Efficient tilings of de Bruijn and Kautz graphs

Kautz and de Bruijn graphs have a high degree of connectivity which makes them ideal candidates for massively parallel computer network topologies. In order to realize a practical computer architecture based on these graphs, it is useful to have a means of constructing a large-scale system from smaller, simpler modules. In this paper we consider the mathematical problem of uniformly tiling a de Bruijn or Kautz graph. This can be viewed as a generalization of the graph bisection problem. We focus on the problem of graph tilings by a set of identical subgraphs. Tiles should contain a maximal number of internal edges so as to minimize the number of edges connecting distinct tiles. We find necessary and sufficient conditions for the construction of tilings. We derive a simple lower bound on the number of edges which must leave each tile, and construct a class of tilings whose number of edges leaving each tile agrees asymptotically in form with the lower bound to within a constant factor. These tilings make possible the construction of large-scale computing systems based on de Bruijn and Kautz graph topologies.Comment: 29 pages, 11 figure

Fail-in-Place Network Design: Interaction Between Topology, Routing Algorithm and Failures

Abstract—The growing system size of high performance com-puters results in a steady decrease of the mean time between failures. Exchanging network components often requires whole system downtime which increases the cost of failures. In this work, we study a fail-in-place strategy where broken network elements remain untouched. We show, that a fail-in-place strategy is feasible for todays networks and the degradation is manageable, and provide guidelines for the design. Our network failure simulation toolchain allows system designers to extrapolate the performance degradation based on expected failure rates, and it can be used to evaluate the current state of a system. In a case study of real-world HPC systems, we will analyze the performance degradation throughout the systems lifetime under the assumption that faulty network components are not repaired, which results in a recommendation to change the used routing algorithm to improve the network performance as well as the fail-in-place characteristic. Keywords—Network design, network simulations, network man-agement, fail-in-place, routing protocols, fault tolerance, availability I

Routing on the Channel Dependency Graph:: A New Approach to Deadlock-Free, Destination-Based, High-Performance Routing for Lossless Interconnection Networks

In the pursuit for ever-increasing compute power, and with Moore's law slowly coming to an end, high-performance computing started to scale-out to larger systems. Alongside the increasing system size, the interconnection network is growing to accommodate and connect tens of thousands of compute nodes. These networks have a large influence on total cost, application performance, energy consumption, and overall system efficiency of the supercomputer. Unfortunately, state-of-the-art routing algorithms, which define the packet paths through the network, do not utilize this important resource efficiently. Topology-aware routing algorithms become increasingly inapplicable, due to irregular topologies, which either are irregular by design, or most often a result of hardware failures. Exchanging faulty network components potentially requires whole system downtime further increasing the cost of the failure. This management approach becomes more and more impractical due to the scale of today's networks and the accompanying steady decrease of the mean time between failures. Alternative methods of operating and maintaining these high-performance interconnects, both in terms of hardware- and software-management, are necessary to mitigate negative effects experienced by scientific applications executed on the supercomputer. However, existing topology-agnostic routing algorithms either suffer from poor load balancing or are not bounded in the number of virtual channels needed to resolve deadlocks in the routing tables. Using the fail-in-place strategy, a well-established method for storage systems to repair only critical component failures, is a feasible solution for current and future HPC interconnects as well as other large-scale installations such as data center networks. Although, an appropriate combination of topology and routing algorithm is required to minimize the throughput degradation for the entire system. This thesis contributes a network simulation toolchain to facilitate the process of finding a suitable combination, either during system design or while it is in operation. On top of this foundation, a key contribution is a novel scheduling-aware routing, which reduces fault-induced throughput degradation while improving overall network utilization. The scheduling-aware routing performs frequent property preserving routing updates to optimize the path balancing for simultaneously running batch jobs. The increased deployment of lossless interconnection networks, in conjunction with fail-in-place modes of operation and topology-agnostic, scheduling-aware routing algorithms, necessitates new solutions to solve the routing-deadlock problem. Therefore, this thesis further advances the state-of-the-art by introducing a novel concept of routing on the channel dependency graph, which allows the design of an universally applicable destination-based routing capable of optimizing the path balancing without exceeding a given number of virtual channels, which are a common hardware limitation. This disruptive innovation enables implicit deadlock-avoidance during path calculation, instead of solving both problems separately as all previous solutions

