paradise.caltech.edu). interleaving scheme includes [10], [12], [15], [16] and =-=[17]-=-, where the underlying graphs on which integers are interleaved include tori, arrays and circulant graphs. In [1], [2] and [3], codewords are interleaved on arrays to correct burst-errors of rectangul... |