Efficient implementation of global computations on a, linear array of processors is complicated due to the small communication bandwidth and the large communication diameter of the array. This paper presents efficient parallel techniques for partitioning, movement, and reduction of data on linear arrays. Also, efficient data structures are used to enable fast sequential access of query points within each processor. This combination of serial and parallel techniques is used to derive an optimal parallel algorithm for computing the convex hull of each connected region in an n × n image. The algorithm takes O(n2/p) time on a linear array with p processors, where 1 ≤ p ≤ n/logn. This result is processor-time optimal since an optimal sequential algorithm takes O(n2) to solve the problem. Thus, a linear array with n/log n processors can solve the above problem in O(nlogn) time. In comparison, a two dimensional mesh-connected array of processors can solve this problem in O(n) time using n2 processors. The processor-time product for the mesh is 0(n3), which is not optimal.