Interconnection networks for parallel and distributed computing

Parallel computers are generally either shared-memory machines or distributed- memory machines. There are currently technological limitations on shared-memory architectures and so parallel computers utilizing a large number of processors tend tube distributed-memory machines. We are concerned solely with distributed-memory multiprocessors. In such machines, the dominant factor inhibiting faster global computations is inter-processor communication. Communication is dependent upon the topology of the interconnection network, the routing mechanism, the flow control policy, and the method of switching. We are concerned with issues relating to the topology of the interconnection network. The choice of how we connect processors in a distributed-memory multiprocessor is a fundamental design decision. There are numerous, often conflicting, considerations to bear in mind. However, there does not exist an interconnection network that is optimal on all counts and trade-offs have to be made. A multitude of interconnection networks have been proposed with each of these networks having some good (topological) properties and some not so good. Existing noteworthy networks include trees, fat-trees, meshes, cube-connected cycles, butterflies, Möbius cubes, hypercubes, augmented cubes, k-ary n-cubes, twisted cubes, n-star graphs, (n, k)-star graphs, alternating group graphs, de Bruijn networks, and bubble-sort graphs, to name but a few. We will mainly focus on k-ary n-cubes and (n, k)-star graphs in this thesis. Meanwhile, we propose a new interconnection network called augmented k-ary n- cubes. The following results are given in the thesis.1. Let k ≥ 4 be even and let n ≥ 2. Consider a faulty k-ary n-cube Q(^k_n) in which the number of node faults f(_n) and the number of link faults f(_e) are such that f(_n) + f(_e) ≤ 2n - 2. We prove that given any two healthy nodes s and e of Q(^k_n), there is a path from s to e of length at least k(^n) - 2f(_n) - 1 (resp. k(^n) - 2f(_n) - 2) if the nodes s and e have different (resp. the same) parities (the parity of a node Q(^k_n) in is the sum modulo 2 of the elements in the n-tuple over 0, 1, ∙∙∙ , k - 1 representing the node). Our result is optimal in the sense that there are pairs of nodes and fault configurations for which these bounds cannot be improved, and it answers questions recently posed by Yang, Tan and Hsu, and by Fu. Furthermore, we extend known results, obtained by Kim and Park, for the case when n = 2.2. We give precise solutions to problems posed by Wang, An, Pan, Wang and Qu and by Hsieh, Lin and Huang. In particular, we show that Q(^k_n) is bi-panconnected and edge-bipancyclic, when k ≥ 3 and n ≥ 2, and we also show that when k is odd, Q(^k_n) is m-panconnected, for m = (^n(k - 1) + 2k - 6’ / ‘_2), and (k -1) pancyclic (these bounds are optimal). We introduce a path-shortening technique, called progressive shortening, and strengthen existing results, showing that when paths are formed using progressive shortening then these paths can be efficiently constructed and used to solve a problem relating to the distributed simulation of linear arrays and cycles in a parallel machine whose interconnection network is Q(^k_n) even in the presence of a faulty processor.3. We define an interconnection network AQ(^k_n) which we call the augmented k-ary n-cube by extending a k-ary n-cube in a manner analogous to the existing extension of an n-dimensional hypercube to an n-dimensional augmented cube. We prove that the augmented k-ary n-cube Q(^k_n) has a number of attractive properties (in the context of parallel computing). For example, we show that the augmented k-ary n-cube Q(^k_n) - is a Cayley graph (and so is vertex-symmetric); has connectivity 4n - 2, and is such that we can build a set of 4n - 2 mutually disjoint paths joining any two distinct vertices so that the path of maximal length has length at most max{{n- l)k- (n-2), k + 7}; has diameter [(^k) / (_3)] + [(^k - 1) /( _3)], when n = 2; and has diameter at most (^k) / (_4) (n+ 1), for n ≥ 3 and k even, and at most [(^k)/ (_4) (n + 1) + (^n) / (_4), for n ^, for n ≥ 3 and k odd.4. We present an algorithm which given a source node and a set of n - 1 target nodes in the (n, k)-star graph S(_n,k) where all nodes are distinct, builds a collection of n - 1 node-disjoint paths, one from each target node to the source. The collection of paths output from the algorithm is such that each path has length at most 6k - 7, and the algorithm has time complexity O(k(^3)n(^4))

