A Theory of the Spatial Computational Domain

Most parallel processing methods developed for geographic analyses bind the design of domain decomposition and task scheduling to specific parallel computer architectures. These methods are not adaptable to emerging distributed computing environments that are based on Grid computing and peer-to-peer technologies. This paper presents a theory to support the development of adaptable parallel processing methods for geographic analyses performed on heterogeneous parallel processing environments. This theory of the spatial computational domain represents the computational intensity of geographic data and analysis methods, and transforms it into a common framework based on transformation theories from earlier cartographic research. The application of the theory is illustrated using an inverse distance weighted interpolation method. We describe the underpinnings of this analysis, and then address the latent parallelism of a conventional sequential algorithm based on spatial computational domain theory. Through the application of this theory, the development of domain decomposition methods is decoupled from specific high performance computer architectures and task scheduling implementations, which makes the design of generic parallel processing geographic analysis solutions feasib le. 1. Existing Problems in Parallel Processing of Geographic Information The state-of-the-practice in parallel processing of geographic information binds the design of domain decomposition and task scheduling to specific conventional parallel computer architectures. This tight-coupling approach is problematic to software design for three reasons: 1.) Domain decomposition and task scheduling methods focus on the characteristics of spatial data and operations performed on them. 2.) Any change to spatial data or operations requires a corresponding change in the design of domain decompositio n and task scheduling methods. 3.) Domain decomposition and task scheduling strategies depend upon specific parallel architectures. These three problems are analogous to those observed in graphics programming before the advent of device-independent software. In this case, these problems hinder the development of portable parallel geographic analysis methods based on computational Grids (see endnote). To eliminate these problems, we introduce a new spatial computational domain theory and illustrate an application using the TeraGrid (TeraGrid, 2005). 2. Spatial Computational Domain Theory The spatial computational domain is formally defined to comprise several two-dimensional computational intensity surfaces. Given a spatial domain projected to a two-dimensional (x and y) Euclidian space, each two-dimensional computational surface can be represented as a vector c = (cij) in the space R, where N = xc × yc, and where xc is the number of cells in the x dimension, and yc is the number of cells in the y dimension of the computational surfaces. Each component of c corresponds to the computational intensity at cell (i, j). The spatial computational domain is similar to an image obtained from an evaluation of the following function: f: I = [0, 1] × [0, 1] ? R (1) at cells (i/xc, j/yc)∈I, i = 1, ..., x c, j = 1, ... , yc . Thus, cij = f (i/x c, j/yc) (2) The value of cij represents the computational burden derived from a new computational transformation concept that is consistent with the role of transformations in other domains of GIScience (Tobler, 1979): • Computing time: the time taken to perform the analysis within cell (i, j); • Memory: the memory required to perform the analysis within cell (i, j); and • I/O: data input/output and transfers to perform analyses within cell (i, j). 3. Computational Transformation A computational transformation elucidates the computational intensity of a particular geographic analysis based on the characteristics of spatial data and operations. Two types of computational transformations are identified: 1.) Data-centric functions transform spatial data characteristics into memory or I/O surfaces. 2.) Operation-centric functions consider the spatial operations that are directly or indirectly related to spatial data characteristics, and transform the characteristics of spatial operations into a computing time surface. 3.1 An Illustrative Example An operation-centric transformation function illustrates the spatial computational domain for Clarke’s inverse distance weighted (IDW) interpolation algorithm (Clarke, 1995). This domain consists of a single computing time surface because both memory and I/O requirements are trivial even if a common desktop PC (e.g., 1G RAM and 2GHz Pentium processor) is used to execute the algorithm. The transformation function IDW-of on the cell (i, j) is defined as follows: IDW-of(i, j) ? timeUnit × eshold densityThr dp ns ne