![]() ![]() ![]() Once we have developed an algorithm (q.v.) for solving a computational problem and analyzed its worst-case time requirements as a function of the size of its input (most usefully, in terms of the O-notation see ALGORITHMS, ANALYSIS OF), it is inevitable to ask the question: "Can we do better?" In a typical problem, we may be able to devise new algorithms for the problem that are more and more efficient. Optimal log-time algorithms for computing row minima of totally monotone matrices will improve our algorithm and enable it to have the same work as the sequential algorithm of T. ![]() A new algorithm for computing the row minima of totally monotone matrices improves our parallel MCOP algorithm to O(nlg 1♵ n) work and polylog time on a CREW PRAM. Next, by using efficient algorithms for computing row minima of totally monotone matrices, this complexity is improved to O(lg 2 n) time with n processors on the EREW PRAM and to O(lg 2 nlglgn) time with n/lglgn processors on a common CRCW PRAM. Here we give several new parallel algorithms including O(lg 3 n)-time and n/lgn-processor algorithms for solving the matrix chain ordering problem and for solving an optimal triangulation problem of convex polygons on the common CRCW PRAM model. The matrix chain ordering problem is to find the cheapest way to multiply a chain of n matrices, where the matrices are pairwise compatible but of varying dimensions. ![]()
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