3.1.1 Triangulating a Monotone Polygon 3.2 Range Minima 2.3.2 Query Retrieval 3.1 All Nearest Smaller Values 2.2 Preex Minima and All Nearest Smaller Values

Parallel algorithms for shared-memory machines. In J. 27] Y. Matias and U. Vishkin. Converting high probability into nearly-constant time|with applications to parallel hashing. 6] O. Berkman and Y. Matias. Fast parallel algorithms for minimum and related problems with small integer inputs. 11] J. Edmonds. Lower bounds with smaller domain size on concurrent write parallel machines. Lower bounds for parallel random-access machines with unbounded shared memory. 15 Proof. The previous theorem showed a lower bound of T 1 3 log log n for s = 2 n 4log logn. A simple calculation shows that T 1 3 log log log s for n suuciently large. 5 Conclusions We have shown that the minima, preex-minima, range-minima, and ansv problems, with input elements taken from the integer domain 1::s], s n, can all be solved in O(log log log s) time using n= log log log s processors (optimal speedup) on the common crcw pram. As an application, we obtain an algorithm with the same bounds for the problem of triangulating a monotone polygon whose coordinates are taken from the integer domain 1::s]. Our results were recently used by 6] to obtain O(log log log s) time algorithms for values of s smaller than n. We also gave a matching lower bound of (log log log s) for 2 2 c 1 logn loglogn s 2 2 log c 2 n , where c 1 is a speciic constant and c 2 an arbitrary constant. Thus, our algorithms cannot be improved when expressed solely in terms of the domain size. This result is somewhat unsatisfying, however, since for the given range of s, log log log s = (log log n). The lower bound is an advance over the previously known bounds 28], which required larger domain sizes, but it would be preferable to show our algorithms are tight for all values of s, particularly those below n. There is evidence, however, that this is not the case. 15] gave a technique which could be applied to nd the minimum of n integers from the range 1::n k ] in O(k) time on a common crcw pram; 5] gave an O(log n) time algorithm on priority for computing the preex-minima when s = O(n). This shows that t = (log log log s) does not give the correct tradeoo between domain size and computation time for all values of s. More work is …