Butterfly Network Research Articles

Improved Upper Bound on the Network Function Computing Capacity

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 not only able to 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.

Towards attack tolerant networks: Concurrent multipath routing and the butterfly network.

It is crucial for large-scale communication networks such as the internet to be resilient against attacks such as censorship and surveillance, which pose a threat to free expression and free association. Self-organized networks such as the internet’s router network typically have heavy-tailed degree distributions, making them highly vulnerable to targeted attacks against central nodes. While cryptographic solutions exist, they fail to address the underlying topological problem, and remain vulnerable to man-in-the-middle attacks and coercion. Coercion-resistant, topological approaches to attack tolerance are needed to address the current vulnerability of communications infrastructure to censorship and surveillance. We present a novel concurrent multipath routing (CMR) algorithm for the wraparound butterfly network topology, as well as a highly attack-tolerant Structured Multipath Fault Tolerance (SMFT) architecture which incorporates the butterfly CMR algorithm. We also identify a previously unexplored relationship between network topology, trust transitivity, and attack-tolerance, and provide a framework for further exploration of this relationship. Our work is the first theoretical demonstration of a point-to-point communication network architecture that can resist coercion and other non-technical attacks, without requiring infinitely transitive trust. To address cases where the network structure cannot be fully controlled, we demonstrate how a snapshot of the internet’s router network can be partially rewired for greater attack-tolerance. More broadly, we hope that this work will serve as a starting point for the evelopment of additional topology-based attack-tolerant communication architectures to guard against the dangers of censorship and surveillance.

Image domain dual material decomposition for dual-energy CT using butterfly network.

Dual-energy CT (DECT) has been increasingly used in imaging applications because of its capability for material differentiation. However, material decomposition suffers from magnified noise from two CT images of independent scans, leading to severe degradation of image quality. Existing algorithms exhibit suboptimal decomposition performance because they fail to fully depict the mapping relationship between DECT images and basis materials under noisy conditions. Convolutional neural network exhibits great promise in the modeling of data coupling and has recently become an important technique in medical imaging application. Inspired by its impressive potential, we developed a new Butterfly network to perform the image domain dual material decomposition. The Butterfly network is derived from the model of image domain DECT decomposition by exploring the geometric relationship between the mapping functions of the data model and network components. The network is designed as the double-entry double-out crossover architecture based on the decomposition formulation. It enters a pair of dual-energy images as inputs and defines the ground true decomposed images as each label. The crossover architecture, which plays an important role in material decomposition, is designed to implement the information exchange between the two material generation pathways in the network. The proposed network is further applied on the digital phantom and clinical data to evaluate its performance. The qualitative and quantitative evaluations of the material decomposition of digital phantoms and clinical data indicate that the proposed network outperforms its counterparts. For the digital phantom, the proposed network reduces the standard deviation (SD) of noise in tissue, bone, and mixture regions by an average of 95.75% and 86.58% compared with the direct matrix inversion and the conventional iterative method, respectively. The line profiles and image biases of the decomposition results of digital phantom indicate that the proposed network provides the decomposition results closest to the ground truth. The proposed network reduces the SD of the noise in decomposed images of clinical head data by over 90% and 80% compared with the direct matrix inversion and conventional iterative method, respectively. As the modulation transfer function decreases to 50%, the proposed network increases the spatial resolution by average factors of 1.34 and 1.17 compared with the direct matrix inversion and conventional iterative methods, respectively. The proposed network is further applied to the clinical abdomen data. Among the three methods, the proposed method received the highest score from six radiologists in the visual inspection of noise suppression in the clinical data. We develop a model-based Butterfly network to perform image domain material decomposition for DECT. The decomposition results of digital phantom validate its capability ofdecomposing two basis materials from DECT images. The proposed approach also leads to higherdecomposition quality in noise suppression on clinical datasets as compared with those using conventional schemes.

Parallel access by butterfly networks for any degree permutation polynomial and ARP interleavers

Using more processors for parallel turbo decoding is an important issue to speed up the processing at the receiver of a communication system. Butterfly networks used to map the addresses of extrinsic values represent an elegant and simple solution in parallel turbo decoding. Recently, it has been shown that quadratic permutation polynomial (QPP) interleavers allow an easy way to compute the control bits for a butterfly network. In this paper we show that not only QPP interleavers, but any degree permutation polynomial (PP) interleavers and almost regular permutation (ARP) interleavers also allow the same easy way to compute the control bits required in butterfly networks. As a consequence, it is useful to apply the butterfly networks in parallel turbo decoding when using these performant algebraic interleavers.

