Fusion Trees Research Articles

Mechanical equipment fault diagnosis based on wireless sensor network data fusion technology

Abstract To save network energy consumption and prolong network life cycle in complex mechanical fault diagnosis, a research method of data fusion routing protocol algorithm based on wireless sensor network (WSN) is proposed. The specific content of the method is as follows: First, the low-energy adaptive clustering hierarchy algorithm is analyzed and discussed. On this basis, the prim route fusion algorithm is proposed to realize the effective utilization of energy and prolong the life of the network. Then, the WSN is abstracted as an undirected graph. From the perspective of saving the energy of the whole network, several current algorithms for building fusion trees are compared. The experimental results show that the prim algorithm consumes energy only after 700 rounds of clustering, while the leach clustering algorithm consumes energy only after 500 rounds. This shows that applying the prim algorithm can reduce the energy consumption of the whole network and prolong the life cycle of the network. However, the algorithm is carried out on the premise of uniform distribution of nodes, and there is a certain gap with the specific application of WSN in mechanical fault diagnosis. In the comparison of node energy consumption, it is found that compared with using the shortest path tree, using the central point of graph algorithm can greatly save the energy consumption of the node and has better performance. Practice has proved that this method can effectively remove redundant data information and solve the problem of unreliable data collected by a single sensor node. It is more suitable for the specific application of WSN in mechanical fault diagnosis.

Reaction-diffusion dynamics in a Fibonacci chain: Interplay between classical and quantum behavior

We study the reaction-diffusion dynamics of Fibonacci anyons in a one dimensional lattice. Due to their non-Abelian nature, besides the position degree of freedom (DOF), these anyons also have a nonlocal internal DOF, which can be characterized by a fusion tree. We first consider a pure-reaction dynamics associated with the internal DOF, which is of intrinsically quantum origin, with either an "all-$\tau$" or "completely random" initial fusion tree. These two fusion trees are unstable and likely stable steady states for the internal DOF, respectively. We obtain the decay rate of the anyon number for these two cases exactly. Still using these two initial fusion trees, we study the full reaction-diffusion dynamics, and find an interesting interplay between classical and quantum behaviors: These two fusion trees are still respectively unstable and likely stable steady states of the internal DOF, while the dynamics of the position DOF can be mapped to a hybrid classical $A+A\rightarrow 0$ and $A+A\rightarrow A$ reaction-diffuson dynamics, with the relative reaction rates of these two classical dynamics determined by the state of the nonlocal internal DOF. In particular, the anyon density at late times are given by $\rho(t)=\frac{c}{\sqrt{8\pi}}(Dt)^{-\Delta}$, where $D$ is a non-universal diffusion constant, $\Delta=1/2$ is superuniversal, and $c$ is universal and can be obtained exactly in terms of the fusion tree structure. Specifically, $c=\frac{2\varphi}{\varphi+1}$ and $c=\frac{2(4\varphi+3)}{5(\varphi+1)}$ for the all-$\tau$ and completely random configuration respectively, where $\varphi=\frac{\sqrt{5}+1}{2}$ is the golden ratio. We also study the two-point correlation functions.

Benchmarking global SU(2) symmetry in two-dimensional tensor network algorithms

We implement and benchmark tensor network algorithms with $SU(2)$ symmetry for systems in two spatial dimensions and in the thermodynamic limit. Specifically, we implement $SU(2)$-invariant versions of the infinite Projected Entangled Pair States (iPEPS) and infinite Projected Entangled Simplex States (iPESS) methods. Our implementation of $SU(2)$ symmetry follows the formalism based on fusion trees from [P. Schmoll, S. Singh, M. Rizzi, R. Or\'us, arXiv:1809.08180]. In order to assess the utility of implementing $SU(2)$ symmetry the algorithms are benchmarked for three models with different local spin: the spin-1 bilinear-biquadratic model on the square lattice, and the Kagome Heisenberg antiferromagnets (KHAF) for spin-1/2 and spin-2. We observe that the implementation of $SU(2)$ symmetry provides better energies in general than non-symmetric simulations, with smooth scalings with respect to the number of parameters in the ansatz, and with the actual improvement depending on the specifics of the model. In particular, for the spin-2 KHAF model, our $SU(2)$ simulations are compatible with a quantum spin liquid ground state.