The effect of limited hip mobility on the lumbar spine in a young adult population

Limited hip mobility is known to affect the lumbar spine. Much of the previous research has utilized a participant population whose hip mobility is compromised due to arthritic or neurological dysfunctions. Such aetiologies may confound the outcomes, as their effects may not be limited to the hip. The purpose of this thesis was to recruit a healthy young adult population with limited hip mobility to further investigate its effect on the lumbar spine, as well as the role of exercise intervention. Several cascading studies were conducted that were unified around a central theme of links between hip and spine function: Study # 1 investigated the normal distribution of passive hip extension and rotation in a group of 77 males (age 19-30). Data was collected using an infra-red motion capture system and compared to goniometric measurements. The resulting angles represent the 5th – 95th percentiles, including the averages and standard deviations. Study # 2 compared movement patterns between groups of males with limited and excessive hip mobility. Participants were required to perform simple functional activities (lunging, twisting, walking, etc) as well as use the elliptical trainer. Resulting hip and spine angles demonstrated that the men with limited hip mobility stood with a more anteriorly tilted pelvis, and assumed a posture with more lumbar and hip flexion on the elliptical trainer, compared to those with greater mobility. This, in turn, resulted in a greater lumbar compression load due to increased back muscle activity. Study #3 involved recruitment of 24 young adult males with limited hip mobility. Their movement patterns were assessed (as in study #2), then they were assigned to one of four intervention groups: hip stretching, spine stabilizing, hip stretching combined with spine stabilization, and control. Participants in the 3 exercise groups attended supervised exercise sessions once/week for 6 weeks, but were expected to exercise a minimum of 4 times/week on their own. At the end of the 6 weeks, intake parameters were re-assessed, and movement pattern assessment repeated. Despite significant increases in available hip flexibility and/or large increases in trunk muscle endurance and trunk motor control, there were few indications that participants were any more adept at decreasing lumbar motion, or utilizing their newfound hip flexibility during functional activities. Study #4 compared those in the 10th and 90th percentiles of available hip rotation, using a frictionless apparatus to investigate passive stiffness properties of the hip. Participants adopted a posture of upright standing, with one leg supported on a turntable apparatus, and upper body and pelvis secured. A an applied rotational moment resulted in passive hip internal and external rotation. Outcomes demonstrate that those with limited hip mobility stand with the leg more externally rotated and require a larger moment to initiate motion. Passive stiffness curves indicate greater stiffness properties in those with limited hip mobility, and more resistance to an external rotation moment than internal rotation. Study #5 investigated passive hip stiffness in the sagittal plane, comparing those with limited and excessive hip extension. Using a frictionless jig, with the participants lying on their left side, the left hip was pulled into extension with knee position varying. Those with limited hip mobility demonstrated increased passive stiffness compared to the more mobile group, and stiffness was greater when the knee was in extension. The group with limited mobility also showed a trend of increased back extension compared to the more mobile group, when the hip and lumbar spine were both free to react to the applied extension moment. Study #6 summarizes the spine/hip kinematics and muscle activation levels produced when using the elliptical trainer, as well as lumbar compressive and shear forces. It differs significantly from walking in that it produces more lumbar motion in flexion/extension and lumbar twist, but less lateral bend. Participants also tended to adopt a greater mean lumbar flexion angle on the elliptical, which in turn resulted in greater muscle activity in the back extensors. Varying hand position, velocity and stride length were all found to significantly affect the amount of lumbar motion. Highly phasic muscle activity is seen, with the gluteal muscles and internal obliques demonstrating the greatest activation levels

Fuzzy EOQ Model with Trapezoidal and Triangular Functions Using Partial Backorder