Quantum Network Communication With a Novel Discrete-Time Quantum Walk

In this paper, we propose a scheme using a new kind of quantum walk to realize quantum communication well. In our scheme, obviously, there are three advantages. First, the initial states can be transferred from the source to the target by finite steps, which will make execution easier in practice. Second, due to fewer restrictions to the network models, our scheme can be adapted to more general network models, such as the butterfly network, the inverted crown network, the grail network, and the generalized butterfly network G k . Finally, arbitrary single-qubit and two-qubit initial states can be transferred to different kinds of network models. At the same time, the fidelity and probability equal 1. Our work promotes more applications of combining quantum computation with quantum communication in the future.

Anti-Noise Quantum Network Coding Protocol Based on Bell States and Butterfly Network Model

How to establish a secure and efficient quantum network coding algorithm is one of important research topics of quantum secure communications. Based on the butterfly network model and the characteristics of easy preparation o... | Find, read and cite all the research you need on Tech Science Press

Efficient quantum state transmission via perfect quantum network coding

Quantum network coding with the assistance of auxiliary resources can achieve perfect transmission of the quantum state. This paper suggests a novel perfect network coding scheme to efficiently solve the quantum $k$-pair problem, in which only a few assisting resources are introduced. Specifically, only one pair of maximally entangled state needs to be pre-shared between two intermediate nodes, and only $O(k)$ of classical information is transmitted though the network. Moreover, the classical communication used in our protocol does not cause transmission congestion, providing better adaptability to large-scale quantum $k$-pair networks.Through relevant analyses and comparisons, we demonstrate that our proposed scheme saves resources and has good application value, thereby showing its high efficiency. Furthermore, the proposed scheme achieves 1-max flow quantum communication, and the achievable rate region result is extended from its counterpart over the butterfly network.

Optimal DoF Region of the MIMO Butterfly Network

A butterfly network is a classical model to demonstrate the throughput advantage of the network coding technique over wireline networks. We are curious about what happens if the wireless network is considered instead, especially for the multiple-antenna scenarios. In this letter, we study this problem and aim to fully characterize the degrees of freedom (DoF) region for the multiple-input-multiple-output (MIMO) butterfly network in the general setting. It turns out that significant DoF gains can be achieved if more advanced interference management schemes than the conventional time-sharing one are employed for the MIMO butterfly network, which is quite similar to the case of its wireline counterpart. Moreover, an existence proof based on the algebraic topology is provided in this letter, which may be of independent interest.

Minimum rank and zero forcing number for butterfly networks

The minimum rank of a simple graph $G$ is the smallest possible rank over all symmetric real matrices $A$ whose nonzero off-diagonal entries correspond to the edges of $G$. Using the zero forcing number, we prove that the minimum rank of the butterfly network is $\frac19\left[(3r+1)2^{r+1}-2(-1)^r\right]$ and that this is equal to the rank of its adjacency matrix.

On the DoF of Two-Way MIMO Butterfly Networks

This paper studies the degrees of freedom (DoF) of two-way multiple-input multiple-output (MIMO) butterfly networks, a class of two-way four-unicast networks, that consists of four source/destination nodes and three relay nodes. We first give upper and lower bounds on the sum DoF of such network with any number of antennas in each node. We also study the DoF of the two-way MIMO butterfly network with caching at the relays. For the special case that each source/destination node has $M$ antennas and each relay node has $N$ antennas, we obtain the exact DoF for some special cases. Specifically, if $N \leq M$ , then without caching, ${\rm DoF}=2N$ , which is equal to the DoF of a one-way MIMO butterfly network, and with relay caching, ${\rm DoF}=4N$ . Moreover, if $N > 2M$ , then with or without relay caching, ${\rm DoF}=4M$ which doubles the DoF of a one-way MIMO butterfly network. Hence, for this MIMO butterfly network, when the number of relay antennas is large compared with the number of source/destination antennas, bidirectional transmission improves the DoF; and when the number of relay antennas is small, relay caching improves the DoF.

