P-complete problems seem to have no parallel algorithm which runs in polylogarithmic time using a polynomial number of processors. A P-complete problem is in the class EP (Efficient and Polynomially fast) if and only if there exists a cost optimal algorithm to solve it in T(n) = O(t(n)ε) (ε < 1) using P(n) processors such that T(n) P(n) = O(t(n)), where t(n) is the time complexity of the fastest sequential algorithm which solves the problem. The goal of our research is to find EP parallel algorithms for some P-complete problems. In this paper first we consider the convex layers problem. We give an algorithm for computing the convex layers of a set S of n points in the plane. Let k be the number of the convex layers of S. When 1 k nε/2 (0 ε < 1) our algorithm runs in O((n log n)/p) time using p processors, where 1 p n1-ε/2, and it is cost optimal. Next, we consider the envelope layers problem of a set S of n line segments in the plane. Let k be the number of the envelope layers of S. When 1 k nε/2 (0 ε < 1), we propose an algorithm for computing the envelope layers of S in O((n α(n) log3 n)/p) time using p processors, where 1 p n1-ε/2, and α(n) is the functional inverse of Ackermann's function which grows extremely slowly. The computational model we use in this paper is the CREW-PRAM. Our first algorithm, for the convex layers problem, belongs to EP, and the second one, for the envelope layers problem, belongs to the class EP if a small factor of log n is ignored.

We study the computational power of PC=P. We give a characterization of the class via single Turing machines. Based on the characterization, we give combinatorial problems that are Pm-complete for the class.