\paran redefined.

diff --git a/macros.tex b/macros.tex
index 19972c873b13541f28532c636d5f025d8640fda3..be78b1f105601d96d0b106ca827774ea8d2822ef 100644 (file)
--- a/macros.tex
+++ b/macros.tex
@@ -376,7 +376,7 @@
 \def\problemn{\problem\labelx}
 \def\remn{\rem\labelx}
 
-\def\paran#1{\para {\sl #1:}}
+\def\paran#1{\para {\sl #1:\/}\enspace}
 
 \def\proof{\noindent {\sl Proof.}\enspace}
 \def\proofsketch{\noindent {\sl Proof sketch.}\enspace}

diff --git a/opt.tex b/opt.tex
index 44a9c8ca5bd562ad350d77a338aef3c35d0df1d8..b0540e3233425ca9a16f0dad77f54955b02db378 100644 (file)
--- a/opt.tex
+++ b/opt.tex
@@ -280,7 +280,7 @@ Let us translate these ideas to real (pseudo)code:
 all queues in the heap, walks the trees and the item lists of all vertices. It records
 all items seen, the corrupted ones are those that different from their \<ckey>.
 
-\paran{Analysis of accuracy}
+\paran{Analysis of accuracy}%
 The description of the operations is now complete, so let us analyse their behavior
 and verify that we have delivered what we promised --- first the accuracy of
 the structure, then the time complexity of operations. In the whole analysis,
@@ -348,7 +348,7 @@ last expression is less than $2^{k-r+2}$. Since the tree contains $n_k=2^k$ blac
 this makes less than $n_k/2^{r-2}$ corrupted items as we asserted.
 \qed
 
-\paran{Analysis of time complexity}
+\paran{Analysis of time complexity}%
 Now we will examine the amortized time complexity of the individual operations.
 We will show that if we charge $\O(r)$ time against every element inserted, it is enough
 to cover the cost of all other operations.
@@ -859,7 +859,7 @@ $\O(\poly(k))$ per object. The time complexity of the whole algorithm is therefo
 $\O(2^{k^2} \cdot 2^{2^{3k^2}} \cdot 2^{k^3} \cdot \poly(k)) = \O(2^{2^{4k^2}})$.
 \qed
 
-\paran{Basic properties of decision trees}
+\paran{Basic properties of decision trees}%
 The following properties will be useful for analysis of algorithms based
 on precomputed decision trees. We will omit some technical details, referring
 the reader to section 5.1 of the Pettie's paper \cite{pettie:optimal}.
@@ -1052,7 +1052,7 @@ The remaining steps of the algorithm can be easily performed in linear time eith
 or in case of the contractions by the bucket-sorting techniques of Section \ref{bucketsort}.
 \qed
 
-\paran{Optimality}
+\paran{Optimality}%
 The properties of decision tree complexity, which we have proven in the previous
 section, will help us show that the time complexity recurrence is satisfied by the
 decision tree complexity $D(m,n)$ itself. This way, we prove the following theorem:
@@ -1080,7 +1080,7 @@ The other inequality is obvious as $D(m,n)$ is an~asymptotic lower bound for
 the time complexity of every comparison-based algorithm.
 \qed
 
-\paran{Complexity of MST}
+\paran{Complexity of MST}%
 As we have already noted, the exact decision tree complexity $D(m,n)$ of the MST problem
 is still open and so is therefore the time complexity of the optimal algorithm. However,
 every time we come up with another comparison-based algorithm, we can use its complexity
@@ -1113,7 +1113,7 @@ independently on the other edges) or $G_{n,m}$ (we draw the graph uniformly at r
 set of all graphs with~$n$ vertices and $m$~edges), it runs in linear time with high probability,
 regardless of the edge weights.
 
-\paran{Models of computation}
+\paran{Models of computation}%
 Another important consequence of the optimal algorithm is that when we aim for a~linear-time
 MST algorithm (or for proving that it does not exist), we do not need to care about computational
 models at all. The elaborate RAM data structures of Chapter \ref{ramchap}, which have helped us