Outage behaviour and SCK‐based power allocation for analogue network coding protocol in cooperative networks

In this study, the authors consider the communication scenario where two sources communicate with the help of a single relay, modelled as a half-duplex butterfly network. Closed-form expressions of the outage probabilities are derived in the high signal-to-noise ratio (SNR) regime for the orthogonal amplify-and-forward and non-orthogonal amplify-and-forward protocols. Then the expressions are approximated to enable power allocation. Closed-form power allocation schemes are proposed for each protocol, where only statistical channel knowledge is required. It is shown that the authors analyses match well with simulation results and the approximated versions are very close to the simulation results. The proposed power allocation schemes achieve a large SNR gain over the equal-power strategies and approach the optimal power allocation schemes.

Exact and heuristic algorithms for Space Information Flow.

Space Information Flow (SIF) is a new promising research area that studies network coding in geometric space, such as Euclidean space. The design of algorithms that compute the optimal SIF solutions remains one of the key open problems in SIF. This work proposes the first exact SIF algorithm and a heuristic SIF algorithm that compute min-cost multicast network coding for N (N ≥ 3) given terminal nodes in 2-D Euclidean space. Furthermore, we find that the Butterfly network in Euclidean space is the second example besides the Pentagram network where SIF is strictly better than Euclidean Steiner minimal tree. The exact algorithm design is based on two key techniques: Delaunay triangulation and linear programming. Delaunay triangulation technique helps to find practically good candidate relay nodes, after which a min-cost multicast linear programming model is solved over the terminal nodes and the candidate relay nodes, to compute the optimal multicast network topology, including the optimal relay nodes selected by linear programming from all the candidate relay nodes and the flow rates on the connection links. The heuristic algorithm design is also based on Delaunay triangulation and linear programming techniques. The exact algorithm can achieve the optimal SIF solution with an exponential computational complexity, while the heuristic algorithm can achieve the sub-optimal SIF solution with a polynomial computational complexity. We prove the correctness of the exact SIF algorithm. The simulation results show the effectiveness of the heuristic SIF algorithm.

Quantum network communication: a discrete-time quantum-walk approach

We study the problem of quantum multi-unicastcommunication over the butterfly network in a quantum-walk architecture,where multiple arbitrary single-qubit states are transmitted simultaneouslybetween multiple source-sink pairs. Here, by introducing quantum walks, wedemonstrate a quantum multi-unicast communication scheme over the butterflynetwork and the inverted crown network, respectively, where the arbitrarysingle-qubit states can be efficiently transferred with both the probabilityand the state fidelity one. The presented result concerns only the butterflynetwork and the inverted crown network, but our techniques can be applied toa more general graph. It paves a way to combine quantum computation andquantum network communication.

A highly efficient reconfigurable rotation unit based on an inverse butterfly network

We propose a reconfigurable control-bit generation algorithm for rotation and sub-word rotation operations. The algorithm uses a self-routing characteristic to configure an inverse butterfly network. In addition to being highly parallelized and inexpensive, the algorithm integrates the rotation-shift, bi-directional rotation-shift, and sub-word rotation-shift operations. To our best knowledge, this is the first scheme to accommodate a variety of rotation operations into the same architecture. We have developed the highly efficient reconfigurable rotation unit (HERRU) and synthesized it into the Semiconductor Manufacturing International Corporation (SMIC)’s 65-nm process. The results show that the overall efficiency (relative area×relative latency) of our HERRU is higher by at least 23% than that of other designs with similar functions. When executing the bi-directional rotation operations alone, HERRU occupies a significantly smaller area with a lower latency than previously proposed designs.

Single-shot secure quantum network coding on butterfly network with free public communication

Quantum network coding on the butterfly network has been studied as a typical example of quantum multiple cast network. We propose a secure quantum network code for the butterfly network with free public classical communication in the multiple unicast setting under restricted eavesdropper’s power. This protocol certainly transmits quantum states when there is no attack. We also show the secrecy with shared randomness as additional resource when the eavesdropper wiretaps one of the channels in the butterfly network and also derives the information sending through public classical communication. Our protocol does not require verification process, which ensures single-shot security.