EOQ fuzzy model is EOQ model that can estimate the cost from existing information. Using trapezoid fuzzy functions can estimate the costs of existing and trapezoid membership functions has some points that have a value of membership . TR ̃C value results of trapezoid fuzzy will be higher than usual TRC value results of EOQ model . This paper aims to determine the optimal amount of inventory in the company, namely optimal Q and optimal V, using the model of partial backorder will be known optimal Q and V for the optimal number of units each time a message . EOQ model effect on inventory very closely by using EOQ fuzzy model with triangular and trapezoid membership functions with partial backorder. Optimal Q and optimal V values for the optimal fuzzy models will have an increase due to the use of trapezoid and triangular membership functions that have a different value depending on the requirements of each membership function value. Therefore, by using a fuzzy model can solve the company's problems in estimating the costs for the next term

Bisecting de Bruijn and Kautz Graphs

this paper appeared at The 2nd Colloquium on Structural Information and Communication Complexity (SIROCCO'95), Olympia, Greece, June 12--14, 1995

Genetic Factors Associated with Variation in Abundance of the Invasive Yellow Crazy Ant (Anoplolepis gracilipes)

A key component of successful invasion is the ability of an introduced population to reach sufficient abundance to persist, spread, and alter or dominate the recipient biological community. Genetic diversity is one of many factors that may contribute to population dynamics, but has important ramifications for biological fitness, and thus invasion success in the long term. I explored genetic factors associated with variation in abundance (i.e., differential invasion success) of the yellow crazy ant Anoplolepis gracilipes in the Indo-Pacific region, primarily focussing on Arnhem Land in Australia's Northern Territory. I explored five aspects that I hypothesised could contribute to variation in the abundance of this ant: 1) I investigated the unusual reproductive mode of A. gracilipes, and tested whether it involved dependent-lineage genetic caste determination (DL GCD) in Arnhem Land. In DL GCD systems populations require hybridisation between genetically distinct groups to produce both reproductive and worker castes. Asymmetry in the ratio of different lineages may result in low abundance and population collapse. I found no evidence for a DL GCD system in A. gracilipes, and thus its abundance in Arnhem Land does not appear to be constrained by any lineage ratio asymmetry. Worker reproduction (either of males or asexual production of other workers) also appeared unlikely. The reproductive mode of the species remains fascinating but enigmatic; 2) I explored whether multiple source populations were responsible for the observed variation in abundance in Arnhem Land (i.e., is abundance associated with propagule pressure, or populations from different sources), and if the population has diverged since introduction. The A. gracilipes population in Arnhem Land stemmed from a single source, and thus propagule pressure was apparently not responsible for variation in abundance. In contrast to many invasive ants, population divergence has occurred since introduction; 3) I tested the hypotheses that genetic variation was associated with variation in abundance in Arnhem Land, and that ecological success was density-dependent. While the population divergence found in Chapter 3 was not related to variation in abundance, genotypic diversity was higher in more abundant nest clusters. These more abundant nest clusters were in turn associated with lower native ant species diversity, and a difference in composition of the invaded ant community (i.e., greater ecological success); 4) I revisited the invasion of the yellow crazy ant in Tokelau to determine whether a haplotype that was linked to greater abundance and dominance of the ant community has increased in distribution. Although ants of the inferred dominant haplotype were implicated in most new invasions, their abundance was substantially lower than previously observed in Tokelau; 5) I conducted a preliminary analysis of the metagenomic diversity of A. gracilipes endogenous parasites and symbionts among populations from Christmas Island, Okinawa, Samoa and Arnhem Land. Bacterial community composition and diversity differed between the study populations. Variation in abundance among A. gracilipes populations in Arnhem Land was not due to parasite load on populations with low abundance. However, low abundance of A. gracilipes was correlated with lower microbial diversity overall, and higher prevalence of some groups, notably two that confer antibiotic properties. Together, my findings suggest that propagule pressure, reproductive mode and haplotype-specific effects do not appear to be associated with variation in A. gracilipes abundance. Other genetic factors I investigated do appear to be associated with the variation in A. gracilipes abundance and effects on the invaded ant communities. Genotypic diversity was positively related to the abundance of A. gracilipes in Arnhem Land, and this relationship may be affected by population divergence through population bottlenecks. In addition, differences in bacterial diversity among populations highlighted several candidate bacteria that could be associated with variation in abundance, which would be a topic of future work. Although genetic factors are often implicated in the successful establishment of invasive species, my thesis demonstrates that genetic factors may also be associated with post-establishment population dynamics