## Optimal File Sharing in Distributed Networks (1991)

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Citations: | 26 - 1 self |

### BibTeX

@MISC{Naor91optimalfile,

author = {Moni Naor and Ron M. Roth},

title = {Optimal File Sharing in Distributed Networks},

year = {1991}

}

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### Abstract

The following le distribution problem is considered: Given a network of processors represented by an undirected graph G = (V; E), and a le size k, an arbitrary le w of k bits is to be distributed among all nodes of G. To this end, each node is assigned a memory device such that, by accessing the memory of its own and of its adjacent nodes, the node can reconstruct the contents of w. The objective is to minimize the total size of memory in the network. This paper presents a le distribution scheme which realizes this objective for k log G, where G stands for the maximum degree in G: For this range of k, the total memory size required by the suggested scheme approaches an integer programming lower bound on that size. The scheme is also constructive in the sense that, given G and k, the memory size at each node in G, as well as the mapping of any le w into the node memory devices, can be computed in time complexity which is polynomial in k and jV j. Furthermore, each node can reconstruct the contents of such a le w in O(k 2) bit operations. Finally, it is shown that the requirement of k being much larger than log G is necessary in order to have total memory size close to the integer programming lower bound.

### Citations

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Citation Context ...ed to as the linear programming bound. For k = 1, Theorem 1 becomes M(G; 1) = J(G; 1). The problem of deciding whether a network graph G has a dominating set of sizess is well-known to be NP-complete =-=[6]-=-. The next corollary immediately follows. Corollary 1. Given an instance of a network graph G and positive integers k and s, the problem of deciding whether there exists a file distribution protocol f... |

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Citation Context ...node u to reconstruct w out of [wB v ] v2\Gamma(u) . We remark that Reed-Solomon codes have been extensively applied to some other reconstruction problems in networks, such as Shamir's secret sharing =-=[18]-=- (see also [10][14]). The file distribution scheme described in this section is not satisfactory when the file size k is, say, O(\Delta G ), in which case the ratio (G; k)=(ae G \Delta k) might be bou... |

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Citation Context ...z = [z u ] u2V be an optimal solution to the linear programming problem LP(G) in (2). Such a vector z can be found in time complexity which is polynomial in jV j (e.g., by using Karmarkar's algorithm =-=[9]-=-). Set h \Delta = dlog 2 (\Delta G \Delta k)e 9 and l \Delta = dk=he, and define the integer vector y = [y u ] u2V by y u \Delta = min n l ; b(l + \Delta G ) \Delta z u c o ; u 2 V : Clearly, kyksae G... |

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Citation Context ...ct w out of [wB v ] v2\Gamma(u) . We remark that Reed-Solomon codes have been extensively applied to some other reconstruction problems in networks, such as Shamir's secret sharing [18] (see also [10]=-=[14]-=-). The file distribution scheme described in this section is not satisfactory when the file size k is, say, O(\Delta G ), in which case the ratio (G; k)=(ae G \Delta k) might be bounded away from 1. T... |

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Citation Context ... (G; k), where we will need to take m slightly larger than k in order to construct the encoding and decoding mappings in Section 4.2. Theorem 3 is proved via a `randomized rounding' argument (see [15]=-=[17]-=-): We first solve the corresponding linear programming problem LP(G) in (2) (say, by Karmarkar's algorithm [9]), and use the rational solution to define a probability measure on integer vectors that a... |

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Citation Context ...ance (G; k), where we will need to take m slightly larger than k in order to construct the encoding and decoding mappings in Section 4.2. Theorem 3 is proved via a `randomized rounding' argument (see =-=[15]-=-[17]): We first solve the corresponding linear programming problem LP(G) in (2) (say, by Karmarkar's algorithm [9]), and use the rational solution to define a probability measure on integer vectors th... |

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Citation Context ... and efficient ones. The problem of file allocation in a network, i.e., of storing a file in a network so that every processor has "easy" access to the file, has been considered in many vari=-=ants (see [4]-=- for a survey). The specific version of reconstruction from adjacent nodes only has received attention in the form of file segmentation, where the task is to partition the file so that, for each node ... |

