Optimal geometric algorithms on fixed-size linear arrays and scan line arrays

Optimal parallel solutions are presented to several geometric problems on an n×n image on a fixed-size linear array with p processors, where 1⩽p⩽n. The array model considered here is an abstraction of several linearly connected parallel computers that have been constructed recently. The authors present O(n²/p) time solutions to several geometric problems which require global transfer of information such as labeling connected regions, computing the convexity and intersections of multiple regions, and computing several distance functions. All the solutions are optimal in the sense that their processor-time product is equal to the sequential complexity of the problems. Limitations of linear arrays in image computations are also discussed by showing that there are certain image problems which can be solved sequentially in O(n²) time, but require Ω(n²3) time on a linear array, irrespective of the number of processors used and the way in which the input image is partitioned among the processors. The authors also show alternate fixed-size array organizations with p processors which can solve the above problems in O(n²/p) time, for 1⩽p⩽n