Distance-layer structure of the De Bruijn and Kautz digraphs: analysis and application to deflection routing

This is the peer reviewed version of the following article: Fàbrega, J.; Martí, J.; Muñoz, X. Distance-layer structure of the De Bruijn and Kautz digraphs: analysis and application to deflection routing. "Networks", 29 Juliol 2023, which has been published in final form at https://onlinelibrary.wiley.com/doi/10.1002/net.22177. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions. This article may not be enhanced, enriched or otherwise transformed into a derivative work, without express permission from Wiley or by statutory rights under applicable legislation. Copyright notices must not be removed, obscured or modified. The article must be linked to Wiley’s version of record on Wiley Online Library and any embedding, framing or otherwise making available the article or pages thereof by third parties from platforms, services and websites other than Wiley Online Library must be prohibited.In this article, we present a detailed study of the reach distance-layer structure of the De Bruijn and Kautz digraphs, and we apply our analysis to the performance evaluation of deflection routing in De Bruijn and Kautz networks. Concerning the distance-layer structure, we provide explicit polynomial expressions, in terms of the degree of the digraph, for the cardinalities of some relevant sets of this structure. Regarding the application to defection routing, and as a consequence of our polynomial description of the distance-layer structure, we formulate explicit expressions, in terms of the degree of the digraph, for some probabilities of interest in the analysis of this type of routing. De Bruijn and Kautz digraphs are fundamental examples of digraphs on alphabet and iterated line digraphs. If the topology of the network under consideration corresponds to a digraph of this type, we can perform, in principle, a similar vertex layer description.Partially supported by the Ministerio de Ciencia e Innovación/Agencia Estatal de Investigación, Spain, and the European Regional Development Fund under project PGC2018-095471-B-I00; and by AGAUR from the Catalan Government under project 2017SGR-1087.Peer ReviewedPostprint (author's final draft

Properties and Algorithms of the KCube Interconnection Networks

The KCube interconnection network was first introduced in 2010 in order to exploit the good characteristics of two well-known interconnection networks, the hypercube and the Kautz graph. KCube links up multiple processors in a communication network with high density for a fixed degree. Since the KCube network is newly proposed, much study is required to demonstrate its potential properties and algorithms that can be designed to solve parallel computation problems. In this thesis we introduce a new methodology to construct the KCube graph. Also, with regard to this new approach, we will prove its Hamiltonicity in the general KC(m; k). Moreover, we will find its connectivity followed by an optimal broadcasting scheme in which a source node containing a message is to communicate it with all other processors. In addition to KCube networks, we have studied a version of the routing problem in the traditional hypercube, investigating this problem: whether there exists a shortest path in a Qn between two nodes 0n and 1n, when the network is experiencing failed components. We first conditionally discuss this problem when there is a constraint on the number of faulty nodes, and subsequently introduce an algorithm to tackle the problem without restrictions on the number of nodes

Small-world interconnection networks for large parallel computer systems

The use of small-world graphs as interconnection networks of multicomputers is proposed and analysed in this work. Small-world interconnection networks are constructed by adding (or modifying) edges to an underlying local graph. Graphs with a rich local structure but with a large diameter are shown to be the most suitable candidates for the underlying graph. Generation models based on random and deterministic wiring processes are proposed and analysed. For the random case basic properties such as degree, diameter, average length and bisection width are analysed, and the results show that a fast transition from a large diameter to a small diameter is experienced when the number of new edges introduced is increased. Random traffic analysis on these networks is undertaken, and it is shown that although the average latency experiences a similar reduction, networks with a small number of shortcuts have a tendency to saturate as most of the traffic flows through a small number of links. An analysis of the congestion of the networks corroborates this result and provides away of estimating the minimum number of shortcuts required to avoid saturation. To overcome these problems deterministic wiring is proposed and analysed. A Linear Feedback Shift Register is used to introduce shortcuts in the LFSR graphs. A simple routing algorithm has been constructed for the LFSR and extended with a greedy local optimisation technique. It has been shown that a small search depth gives good results and is less costly to implement than a full shortest path algorithm. The Hilbert graph on the other hand provides some additional characteristics, such as support for incremental expansion, efficient layout in two dimensional space (using two layers), and a small fixed degree of four. Small-world hypergraphs have also been studied. In particular incomplete hypermeshes have been introduced and analysed and it has been shown that they outperform the complete traditional implementations under a constant pinout argument. Since it has been shown that complete hypermeshes outperform the mesh, the torus, low dimensional m-ary d-cubes (with and without bypass channels), and multi-stage interconnection networks (when realistic decision times are accounted for and with a constant pinout), it follows that incomplete hypermeshes outperform them as well

Large Propulsion Demands Increase Locomotor Adaptation at the Expense of Step Length Symmetry

