Abstract: Final fixes.

diff --git a/abstract.tex b/abstract.tex
index ce4baa0a662a3a43ebeeffacebc739121cd91fcb..27a0870849607b29cad18e3cf7103d3c0a299126 100644 (file)
--- a/abstract.tex
+++ b/abstract.tex
@@ -31,7 +31,7 @@ with the intricate details of various models of computation and even of arithmet
 itself.
 
 We have tried to cover all known important results on both problems and unite them
 itself.
 
 We have tried to cover all known important results on both problems and unite them

-in a~single coherent theory. At many places, we have attempted to contribute my own
+in a~single coherent theory. At many places, we have attempted to contribute our own

 little stones to this mosaic: several new results, simplifications of existing
 ones, and last, but not least filling in important details where the original
 authors have missed some.
 little stones to this mosaic: several new results, simplifications of existing
 ones, and last, but not least filling in important details where the original
 authors have missed some.


@@ -615,7 +615,7 @@ the expense of \df{corrupting} a~fraction of the inserted elements by raising
 their values (the values are however never lowered). This allows for
 a~trade-off between accuracy and speed, controlled by a~parameter~$\varepsilon$.
 
 their values (the values are however never lowered). This allows for
 a~trade-off between accuracy and speed, controlled by a~parameter~$\varepsilon$.
 

-\>In the thesis, we describe the exact mechanics of the soft heaps and analyse its complexity.
+In the thesis, we describe the exact mechanics of the soft heaps and analyse its complexity.

 The important properties are characterized by the following theorem:
 
 \thmn{Performance of soft heaps, Chazelle \cite{chazelle:softheap}}\id{softheap}%
 The important properties are characterized by the following theorem:
 
 \thmn{Performance of soft heaps, Chazelle \cite{chazelle:softheap}}\id{softheap}%


@@ -726,7 +726,7 @@ complexity is therefore an~obvious lower bound on the time complexity of the
 problem in all other comparison-based models.
 
 The downside is that we do not know any explicit construction of the optimal
 problem in all other comparison-based models.
 
 The downside is that we do not know any explicit construction of the optimal

-decision trees, or at least a~non-constructive proof of their complexity.
+decision trees, nor even a~non-constructive proof of their complexity.

 On the other hand, the complexity of any existing comparison-based algorithm
 can be used as an~upper bound on the decision tree complexity. Also, we can
 construct an~optimal decision tree using brute force:
 On the other hand, the complexity of any existing comparison-based algorithm
 can be used as an~upper bound on the decision tree complexity. Also, we can
 construct an~optimal decision tree using brute force:


@@ -1356,7 +1356,7 @@ We have seen the many facets of the minimum spanning tree problem. It has
 turned out that while the major question of the existence of a~linear-time
 MST algorithm is still open, backing off a~little bit in an~almost arbitrary
 direction leads to a~linear solution. This includes classes of graphs with edge
 turned out that while the major question of the existence of a~linear-time
 MST algorithm is still open, backing off a~little bit in an~almost arbitrary
 direction leads to a~linear solution. This includes classes of graphs with edge

-density at least $\lambda_k(n)$ for an~arbitrary fixed~$k$,
+density at least $\lambda_k(n)$ (the $k$-th row inverse of the Ackermann's function) for an~arbitrary fixed~$k$,

 minor-closed classes, and graphs whose edge weights are
 integers. Using randomness also helps, as does having the edges pre-sorted.
 
 minor-closed classes, and graphs whose edge weights are
 integers. Using randomness also helps, as does having the edges pre-sorted.