Sum-Networks: Min-Cut = 2 Does Not Guarantee Solvability

Kumar, Brijesh; Das, Niladri

doi:10.1109/lcomm.2013.101413.131605

Cited by 5 publications

(6 citation statements)

References 10 publications

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“…To our knowledge, the first non-identity function computation problem over a directed acyclic network is the following so-called sum-network problem [28]- [33]. In a directed acyclic network, the multiple sink nodes are required to compute an algebraic sum of the messages observed by all the source nodes over a finite field (e.g., the foregoing modulo 2 sum is an algebraic sum over the finite field F 2 ).…”

Section: A Related Workmentioning

confidence: 99%

“…2 Instead, it is sufficient if every pair of source and sink nodes can be connected by 2 edge-disjoint paths. 3 However, Rai and Das [33] showed by a counterexample that even this condition is not always sufficient if there are 7 source nodes and 7 sink nodes in the network.…”

Section: A Related Workmentioning

confidence: 99%

“…We use Cl (k) [ a J ] to denote an (I, a J )-equivalence class. Since all (I, a J )-equivalence classes form a partition of A k×I , we can write the right hand side of (33) as…”

Section: Proof Of Theoremmentioning

confidence: 99%

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Improved Upper Bound on the Network Function Computing Capacity

Guang

Yeung

Yang

et al. 2019

IEEE Trans. Inform. Theory

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The problem of network function computation over a directed acyclic network is investigated in this paper. In such a network, a sink node desires to compute with zero error a target function, of which the inputs are generated at multiple source nodes. The edges in the network are assumed to be error-free and have limited capacity. The nodes in the network are assumed to have unbounded computing capability and be able to perform network coding. The computing rate of a network code that can compute the target function over the network is the average number of times that the target function is computed with zero error for one use of the network. In this paper, we obtain an improved upper bound on the computing capacity, which is applicable to arbitrary target functions and arbitrary network topologies.This improved upper bound not only is an enhancement of the previous upper bounds but also is the first tight upper bound on the computing capacity for computing an arithmetic sum over a certain non-tree network, which has been widely studied in the literature. We also introduce a multi-dimensional array approach that facilitates evaluation of the improved upper bound. Furthermore, we apply this bound to the problem of computing a vector-linear function over a network. With this bound, we are able to not only enhance a previous result on computing a vector-linear function over a network but also simplify the proof significantly. Finally, we prove that for computing the binary maximum function over the reverse butterfly network, our improved upper bound is not achievable. This result establishes that in general our improved upper bound is non achievable, but whether it is asymptotically achievable or not remains open.

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Section: A Related Workmentioning

confidence: 99%

Section: A Related Workmentioning

confidence: 99%

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Improved Upper Bound on the Network Function Computing Capacity

Guang

Yeung

Yang

et al. 2019

IEEE Trans. Inform. Theory

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“…In reference [9], the authors also showed that if the sumnetwork having three source and three terminal has min-cut equal to two between any source and any terminal, then the linear coding capacity of the network is at least 1. However, this property also fails to hold for larger sum-networks [10]. For a class of sum-networks having 3 sources and n (≥ 3) terminals or m (≥ 3) sources and 3 terminals, it was shown that there exist networks which have linear coding capacity of the form k k+1 , where k is a positive integer greater than or equal to 2 [11].…”

Section: Introductionmentioning

confidence: 99%

Sum-networks: Dependency on characteristic of the finite field under linear network coding

Das

Kumar

2017

2017 Twenty-Third National Conference on Communications (NCC)

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Abstract-Sum-networks are networks where all the terminals demand the sum of the symbols generated at the sources. It has been shown that for any finite set/co-finite set of prime numbers, there exists a sum-network which has a vector linear solution if and only if the characteristic of the finite field belongs to the given set. It has also been shown that for any positive rational number k/n, there exists a sum-network which has capacity equal to k/n. It is a natural question whether, for any positive rational number k/n, and for any finite set/co-finite set of primes {p1, p2, . . . , p l }, there exists a sum-network which has a capacity achieving rate k/n fractional linear network coding solution if and only if the characteristic of the finite field belongs to the given set. We show that indeed there exists such a sum-network by constructing such a sum-network.

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“…The study of network coding for sum-networks began with A. Ramamoorthy [12] and was investigated from several aspects recently [12]- [20]. In [12], it was shown that for a ks/2t or 2s/nt sum-network with unit capacity edges and independent, unit-entropy sources, it is solvable if and only if every source-terminal pair is connected.…”

Section: Introductionmentioning

confidence: 99%

On the Solvability of 3s/nt Sum-Network---A Region Decomposition and Weak Decentralized Code Method

Song,

Cai,

Yuen

et al. 2015

Preprint

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We study the network coding problem of sum-networks with 3 sources and n terminals (3s/nt sum-network), for an arbitrary positive integer n, and derive a sufficient and necessary condition for the solvability of a family of so-called "terminal-separable" sum-network. Both the condition of "terminal-separable" and the solvability of a terminal-separable sum-network can be decided in polynomial time. Consequently, we give another necessary and sufficient condition, which yields a faster (O(|E|) time) algorithm than that of Shenvi and Dey ([18], (O(|E| 3 ) time), to determine the solvability of the 3s/3t sum-network.To obtain the results, we further develop the region decomposition method in [22], [23] and generalize the decentralized coding method in [21]. Our methods provide new efficient tools for multiple source multiple sink network coding problems.

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Sum-Networks: Min-Cut = 2 Does Not Guarantee Solvability

Cited by 5 publications

References 10 publications

Improved Upper Bound on the Network Function Computing Capacity

Improved Upper Bound on the Network Function Computing Capacity

Sum-networks: Dependency on characteristic of the finite field under linear network coding

On the Solvability of 3s/nt Sum-Network---A Region Decomposition and Weak Decentralized Code Method

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