Constant-Time Algorithms for Minimum Spanning Tree and Related Problems on Processor Array with Reconfigurable Bus Systems

[[abstract]]A processor array with a reconfigurable bus system is a parallel computation model that consists of a processor array and a reconfigurable bus system. In this paper, a constant-time algorithm is proposed on this model for finding the cycles in an undirected graph. We can use this algorithm to decide whether a specified edge belongs to the minimum spanning tree of the graph or not. This cycle-finding algorithm is designed on a two-dimensional $n\times n$ processor array with a reconfigurable bus system, where $n$ is the number of vertices in the graph. Based on this cycle-finding algorithm, the minimum spanning tree problem and the spanning tree problem can be solved in O(1) time by using fewer processors than before, O($n\times m\times n$) and O($n^3$) processors respectively. This is a substantial improvement over previous known results. Moreover, we also propose two constant-time algorithms for solving the minimum spanning tree verification problem and spanning tree verification problem by using O($n^3$) and O($n^2$) processors, respectively.

Sublogarithmic deterministic selection on arrays with a reconfigurable optical bus

The linear array with a reconfigurable pipelined bus system (LARPBS) is a newly introduced parallel computational model, where processors are connected by a reconfigurable optical bus. In this paper, we show that the selection problem can be solved on the LARPBS model deterministically in O((loglogN)2/ log log log N) time. To our best knowledge, this is the best deterministic selection algorithm on any model with a reconfigurable optical bus.Yijie Han, Yi Pan and Hong She

Scaling Simulations of Reconfigurable Meshes.

This dissertation deals with reconfigurable bus-based models, a new type of parallel machine that uses dynamically alterable connections between processors to allow efficient communication and to perform fast computations. We focus this work on the Reconfigurable Mesh (R-Mesh), one of the most widely studied reconfigurable models. We study the ability of the R-Mesh to adapt an algorithm instance of an arbitrary size to run on a given smaller model size without significant loss of efficiency. A scaling simulation achieves this adaptation, and the simulation overhead expresses the efficiency of the simulation. We construct a scaling simulation for the Fusing-Restricted Reconfigurable Mesh (FR-Mesh), an important restriction of the R-Mesh. The overhead of this simulation depends only on the simulating machine size and not on the simulated machine size. The results of this scaling simulation extend to a variety of concurrent write rules and also translate to an improved scaling simulation of the R-Mesh itself. We present a bus linearization procedure that transforms an arbitrary non-linear bus configuration of an R-Mesh into an equivalent acyclic linear bus configuration implementable on an Linear Reconfigurable Mesh (LR-Mesh), a weaker version of the R-Mesh. This procedure gives the algorithm designer the liberty of using buses of arbitrary shape, while automatically translating the algorithm to run on a simpler platform. We illustrate our bus linearization method through two important applications. The first leads to a faster scaling simulation of the R-Mesh. The second application adapts algorithms designed for R-Meshes to run on models with pipelined optical buses. We also present a simulation of a Directional Reconfigurable Mesh (DR-Mesh) on an LR-Mesh. This simulation has a much better efficiency compared to previous work. In addition to the LR-Mesh, this simulation also runs on models that use pipelined optical buses

A constant time parallel algorithm for the triangularization of a sparse matrix using CD-PARBS

An algorithm for the triangularization of a matrix whose graph is a directed acyclic graph, popularly known as dag, is presented. One of the algorithms for obtaining this special form has been given by Sargent and Westerberg. Their approach is practically good but sequential in nature and cannot be parallelised easily. In this work we present a parallel algorithm which is based on the observation that, if we find the transitive closure matrix of a directed acyclic graph, count the number of entries in each row, sort them in the ascending order of their values and rank them accordingly, we get a lower triangular matrix. We show that all these operations can be done using 3-d CD- PARBS(Complete Directed PARBS) in constant time. The same approach can be used for the block cases, producing the same relabelling as produced by Tarjan’s algorithm, in constant time. To the best of our knowledge, it is the first approach to solve such problems using directed PARBS

Efficient parallel computation on multiprocessors with optical interconnection networks

