Comparison-sorting and selecting m totally monotone matrices

Noga Alon, Yossi Azar

Research output: Chapter in Book/Report/Conference proceedingConference contribution

4 Scopus citations


An m X n matrix A is called totally monotone if for all i1 < i2 and J1 < J2, A[i1, j1] > A[i2, 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 Ω(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.

Original languageEnglish (US)
Title of host publicationProceedings of the 3rd Annual ACM-SIAM Symposium on Discrete Algorithms. SODA 1992
PublisherAssociation for Computing Machinery
Number of pages6
ISBN (Electronic)089791466X
StatePublished - Sep 1 1992
Externally publishedYes
Event3rd Annual ACM-SIAM Symposium on Discrete Algorithms. SODA 1992 - Orlando, United States
Duration: Jan 27 1992Jan 29 1992

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
VolumePart F129721


Other3rd Annual ACM-SIAM Symposium on Discrete Algorithms. SODA 1992
Country/TerritoryUnited States

All Science Journal Classification (ASJC) codes

  • Software
  • General Mathematics


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