Efficient Approximation of Two-Terminal Networks Reliability Polynomials Using Cubic Splines

Cristescu, Gabriela; Drăgoi, Vlad

doi:10.1109/tr.2021.3049957

Cited by 13 publications

(34 citation statements)

References 25 publications

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“…Hence, it might be less efficient than polar codes designed for a particular value of p, but it has the best performance on average. The preorder u ≤ Avr v ⇔ Avr(W u ) ≤ Avr(W v ) induces a complementarity property with respect to the integral operator over [0, 1], as defined in [32,33] in the case of twoterminal networks. We retrieve a similar property, i.e., Avr(W u ) = 1 − Avr W u , where u is the bit-wise complement of u, in the context of monomial codes.…”

Section: Average Reliability Of Synthetic Channelsmentioning

confidence: 99%

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Bhattacharyya Parameter of Monomial Codes for the Binary Erasure Channel: From Pointwise to Average Reliability

Drăgoi

Cristescu

2021

Sensors

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Monomial codes were recently equipped with partial order relations, a fact that allowed researchers to discover structural properties and efficient algorithm for constructing polar codes. Here, we refine the existing order relations in the particular case of the binary erasure channel. The new order relation takes us closer to the ultimate order relation induced by the pointwise evaluation of the Bhattacharyya parameter of the synthetic channels, which is still a partial order relation. To overcome this issue, we appeal to a related technique from network theory. Reliability network theory was recently used in the context of polar coding and more generally in connection with decreasing monomial codes. In this article, we investigate how the concept of average reliability is applied for polar codes designed for the binary erasure channel. Instead of minimizing the error probability of the synthetic channels, for a particular value of the erasure parameter p, our codes minimize the average error probability of the synthetic channels. By means of basic network theory results, we determine a closed formula for the average reliability of a particular synthetic channel, that recently gain the attention of researchers.

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Section: Average Reliability Of Synthetic Channelsmentioning

confidence: 99%

“…The preorder

induces a complementarity property with respect to the integral operator over

, as defined in [ 32 , 33 ] in the case of two-terminal networks. We retrieve a similar property, i.e.,

where

is the bit-wise complement of

, in the context of monomial codes.…”

Section: Introductionmentioning

confidence: 99%

Bhattacharyya Parameter of Monomial Codes for the Binary Erasure Channel: From Pointwise to Average Reliability

Drăgoi

Cristescu

2021

Sensors

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“…The less investigated topics related to network reliability are the analytical properties such as shape properties of the reliability polynomials, including convexity, the number of real roots and their density, etc. In order to compensate the complexity problems of computing the coefficients of the reliability polynomial of a two-terminal network, the authors of [6,7] proposed to approximate the polynomials using structural properties of the networks. In particular, duality is a characteristic that induces complementary properties on the coefficients, which are considered in the approximations.…”

Section: Introductionmentioning

confidence: 99%

“…In [8], Hermite interpolation is used for hammock networks based on previous results on the shape [9]. Cubic splines are proposed in [6,7] and are suitable for any two-terminal networks. In [7], two methods of producing cubic splines are compared-Lagrange-type interpolation procedures and Bernstein approximation operator, emphasizing the accuracy of the methods.…”

Section: Introductionmentioning

confidence: 99%

“…Cubic splines are proposed in [6,7] and are suitable for any two-terminal networks. In [7], two methods of producing cubic splines are compared-Lagrange-type interpolation procedures and Bernstein approximation operator, emphasizing the accuracy of the methods. There are several advantages for taking duality into account in the context of approximations:…”

Section: Introductionmentioning

confidence: 99%

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Generalized Convexity Properties and Shape-Based Approximation in Networks Reliability

2021

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Some properties of generalized convexity for sets and functions are identified in case of the reliability polynomials of two dual minimal networks. A method of approximating the reliability polynomials of two dual minimal network is developed based on their mutual complementarity properties. The approximating objects are from the class of quadratic spline functions, constructed based on both interpolation conditions and shape knowledge. It is proved that the approximant objects preserve both the high-order convexity and some extremum properties of the exact reliability polynomials. It leads to pointing out the area of the network where the maximum number of paths is achieved. Numerical examples and simulations show the performance of the algorithm, both in terms of low complexity, small error and shape preserving. Possibilities of increasing the accuracy of approximation are discussed.

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An optimization‐based Monte Carlo method for estimating the two‐terminal survival signature of networks with two component classes

Lopes da Silva,

Sullivan

2024

Naval Research Logistics

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Evaluating two‐terminal network reliability is a classical problem with numerous applications. Because this problem is ‐Complete, practical studies involving large systems commonly resort to approximating or estimating system reliability rather than evaluating it exactly. Researchers have characterized signatures, such as the destruction spectrum and survival signature, which summarize the system's structure and give rise to procedures for evaluating or approximating network reliability. These procedures are advantageous if the signature can be computed efficiently; however, computing the signature is challenging for complex systems. With this motivation, we consider the use of Monte Carlo (MC) simulation to estimate the survival signature of a two‐terminal network in which there are two classes of i.i.d. components. In this setting, we prove that each MC replication to estimate the signature of a multi‐class system entails solving a multi‐objective maximum capacity path problem. For the case of two classes of components, we adapt a Dijkstra's‐like bi‐objective shortest path algorithm from the literature for the purpose of solving the resulting bi‐objective maximum capacity path problem. We perform computational experiments to compare our method's efficiency against intuitive benchmark approaches. Our computational results demonstrate that the bi‐objective optimization approach consistently outperforms the benchmark approaches, thereby enabling a larger number of MC replications and improved accuracy of the reliability estimation. Furthermore, the efficiency gains versus benchmark approaches appear to become more significant as the network increases in size.

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Efficient Approximation of Two-Terminal Networks Reliability Polynomials Using Cubic Splines

Cited by 13 publications

References 25 publications

Bhattacharyya Parameter of Monomial Codes for the Binary Erasure Channel: From Pointwise to Average Reliability

Bhattacharyya Parameter of Monomial Codes for the Binary Erasure Channel: From Pointwise to Average Reliability

Generalized Convexity Properties and Shape-Based Approximation in Networks Reliability

An optimization‐based Monte Carlo method for estimating the two‐terminal survival signature of networks with two component classes

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