A programming guide for tensor networks with global [formula omitted] symmetry

This paper is a manual with tips and tricks for programming tensor network algorithms with global SU(2) symmetry. We focus on practical details that are many times overlooked when it comes to implementing the basic building blocks of codes, such as useful data structures to store the tensors, practical ways of manipulating them, and adapting typical functions for symmetric tensors. Here we do not restrict ourselves to any specific tensor network method, but keep always in mind that the implementation should scale well for simulations of higher-dimensional systems using, e.g., Projected Entangled Pair States, where tensors with many indices may show up. To this end, the structural tensors (or intertwiners) that arise in the usual decomposition of SU(2)-symmetric tensors are never explicitly stored throughout the simulation. Instead, we store and manipulate the corresponding fusion trees – an algebraic specification of the symmetry constraints on the tensor – in order to implement basic SU(2)-symmetric tensor operations. This fusion tree approach is readily extensible to anyonic systems, as we demonstrate for a chain of Fibonacci anyons.

Non-Abelian statistics in one dimension: Topological momentum spacings and SU(2) level-kfusion rules

We use a family of critical spin chain models discovered recently by one of us [M. Greiter, Mapping of Parent Hamiltonians, Springer, Berlin/Heidelberg 2011] to propose and elaborate that non-Abelian, SU(2) level $k=2S$ anyon statistics manifests itself in one dimension through topological selection rules for fractional shifts in the spacings of linear momenta, which yield an internal Hilbert space of, in the thermodynamic limit degenerate states. These shifts constitute the equivalent to the fractional shifts in the relative angular momenta of anyons in two dimensions. We derive the rules first for Ising anyons, and then generalize them to SU(2) level $k$ anyons. We establish a one-to-one correspondence between the topological choices for the momentum spacings and the fusion rules of spin \half spinons in the SU(2) level $k$ Wess--Zumino--Witten model, where the internal Hilbert space is spanned by the manifold of allowed fusion trees in the Bratelli diagrams. Finally, we show that the choices in the fusion trees may be interpreted as the choices between different domain walls between the $2S+1$ possible, degenerate dimer configurations of the spin $S$ chains at the multicritical point.

Neighbor Dependency-Based Dynamic Fusion Tree Generation for a Multi-radar System

This paper proposes a fusion tree generation (FTG) algorithm for a two-tier fusion process in a multi-radar system. The two-tier fusion process divides the fusion process into local and global parts. The fusion workload of a multi-radar system at the central server can be reduced by applying two-tier approach. However, the two-tier fusion process requires a balanced fusion tree to increase the number of processed tracks. This paper presents a dynamic fusion tree generation (D-FTG) algorithm based on a clustering scheme. We developed a novel clustering scheme, which can be used in generating balanced fusion trees in a multi-radar system. The developed clustering scheme is able to generate a balanced fusion tree with the neighbor dependency-based scoring method, manage the dynamic changes of the distribution of targets with the pruning-and-rejoining strategy, and cope with the failure of radar nodes by repeating initial clustering. Simulation results show that D-FTG outperforms existing clustering methods when used to generate balanced fusion trees in a dynamic multi-radar system.

Experiments and Models for Decision Fusion by Humans in Inference Networks

With the advent of the Internet of Things (IoT) and a rapid deployment of smart devices and wireless sensor networks (WSNs), humans interact extensively with machine data. These human decision makers use sensors that provide information through a sociotechnical network. The sensors can be other human users or they can be IoT devices. The decision makers themselves are also part of the network, and there is a need to understand how they will behave. In this paper, the decision fusion behavior of humans is analyzed on the basis of behavioral experiments. The data collected from these experiments demonstrate that people perform decision fusion in a stochastic manner dependent on various factors, unlike machines that perform this task in a deterministic manner. A Bayesian hierarchical model is developed to characterize the observed stochastic human behavior. This hierarchical model captures the differences observed in people at individual, crowd, and population levels. The implications of such a model on designing large-scale inference systems are presented by developing optimal decision fusion trees with both human and machine agents.

