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
-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.
@@ -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~$\varepsilon$.
 
-\>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}%
@@ -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
-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:
@@ -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
-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.