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Citation Context ...struct w out of [wB v ] v2\Gamma(u) . We remark that Reed-Solomon codes have been extensively applied to some other reconstruction problems in networks, such as Shamir's secret sharing [18] (see also =-=[10]-=-[14]). The file distribution scheme described in this section is not satisfactory when the file size k is, say, O(\Delta G ), in which case the ratio (G; k)=(ae G \Delta k) might be bounded away from ... |

38 |
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Citation Context ...elta Ys(1 \Gamma ffi) o / e \Gammaffi (1 \Gamma ffi) 1\Gammaffi !s; (b) for every ffis0 andsa \Delta p, Prob n a \Delta Ys(1 + ffi) o / e ffi (1 + ffi) 1+ffi !s: Proof. Lemma 11 is proved in [15] and =-=[16]-=-. Part (b) of the lemma appears as is in [15] and for the sake of completeness we include the proof of part (a) here. For a real random variable Z and constants fls0 and b, we have Prob n Zsb osE i e ... |

36 |
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Citation Context ...the form of file segmentation, where the task is to partition the file so that, for each node u in the network, the union of the file segments stored at nodes adjacent to u is the complete file [4][8]=-=[13]-=-. As we shall see, allowing more general reconstruction procedures than simply taking the union of file segments at adjacent nodes can result in a considerable savings of the total amount of memory re... |

34 | An algorithmic approach to the Lovász - Beck - 1991 |

11 |
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(Show Context)
Citation Context ...m variables Y u over f0; 1g such that Prob n Y u = 1 o = p u ; u 2 V ; (6) and let X = [X u ] u2V be a random vector defined by X \Delta = s + Y : (7) Fix a to be a real vector in the unit hyper-cube =-=[0; 1]-=- jV j such that a \Delta zs1. Since the expectation vector E i Y j is equal to p, we have E i a \Delta X j = a \Delta s + a \Delta p = ` \Delta a \Delta zs` : In particular, if z is a rational vector ... |

6 |
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Citation Context ...in the form of file segmentation, where the task is to partition the file so that, for each node u in the network, the union of the file segments stored at nodes adjacent to u is the complete file [4]=-=[8]-=-[13]. As we shall see, allowing more general reconstruction procedures than simply taking the union of file segments at adjacent nodes can result in a considerable savings of the total amount of memor... |

3 |
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Citation Context ...og 2 \Delta G ; on the other hand, one can construct an infinite family of network graphs fG l g l (such as the ones presented in Section 5) for which J(G l ; 1)s1 4 ae G l log 2 \Delta G l (see also =-=[7]-=-). In terms of file segmentation schemes (Example 1) this means that there always exists a file distribution protocol for (G; k) based on segmentation whose memory size, k \Delta J(G; 1), is within a ... |

3 |
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(Show Context)
Citation Context ...ore, whether it is NP-complete) since it is unclear how to verify (1) in polynomial-time, even when the encoding and decoding mappings are computable in polynomial-time. Remark 3. A result of Lov'asz =-=[11]-=- states that J(G; 1)sae G log 2 \Delta G ; on the other hand, one can construct an infinite family of network graphs fG l g l (such as the ones presented in Section 5) for which J(G l ; 1)s1 4 ae G l ... |

2 |
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(Show Context)
Citation Context ...ncoding mappings E u : w 7! wB u ; u 2 V . The techniques used will turn out to be useful in Section refvariations. To this end, we make use of the following lemma. Lemma 5 . (The Lov'asz Local Lemma =-=[5]-=-[19]). Let A 1 ; A 2 ; : : : ; A n be events in an arbitrary probability space. Suppose that each event A i is mutually independent of a set of all, but at most ffi, events A j and that Prob fA i gsp ... |

2 | Comparative models of the le assignment problem - Dowdy, Foster - 1982 |