Design of adaptive constellations and error protection coding for wireless network coding in 5-node butterfly networks

In this paper, we focus on the design of a reliable communication scheme for generic 5-node wireless butterfly networks (WBN). Since the WBN represents a natural extension of the basic two-way relay communication channel, we follow the paradigm of Wireless Physical Layer Network Coding and we show that the design of novel adaptive constellations is desirable to exploit the promising performance of wireless network coding in WBN. We introduce a systematic constellation design algorithm, and we show that the proposed constellations are capable of outperforming conventional point-to-point modulations in WBN over the whole range of channel signal to noise ratios. To further support the applicability of the proposed constellations in a practical system, we integrate a simple binary channel coding scheme, providing a robust adaptive modulation and coding scheme for relaying in symmetric WBNs with arbitrary channel conditions.

An Optimal Single-Path Routing Algorithm in the Datacenter Network DPillar

DPillar has recently been proposed as a server-centric datacenter network and is combinatorially related to (but distinct from) the well-known wrapped butterfly network. We explain the relationship between DPillar and the wrapped butterfly network before proving that the underlying graph of DPillar is a Cayley graph; hence, the datacenter network DPillar is node-symmetric. We use this symmetry property to establish a single-path routing algorithm for DPillar that computes a shortest path and has time complexity $O(k)$, where $k$ parameterizes the dimension of DPillar (we refer to the number of ports in its switches as $n$). Our analysis also enables us to calculate the diameter of DPillar exactly. Moreover, our algorithm is trivial to implement, being essentially a conditional clause of numeric tests, and improves significantly upon a routing algorithm earlier employed for DPillar. Furthermore, we provide empirical data in order to demonstrate this improvement. In particular, we empirically show that our routing algorithm improves the average length of paths found, the aggregate bottleneck throughput, and the communication latency. A secondary, yet important, effect of our work is that it emphasises that datacenter networks are amenable to a closer combinatorial scrutiny that can significantly improve their computational efficiency and performance.

On quadratic permutation polynomials, turbo codes, and butterfly networks

The use of quadratic permutation polynomials as interleavers for turbo codes is justified, among other things, by the fact that the quadratic permutation polynomials provide turbo codes of very good performance and simultaneously they support a specific conflict-free parallel access to extrinsic values in two different access orders. It is rather easy to write and read in parallel to and from multiple memories by one access order. Nevertheless, turbo decoders have to access the multiple memories at least by two different access orders. Then, parallel writes and reads lead to conflicts in accessing the multiple memories. Having only one conflict-free parallel access method for quadratic permutation polynomials restricts to design efficient flexible (multistandard) decoders for turbo codes at high data rates. We show that quadratic permutation polynomials with butterfly networks as interconnection networks between decoder units and memories support many kinds of flexible and variant conflict-free parallel access methods for turbo codes that are not known today. This result extends possibilities to implement high speed turbo decoders not only by application specific circuits but also by (general purpose) graphics processor units for (multistandard) modems.

Chaotic Image Encryption Algorithm Based on Bit Permutation and Dynamic DNA Encoding.

With the help of the fact that chaos is sensitive to initial conditions and pseudorandomness, combined with the spatial configurations in the DNA molecule's inherent and unique information processing ability, a novel image encryption algorithm based on bit permutation and dynamic DNA encoding is proposed here. The algorithm first uses Keccak to calculate the hash value for a given DNA sequence as the initial value of a chaotic map; second, it uses a chaotic sequence to scramble the image pixel locations, and the butterfly network is used to implement the bit permutation. Then, the image is coded into a DNA matrix dynamic, and an algebraic operation is performed with the DNA sequence to realize the substitution of the pixels, which further improves the security of the encryption. Finally, the confusion and diffusion properties of the algorithm are further enhanced by the operation of the DNA sequence and the ciphertext feedback. The results of the experiment and security analysis show that the algorithm not only has a large key space and strong sensitivity to the key but can also effectively resist attack operations such as statistical analysis and exhaustive analysis.

More agents may decrease global work: A case in butterfly decontamination

This paper is a contribution to network decontamination with a view inherited from parallel processing. At the beginning some or all the vertices may be contaminated. The network is visited by a group of decontaminating agents. When a decontaminated vertex is left by the agents, it can be re-contaminated only if the number of infected neighbors exceeds a certain immunity threshold m. The main goal of the studies in this line is to minimize the number A of agents needed to do the job and, for a minimum team, to minimize the number M of agent moves. Instead of M we consider the number T of steps (i.e. parallel moves) as a measure of time, and evaluate the quality of a protocol on the basis of its workW=AT. Taking butterfly networks as an example, we compare different protocols and show that, for some values of m, a larger team of agents may require smaller work.