Corrections to K-best.

diff --git a/dyn.tex b/dyn.tex
index 221472cf80a0eb53d3ec7104d1be1abf7a17824f..c31c63e393c7550bb416b0b6f6c1e0d04975dd4a 100644 (file)
--- a/dyn.tex
+++ b/dyn.tex
@@ -765,15 +765,22 @@ exchange in this sequence therefore obtains~$T_i$ from a~tree of the desired pro
 
 \para
 This lemma implies that the second best spanning tree~$T_2$ differs from~$T_1$ by a~single
 
 \para
 This lemma implies that the second best spanning tree~$T_2$ differs from~$T_1$ by a~single

-edge. It remains to find which exchange it is. Let us consider the exchange
+edge exchange. It remains to find which exchange it is. Let us consider the exchange

 of an~edge $f\in E\setminus T_1$ with an~edge $e\in T_1[f]$. We get a~tree $T_1-e+f$
 of weight $w(T_1)-w(e)+w(f)$. To obtain~$T_2$, we have to find~$e$ and~$f$ such that the
 difference $w(f)-w(e)$ is the minimum possible. Thus for every~$f$, the edge $e$~must be always
 the heaviest on the path $T_1[f]$. We can now apply the algorithm from Corollary \ref{rampeaks}
 of an~edge $f\in E\setminus T_1$ with an~edge $e\in T_1[f]$. We get a~tree $T_1-e+f$
 of weight $w(T_1)-w(e)+w(f)$. To obtain~$T_2$, we have to find~$e$ and~$f$ such that the
 difference $w(f)-w(e)$ is the minimum possible. Thus for every~$f$, the edge $e$~must be always
 the heaviest on the path $T_1[f]$. We can now apply the algorithm from Corollary \ref{rampeaks}

-to find the heaviest edges (peaks) of all such paths and thus examine all possible choices of~$f$
-in linear time. This implies the following:
+and find the heaviest edges (peaks) of all such paths and thus examine all possible choices of~$f$
+in linear time. So we get:

 
 \lemma
 
 \lemma

+For every graph~$H$ and a~MST $T$ of~$H$, linear time is sufficient to find
+edges $e\in T$ and $f\in H\setminus T$ such that $w(f)-w(e)$ is minimum.
+
+\nota
+We will call this \df{finding the best exchange in $(H,T)$.}
+
+\cor

 Given~$G$ and~$T_1$, we can find~$T_2$ in time $\O(m)$.
 
 \paran{Third lightest spanning tree}%
 Given~$G$ and~$T_1$, we can find~$T_2$ in time $\O(m)$.
 
 \paran{Third lightest spanning tree}%


@@ -808,7 +815,9 @@ on both~$G_1$ and~$G_2$ and find~$T_3$ again in time $\O(m)$.
 \paran{Further spanning trees}%
 The construction of auxiliary graphs can be iterated to obtain $T_1,\ldots,T_K$
 for an~arbitrary~$K$. We will build a~\df{meta-tree} of auxiliary graphs. Each node of this meta-tree
 \paran{Further spanning trees}%
 The construction of auxiliary graphs can be iterated to obtain $T_1,\ldots,T_K$
 for an~arbitrary~$K$. We will build a~\df{meta-tree} of auxiliary graphs. Each node of this meta-tree

-is assigned a~minor of~$G$ and its minimum spanning tree. The root node contains~$(G,T_1)$,
+is assigned a~graph\foot{This graph is always derived from~$G$ by a~sequence of edge deletions
+and contractions. It is tempting to say that it is a~minor of~$G$, but this is not true as we
+preserve multiple edges.} and its minimum spanning tree. The root node contains~$(G,T_1)$,

 its sons have $(G_1,T_1/e)$ and $(G_2,T_2)$. When $T_3$ is obtained by an~exchange
 in one of these sons, we attach two new leaves to that son and we assign to them the two auxiliary
 graphs derived by contracting or deleting the exchanged edge. Then we find the best
 its sons have $(G_1,T_1/e)$ and $(G_2,T_2)$. When $T_3$ is obtained by an~exchange
 in one of these sons, we attach two new leaves to that son and we assign to them the two auxiliary
 graphs derived by contracting or deleting the exchanged edge. Then we find the best


@@ -829,7 +838,7 @@ them into the heap. The algorithm is now straightforward and so will be its anal
 \:$R\=$ a~meta tree whose vertices carry triples $(G',T',F')$. Initially
   it contains just a~root with $(G,T_1,\emptyset)$.
   \hfil\break
 \:$R\=$ a~meta tree whose vertices carry triples $(G',T',F')$. Initially
   it contains just a~root with $(G,T_1,\emptyset)$.
   \hfil\break

-  \cmt{$G'$ is a~minor of~$G$, $T'$~is its MST, and~$F'$ is a~set of edges of~$G$
+  \cmt{$G'$ is a~graph, $T'$~is its MST, and~$F'$ is a~set of edges of~$G$

   that are contracted in~$G'$.}
 \:$H\=$ a~heap of quadruples $(\delta,r,e,f)$ ordered on~$\delta$, initially empty.
   \hfil\break
   that are contracted in~$G'$.}
 \:$H\=$ a~heap of quadruples $(\delta,r,e,f)$ ordered on~$\delta$, initially empty.
   \hfil\break