On Longest Repeat Queries Using GPU

Repeat finding in strings has important applications in subfields such as computational biology. The challenge of finding the longest repeats covering particular string positions was recently proposed and solved by \.{I}leri et al., using a total of the optimal $O(n)$ time and space, where $n$ is the string size. However, their solution can only find the \emph{leftmost} longest repeat for each of the $n$ string position. It is also not known how to parallelize their solution. In this paper, we propose a new solution for longest repeat finding, which although is theoretically suboptimal in time but is conceptually simpler and works faster and uses less memory space in practice than the optimal solution. Further, our solution can find \emph{all} longest repeats of every string position, while still maintaining a faster processing speed and less memory space usage. Moreover, our solution is \emph{parallelizable} in the shared memory architecture (SMA), enabling it to take advantage of the modern multi-processor computing platforms such as the general-purpose graphics processing units (GPU). We have implemented both the sequential and parallel versions of our solution. Experiments with both biological and non-biological data show that our sequential and parallel solutions are faster than the optimal solution by a factor of 2--3.5 and 6--14, respectively, and use less memory space.Comment: 14 page

Impact of delay on HIV-1 dynamics of fighting a virus with another virus

In this paper, we propose a mathematical model for HIV-1 infection with intracellular delay. The model examines a viral-therapy for controlling infections through recombining HIV-1 virus with a genetically modified virus. For this model, the basic reproduction number $\mathcal{R}_0$ are identified and its threshold properties are discussed. When $\mathcal{R}_0 < 1$, the infection-free equilibrium $E_0$ is globally asymptotically stable. When $\mathcal{R}_0 > 1$, $E_0$ becomes unstable and there occurs the single-infection equilibrium $E_s$, and $E_0$ and $E_s$ exchange their stability at the transcritical point $\mathcal{R}_0 =1$. If $1< \mathcal{R}_0 < R_1$, where $R_1$ is a positive constant explicitly depending on the model parameters, $E_s$ is globally asymptotically stable, while when $\mathcal{R}_0 > R_1$, $E_s$ loses its stability to the double-infection equilibrium $E_d$. There exist a constant $R_2$ such that $E_d$ is asymptotically stable if $R_1<\mathcal R_0 < R_2$, and $E_s$ and $E_d$ exchange their stability at the transcritical point $\mathcal{R}_0 =R_1$. We use one numerical example to determine the largest range of $\mathcal R_0$ for the local stability of $E_d$ and existence of Hopf bifurcation. Some simulations are performed to support the theoretical results. These results show that the delay plays an important role in determining the dynamic behaviour of the system. In the normal range of values, the delay may change the dynamic behaviour quantitatively, such as greatly reducing the amplitudes of oscillations, or even qualitatively changes the dynamical behaviour such as revoking oscillating solutions to equilibrium solutions. This suggests that the delay is a very important fact which should not be missed in HIV-1 modelling

Bifurcation of Limit Cycles in Smooth and Non-smooth Dynamical Systems with Normal Form Computation

This thesis contains two parts. In the first part, we investigate bifurcation of limit cycles around a singular point in planar cubic systems and quadratic switching systems. For planar cubic systems, we study cubic perturbations of a quadratic Hamiltonian system and obtain 10 small-amplitude limit cycles bifurcating from an elementary center, for which up to 5th-order Melnikov functions are used. Moreover, we prove the existence of 12 small-amplitude limit cycles around a singular point in a cubic system by computing focus values. For quadratic switching system, we develop a recursive algorithm for computing Lyapunov constants. With this efficient algorithm, we obtain a complete classification of the center conditions for a switching Bautin system. Moreover, we construct a concrete example of switching system to obtain 10 small-amplitude limit cycles bifurcating from a center. In the second part, we derive two explicit, computationally explicit, recursive formulas for computing the normal forms, center manifolds and nonlinear transformations for general n-dimensional systems, associated with Hopf and semisimple singularities, respectively. Based on the formulas, we develop Maple programs, which are very convenient for an end-user who only needs to prepare an input file and then execute the program to “automatically” generate the results. Several examples are presented to demonstrate the computational efficiency of the algorithms. In addition, we show that a simple 3-dimensional quadratic vector field can have 7 small-amplitude limit cycles, bifurcating from a Hopf singular point. This result is surprising higher than the Bautin’s result for quadratic planar vector fields which can only have 3 small-amplitude limit cycles around an elementary focus or an elementary center

CloudTree: A Library to Extend Cloud Services for Trees

In this work, we propose a library that enables on a cloud the creation and management of tree data structures from a cloud client. As a proof of concept, we implement a new cloud service CloudTree. With CloudTree, users are able to organize big data into tree data structures of their choice that are physically stored in a cloud. We use caching, prefetching, and aggregation techniques in the design and implementation of CloudTree to enhance performance. We have implemented the services of Binary Search Trees (BST) and Prefix Trees as current members in CloudTree and have benchmarked their performance using the Amazon Cloud. The idea and techniques in the design and implementation of a BST and prefix tree is generic and thus can also be used for other types of trees such as B-tree, and other link-based data structures such as linked lists and graphs. Preliminary experimental results show that CloudTree is useful and efficient for various big data applications

Partial Replica Location And Selection For Spatial Datasets

As the size of scientific datasets continues to grow, we will not be able to store enormous datasets on a single grid node, but must distribute them across many grid nodes. The implementation of partial or incomplete replicas, which represent only a subset of a larger dataset, has been an active topic of research. Partial Spatial Replicas extend this functionality to spatial data, allowing us to distribute a spatial dataset in pieces over several locations. We investigate solutions to the partial spatial replica selection problems. First, we describe and develop two designs for an Spatial Replica Location Service (SRLS), which must return the set of replicas that intersect with a query region. Integrating a relational database, a spatial data structure and grid computing software, we build a scalable solution that works well even for several million replicas. In our SRLS, we have improved performance by designing a R-tree structure in the backend database, and by aggregating several queries into one larger query, which reduces overhead. We also use the Morton Space-filling Curve during R-tree construction, which improves spatial locality. In addition, we describe R-tree Prefetching(RTP), which effectively utilizes the modern multi-processor architecture. Second, we present and implement a fast replica selection algorithm in which a set of partial replicas is chosen from a set of candidates so that retrieval performance is maximized. Using an R-tree based heuristic algorithm, we achieve O(n log n) complexity for this NP-complete problem. We describe a model for disk access performance that takes filesystem prefetching into account and is sufficiently accurate for spatial replica selection. Making a few simplifying assumptions, we present a fast replica selection algorithm for partial spatial replicas. The algorithm uses a greedy approach that attempts to maximize performance by choosing a collection of replica subsets that allow fast data retrieval by a client machine. Experiments show that the performance of the solution found by our algorithm is on average always at least 91% and 93.4% of the performance of the optimal solution in 4-node and 8-node tests respectively