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Comparison-Sorting and Selecting in Totally Monotone Matrices
 

Summary: Chapter 1
Comparison-Sorting and Selecting in
Totally Monotone Matrices
Noga Alon
Yossi Azar
Abstract
An mn matrix A is called totally monotone if for all i1 < i2
and j1 < j2, 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

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics