Deterministic selection in O(loglog N) parallel time
STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
Sorting and selecting in rounds
SIAM Journal on Computing
The concave least-weight subsequence problem revisited
Journal of Algorithms
Sorting, approximate sorting, and searching in rounds
SIAM Journal on Discrete Mathematics
Discrete Applied Mathematics - Computational combinatiorics
An efficient parallel algorithm for the row minima of a totally monotone matrix
SODA '91 Proceedings of the second annual ACM-SIAM symposium on Discrete algorithms
Selection and sorting in totally monotone arrays
SODA '90 Proceedings of the first annual ACM-SIAM symposium on Discrete algorithms
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An m x n matrix A is called totally monotone if for all i1 2 and j1 2, A[i1, j1] A[i1, j2 implies A[i2, j1] A[i2, j2]. We consider the complexity of comparison-based selection and sorting algorithms in such matrices. Although our selection algorithm counts only comparisons its advantage on all previous work is that it can also handle selection of elements of different (and arbitrary) ranks in different rows (or even selection of elements of several ranks in each row), in time which is slightly better than that of the best known algorithm for selecting elements of the same rank in each row. We also determine the decision tree complexity of sorting each row of a totally monotone matrix up to a factor of at most log n by proving a quadratic lower bound and by slightly improving the upper bound. No nontrivial lower bound was previously known for this problem. In particular for the case m = n we prove a tight &OHgr;(n2) lower bound. This bound holds for any decision-tree algorithm, and not only for a comparison-based algorithm. The lower bound is proved by an exact characterization of the bitonic totally monotone matrices, whereas our new algorithms depend on techniques from parallel comparison algorithms.