 
Summary: Chapter 1
ComparisonSorting and Selecting in
Totally Monotone Matrices
Noga Alon \Lambda Yossi Azar y
Abstract
An m \Theta n matrix A is called totally monotone if for all i 1 ! i 2
and j1 ! j2 , A[i1 ; j1 ] ? A[i1 ; j2 ] implies A[i2 ; j1 ] ? A[i2 ; j2 ].
We consider the complexity of comparisonbased 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