Cost-Effective Robustness in Clock Networks Using Near-Tree Structures

Clock trees are commonly used to deliver clock signals to sequential elements in circuits. However, by construction, tree structures are inherently prone to failure caused by variations. The robustness of a clock tree can be improved by inserting redundancy in the form of cross links or multilevel fusion trees. Such near-tree structures can provide robustness at low cost. In this paper, we establish that the locations of the inserted redundancy are crucial in providing cost-effective robustness. We present two methods to systematically insert redundancy. The redundancy is realized by either inserting cross links or performing local merges. Moreover, we present a vertex reduction method that reduces the amount of redundancy that needs to be inserted in our near-tree structures. Empirical results show that our structures are more robust to variations and have lower power consumption compared to the state-of-the-art clock networks. Furthermore, our near-tree structures provide smooth trade-offs between cost and robustness, reducing clock skews by 11%-39% at an expense of 3%-68% higher power consumption.

Fusion trees for fast and accurate classification of hyperspectral data with ensembles of $$\gamma$$ γ -divergence-based RBF networks

Ensembles of RBF networks trained with $$\gamma$$ ? -divergence-based similarity measures can improve classification accuracy of hyperspectral imaging data significantly compared to any single RBF network as well as to RBF ensembles based on the Euclidian distance. So far, the drawback of using classifier ensembles is the need to compute the results of a typically large number of RBF networks prior to combination. In this paper, a modified approach to the fusion of classifier outputs is proposed which is based on decision trees. It is shown for several real-world datasets that a small subset of the RBF networks contributes to the decisions in the average case. Hence, for any decision, a conditional computation of required RBF network outputs yields a significant decrease in the computational costs. Additionally, a selection scheme for subsets of RBF classifiers based on their relevance in the fusion process is proposed. This alternative approach can be used, if analysis requires fixed settings, e.g., to meet time constraints.

Limit theorem for Fibonacci anyons in a pentagon underFmoves

In this paper we present an introduction to the theory of anyons and discuss how basis sets and matrix representation of the interaction terms can be obtained, using non-Abelian Fibonacci anyons. We show that the change of basis matrix for certain fusion trees called pentagons is given by a certain unitary matrix which is introduced in Sec. I, and denoted ${F}_{\ensuremath{\tau}}^{\ensuremath{\tau},\ensuremath{\tau},\ensuremath{\tau}}$ throughout the remainder of the paper. We perform a quantum walk with the Fibonacci anyons governed by the unitary matrix ${F}_{\ensuremath{\tau}}^{\ensuremath{\tau},\ensuremath{\tau},\ensuremath{\tau}}$ and obtain the limiting distribution of the walk.

Data Fusion Trees for Detection: Does Architecture Matter?

We consider the problem of decentralized detection in a network consisting of a large number of nodes arranged as a tree of bounded height, under the assumption of conditionally independent and identically distributed (i.i.d.) observations. We characterize the optimal error exponent under a Neyman-Pearson formulation. We show that the Type II error probability decays exponentially fast with the number of nodes, and the optimal error exponent is often the same as that corresponding to a parallel configuration. We provide sufficient, as well as necessary, conditions for this to happen. For those networks satisfying the sufficient conditions, we propose a simple strategy that nearly achieves the optimal error exponent, and in which all non-leaf nodes need only send 1-bit messages.

Novel Transformation Techniques Using Q-Heaps with Applications to Computational Geometry

Using the notions of Q-heaps and fusion trees developed by Fredman and Willard, we develop general transformation techniques to reduce a number of computational geometry problems to their special versions in partially ranked spaces. In particular, we develop a fast fractional cascading technique, which uses linear space and enables sublogarithmic iterative search on catalog trees in the case when the degree of each node is bounded by $O(\log^{\epsilon}n)$ for some constant $\epsilon >0$, where $n$ is the total size of all the lists stored in the tree. We apply the fast fractional cascading technique in combination with the other techniques to derive the first linear-space sublogarithmic time algorithms for two fundamental geometric retrieval problems: orthogonal segment intersection and rectangular point enclosure.

Examining Computational Geometry, Van Emde Boas Trees, and Hashing from the Perspective of the Fusion Tree

This article illustrates several examples of computer science problems whose performance can be improved with the use of either the fusion trees [Fredman and Willard, J. Comput. System Sci., 47 (1993), pp. 424--436; Fredman and Willard, J. Comput. System Sci., 48 (1994), pp. 533--551] or one of several recent improvements to this data structure. It is likely that many other data structures can also have their performance improved with fusion trees. The examples here are only illustrative.

Fusion trees can be implemented with AC 0 instructions only

Addressing a problem of Fredman and Willard, we implement fusion trees in deterministic linear space using AC 0 instructions only. More precisely, we show that a subset of {0,…,2 w - 1} of size n can be maintained using linear space under insertion, deletion, predecessor, and successor queries, with O( log n log log n ) amortized time per operation on a RAM with word size w, where the only computational instructions allowed on the RAM are functions in AC 0 . The AC 0 instructions used are not all available on today's computers.

Sorting in Linear Time?

We show that a unit-cost RAM with a word length ofwbits can sortnintegers in the range 0…2w−1 inO(nloglogn) time for arbitraryw⩾logn, a significant improvement over the bound ofO(nlogn) achieved by the fusion trees of Fredman and Willard. Provided thatw⩾(logn)2+εfor some fixedε>0, the sorting can even be accomplished in linear expected time with a randomized algorithm. Both of our algorithms parallelize without loss on a unit-cost PRAM with a word length ofwbits. The first one yields an algorithm that usesO(logn) time andO(nloglogn) operations on a deterministic CRCW PRAM. The second one yields an algorithm that usesO(logn) expected time andO(n) expected operations on a randomized EREW PRAM, provided thatw⩾(logn)2+εfor some fixedε>0. Our deterministic and randomized sequential and parallel algorithms generalize to the lexicographic sorting of multiple-precision integers represented in several words.

On the difficulty of range searching

We consider the general problem of (2-dimensional) range reporting allowing arbitrarily convex queries. We show that using a traditional approach, even when incorporating techniques like those used in fusion trees, a polylogarithmic query time cannot be achieved unless more than linear space is used. Our arguments are based on a new non-trivial lower bound in a model of computation which, in contrast to the pointer machine model, allows for the use of arrays and bit manipulation. The crucial property of our model, Layered partitions, is that it can be used to describe all known algorithms for processing range queries, as well as many other data structures used to represent multi-dimensional data. We show that Ω( logn logT(n) ) partitions must be used to allow queries in O( T( n) + k) time, for n total and k reported elements, and for any growing function T( n).

Improved Parallel Integer Sorting without Concurrent Writing

We show thatnintegers in the range 1, …, ncan be sorted stably on an EREW PRAM usingO(t) time andO(n(lognloglogn+(log n)2/t)) operations, for arbitrary givent⩾log n log log n, and on a CREW PRAM usingO(t) time andO(n(logn+log n/2t/log n)) operations, for arbitrary givent⩾log n. In addition, we are able to sortnarbitrary integers on a randomized CREW PRAM within the same resource bounds with high probability. In each case our algorithm is a factor of almostΘ:(logn) closer to optimality than all previous algorithms for the stated problem in the stated model, and our third result matches the operation count of the best previous sequential algorithm. We also show thatnintegers in the range 1, …, mcan be sorted inO((log n)2) time withO(n) operations on an EREW PRAM using a nonstandard word length ofO(log n log log n log m) bits, thereby greatly improving the upper bound on the word length necessary to sort integers with a linear time–processor product, even sequentially. Our algorithms were inspired by, and in one case directly use, the fusion trees of Fredman and Willard.

Fusion Trees can be Implemented with AC0 Instructions only

Addressing a problem of Fredman and Willard, we implement fusion<br />trees in deterministic linear space using AC0 instructions only.

Surpassing the information theoretic bound with fusion trees

This paper introduces a new sublogarithmic data structure for searching, the fusion tree. These trees lead to improved worst-case algorithms for sorting and searching, surpassing the limitations of the information theoretic lower bound.

Computing 'Fusion Trees' to Explode Barriers

Computing 'Fusion Trees' to Explode Barriers