This dissertation studies optical interconnection networks, their architecture, address schemes, and computation and communication capabilities. We focus on a simple but powerful optical interconnection network model - the Linear Array with Reconfigurable pipelined Bus System (LARPBS). We extend the LARPBS model to a simplified higher dimensional LAPRBS and provide a set of basic computation operations. We then study the following two groups of parallel computation problems on both one dimensional LARPBS\u27s as well as multi-dimensional LARPBS\u27s: parallel comparison problems, including sorting, merging, and selection; Boolean matrix multiplication, transitive closure and their applications to connected component problems. We implement an optimal sorting algorithm on an n-processor LARPBS. With this optimal sorting algorithm at disposal, we study the sorting problem for higher dimensional LARPBS\u27s and obtain the following results: • An optimal basic Columnsort algorithm on a 2D LARPBS. • Two optimal two-way merge sort algorithms on a 2D LARPBS. • An optimal multi-way merge sorting algorithm on a 2D LARPBS. • An optimal generalized column sort algorithm on a 2D LARPBS. • An optimal generalized column sort algorithm on a 3D LARPBS. • An optimal 5-phase sorting algorithm on a 3D LARPBS. Results for selection problems are as follows: • A constant time maximum-finding algorithm on an LARPBS. • An optimal maximum-finding algorithm on an LARPBS. • An O((log log n)2) time parallel selection algorithm on an LARPBS. • An O(k(log log n)2) time parallel multi-selection algorithm on an LARPBS. While studying the computation and communication properties of the LARPBS model, we find Boolean matrix multiplication and its applications to the graph are another set of problem that can be solved efficiently on the LARPBS. Following is a list of results we have obtained in this area. • A constant time Boolean matrix multiplication algorithm. • An O(log n)-time transitive closure algorithm. • An O(log n)-time connected components algorithm. • An O(log n)-time strongly connected components algorithm. The results provided in this dissertation show the strong computation and communication power of optical interconnection networks

Design and Analysis of Optical Interconnection Networks for Parallel Computation.

In this doctoral research, we propose several novel protocols and topologies for the interconnection of massively parallel processors. These new technologies achieve considerable improvements in system performance and structure simplicity. Currently, synchronous protocols are used in optical TDM buses. The major disadvantage of a synchronous protocol is the waste of packet slots. To offset this inherent drawback of synchronous TDM, a pipelined asynchronous TDM optical bus is proposed. The simulation results show that the performance of the proposed bus is significantly better than that of known pipelined synchronous TDM optical buses. Practically, the computation power of the plain TDM protocol is limited. Various extensions must be added to the system. In this research, a new pipelined optical TDM bus for implementing a linear array parallel computer architecture is proposed. The switches on the receiving segment of the bus can be dynamically controlled, which make the system highly reconfigurable. To build large and scalable systems, we need new network architectures that are suitable for optical interconnections. A new kind of reconfigurable bus called segmented bus is introduced to achieve reduced structure simplicity and increased concurrency. We show that parallel architectures based on segmented buses are versatile by showing that it can simulate parallel communication patterns supported by a wide variety of networks with small slowdown factors. New kinds of interconnection networks, the hypernetworks, have been proposed recently. Compared with point-to-point networks, they allow for increased resource-sharing and communication bandwidth utilization, and they are especially suitable for optical interconnects. One way to derive a hypernetwork is by finding the dual of a point-to-point network. Hypercube Q\sb{n}, where n is the dimension, is a very popular point-to-point network. It is interesting to construct hypernetworks from the dual Q\sbsp{n}{*} of hypercube of Q\sb{n}. In this research, the properties of Q\sbsp{n}{*} are investigated and a set of fundamental data communication algorithms for Q\sbsp{n}{*} are presented. The results indicate that the Q\sbsp{n}{*} hypernetwork is a useful and promising interconnection structure for high-performance parallel and distributed computing systems

Simulations and Algorithms on Reconfigurable Meshes With Pipelined Optical Buses.

Recently, many models using reconfigurable optically pipelined buses have been proposed in the literature. A system with an optically pipelined bus uses optical waveguides, with unidirectional propagation and predictable delays, instead of electrical buses to transfer information among processors. These two properties enable synchronized concurrent access to an optical bus in a pipelined fashion. Combined with the abilities of the bus structure to broadcast and multicast, this architecture suits many communication-intensive applications. We establish the equivalence of three such one-dimensional optical models, namely the LARPBS, LPB, and POB. This implies an automatic translation of algorithms (without loss of speed or efficiency) among these models. In particular, since the LPB is the same as an LARPBS without the ability to segment its buses, their equivalence establishes reconfigurable delays (rather than segmenting ability) as the key to the power of optically pipelined models. We also present simulations for a number of two-dimensional optical models and establish that they possess the same complexity, so that any of these models can simulate a step of one of the other models in constant time with a polynomial increase in size. Specifically, we determine the complexity of three two-dimensional optical models (the PR-Mesh, APPBS, and AROB) to be the same as the well known LR-Mesh and the cycle-free LR-Mesh. We develop algorithms for the LARPBS and PR-Mesh that are more efficient than existing algorithms in part by exploiting the pipelining, segmenting, and multicasting characteristics of these models. We also consider the implications of certain physical constraints placed on the system by restricting the distance over which two processors are able to communicate. All algorithms developed for these models assume that a healthy system is available. We present some fundamental algorithms that are able to tolerate up to N/2 faults on an N-processor LARPBS. We then extend these results to apply to other algorithms in the areas of image processing and matrix operations