There is an interest to identify factors facilitating locomotor adaptation induced by split-belt walking (i.e., legs moving at different speeds) because of its clinical potential. We hypothesized that augmenting braking forces, rather than propulsion forces, experienced at the feet would increase locomotor adaptation during and after split-belt walking. To test this, forces were modulated during split-belt walking with distinct slopes: incline (larger propulsion than braking), decline (larger braking than propulsion), and flat (similar propulsion and braking). Step length asymmetry was compared between groups because it is a clinically relevant measure robustly adapted on split-belt treadmills. Unexpectedly, the group with larger propulsion demands (i.e., the incline group) changed their gait the most during adaptation, reached their final adapted state more quickly, and had larger after-effects when the split-belt perturbation was removed. We also found that subjects who experienced larger disruptions of propulsion forces in early adaptation exhibited greater after-effects, which further highlights the catalytic role of propulsion forces on locomotor adaptation. The relevance of mechanical demands on shaping our movements was also indicated by the steady state split-belt behavior, during which each group recovered their baseline leg orientation to meet leg-specific force demands at the expense of step length symmetry. Notably, the flat group was nearly symmetric, whereas the incline and decline group overshot and undershot step length symmetry, respectively. Taken together, our results indicate that forces propelling the body facilitate gait changes during and after split-belt walking. Therefore, the particular propulsion demands to walk on a split-belt treadmill might explain the gait symmetry improvements in hemiparetic gait following split-belt training

A graph theoretic approach to transputer network design for computer vision

The work in this thesis is concerned with parallel architectures based on the Inmos transputer-type processors and parallelisation of some computer vision tasks chosen from low to high level. The transputer is a microprocessor with a micro-programmed scheduler and four serial communication links. It directly supports parallel processing since several transputers can be connected through their links to co-operate on solving a problem. Also several processes can be run on the same transputer. A major issue in parallel processing is the communication overhead introduced by parallelising a given task. This overhead is not present in sequential processing and must be curbed if the implementation of a task on a parallel machine is to be successful. The interconnection network underlying the architecture of a parallel computer is therefore of the utmost importance. Computer Vision consists of a hierarchy of tasks ranging from low-level operations dealing with large amounts of relatively simple data to high level operations handling increasingly complex structures. In this work a novel edge detector based on adaptive filtering and an edge detector operating on colour images are presented and implemented on a number of transputers. These parallel implementations together with implementations of vector quantisation, Fourier descriptors for shape discrimination, the Hough transform and the Maximum clique algorithm, offer a notable performance increase when compared with sequential implementations. However, every algorithm required the design of a specific network of transputers to take advantage of the parallelism and data dependencies inherent in each. Consequently, attention is focused on the topology of interconnection networks. In particular, the communication requirements of computer vision algorithms as identified by the various computer vision tasks are analysed. These requirements together with graph theoretical considerations are then used to suggest a topology for large transputer networks. The latter is based on sub-graphs, with proven performance when used to implement interconnection networks, combined to form an architecture with improved performance. This architecture consists of a fixed structure supplemented with a dynamically reconfigured network. After describing this topology, a routing algorithm that conveys messages along shortest paths in the network is given and implemented. And finally, some practical issues in the use of transputers are considered and solutions proposed

System Identification of Bipedal Locomotion in Robots and Humans

The ability to perform a healthy walking gait can be altered in numerous cases due to gait disorder related pathologies. The latter could lead to partial or complete mobility loss, which affects the patients’ quality of life. Wearable exoskeletons and active prosthetics have been considered as a key component to remedy this mobility loss. The control of such devices knows numerous challenges that are yet to be addressed. As opposed to fixed trajectories control, real-time adaptive reference generation control is likely to provide the wearer with more intent control over the powered device. We propose a novel gait pattern generator for the control of such devices, taking advantage of the inter-joint coordination in the human gait. Our proposed method puts the user in the control loop as it maps the motion of healthy limbs to that of the affected one. To design such control strategy, it is critical to understand the dynamics behind bipedal walking. We begin by studying the simple compass gait walker. We examine the well-known Virtual Constraints method of controlling bipedal robots in the image of the compass gait. In addition, we provide both the mechanical and control design of an affordable research platform for bipedal dynamic walking. We then extend the concept of virtual constraints to human locomotion, where we investigate the accuracy of predicting lower limb joints angular position and velocity from the motion of the other limbs. Data from nine healthy subjects performing specific locomotion tasks were collected and are made available online. A successful prediction of the hip, knee, and ankle joints was achieved in different scenarios. It was also found that the motion of the cane alone has sufficient information to help predict good trajectories for the lower limb in stairs ascent. Better estimates were obtained using additional information from arm joints. We also explored the prediction of knee and ankle trajectories from the motion of the hip joints