Parallel B-Spline Surface Fitting on Mesh-Connected Computers

sented, where cost is defined to be the product of time and the number of processors used [19]. Using a newly proposed computational model called the mesh-of-unshuffle network, Chung and Lin [7] presented an O(log m log n)-time parallel algorithm using O(mn/(log m log n)) processors. On a hypercube network, a similar result was presented by Lin and Chung [20]. Using the pipelining strategy [11], an O(log n9)-time parallel algorithm was presented on the same network with O(mn/log n9) processors [13], where n9 5 max(m, n). Cheng et al. [6] presented the first constant-time parallel algorithm on a mesh-connected computer with O(mn) processors based on the Chebyshev iterative method; they also gave the corresponding relative error analysis. Note that their algorithm [6] is timeand cost-optimal. Based on the matrix perturbation method, this paper first presents a novel approximate O(n/p)-time parallel Bspline curve fitting algorithm for finding the corresponding n control points that interpolate those n data points on a linear array processor with p processors, where p # n. Given m 3 n data points, we then present an O(mn/ (p1 p2))-time parallel algorithm for B-spline surface fitting on a p1 3 p2 mesh-connected computer, where p1 # m and p2 # n. The relative error analyses of our two stable and cost-optimal parallel solvers are also given. When setting p1 5 m and p2 5 n, a timeand cost-optimal parallel solver can be derived; our result is a direct method vs. the result of Cheng et al. [6].