A Field Programmable Gate Array Architecture for Two-Dimensional Partial Reconfiguration

Reconfigurable machines can accelerate many applications by adapting to their needs through hardware reconfiguration. Partial reconfiguration allows the reconfiguration of a portion of a chip while the rest of the chip is busy working on tasks. Operating system models have been proposed for partially reconfigurable machines to handle the scheduling and placement of tasks. They are called OS4RC in this dissertation. The main goal of this research is to address some problems that come from the gap between OS4RC and existing chip architectures and the gap between OS4RC models and practical applications. Some existing OS4RC models are based on an impractical assumption that there is no data exchange channel between IP (Intellectual Property) circuits residing on a Field Programmable Gate Array (FPGA) chip and between an IP circuit and FPGA I/O pins. For models that do not have such an assumption, their inter-IP communication channels have severe drawbacks. Those channels do not work well with 2-D partial reconfiguration. They are not suitable for intensive data stream processing. And frequently they are very complicated to design and very expensive. To address these problems, a new chip architecture that can better support inter-IP and IP-I/O communication is proposed and a corresponding OS4RC kernel is then specified. The proposed FPGA architecture is based on an array of clusters of configurable logic blocks, with each cluster serving as a partial reconfiguration unit, and a mesh of segmented buses that provides inter-IP and IP-I/O communication channels. The proposed OS4RC kernel takes care of the scheduling, placement, and routing of circuits under the constraints of the proposed architecture. Features of the new architecture in turns reduce the kernel execution times and enable the runtime scheduling, placement and routing. The area cost and the configuration memory size of the new chip architecture are calculated and analyzed. And the efficiency of the OS4RC kernel is evaluated via simulation using three different task models

Cryptarray A Scalable And Reconfigurable Architecture For Cryptographic Applications

Cryptography is increasingly viewed as a critical technology to fulfill the requirements of security and authentication for information exchange between Internet applications. However, software implementations of cryptographic applications are unable to support the quality of service from a bandwidth perspective required by most Internet applications. As a result, various hardware implementations, from Application-Specific Integrated Circuits (ASICs), Field-Programmable Gate Arrays (FPGAs), to programmable processors, were proposed to improve this inadequate quality of service. Although these implementations provide performances that are considered better than those produced by software implementations, they still fall short of addressing the bandwidth requirements of most cryptographic applications in the context of the Internet for two major reasons: (i) The majority of these architectures sacrifice flexibility for performance in order to reach the performance level needed for cryptographic applications. This lack of flexibility can be detrimental considering that cryptographic standards and algorithms are still evolving. (ii) These architectures do not consider the consequences of technology scaling in general, and particularly interconnect related problems. As a result, this thesis proposes an architecture that attempts to address the requirements of cryptographic applications by overcoming the obstacles described in (i) and (ii). To this end, we propose a new reconfigurable, two-dimensional, scalable architecture, called CRYPTARRAY, in which bus-based communication is replaced by distributed shared memory communication. At the physical level, the length of the wires will be kept to a minimum. CRYPTARRAY is organized as a chessboard in which the dark and light squares represent Processing Elements (PE) and memory blocks respectively. The granularity and resource composition of the PEs is specifically designed to support the computing operations encountered in cryptographic algorithms in general, and symmetric algorithms in particular. Communication can occur only between neighboring PEs through locally shared memory blocks. Because of the chessboard layout, the architecture can be reconfigured to allow computation to proceed as a pipelined wave in any direction. This organization offers a high computational density in terms of datapath resources and a large number of distributed storage resources that easily support a high degree of parallelism and pipelining. Experimental prototyping a small array on FPGA chips shows that this architecture can run at 80.9 MHz producing 26,968,716 outputs every second in static reconfiguration mode and 20,226,537 outputs every second in dynamic reconfiguration mode