In the definiton of the soft queue, we should better explicitly mention that

both sons always keep the same rank.

diff --git a/opt.tex b/opt.tex
index 067ac93a268cc556e548fbb371b596b71acba4d7..202e52b72bdb449277d0d851efa4faff8adce8b1 100644 (file)
--- a/opt.tex
+++ b/opt.tex
@@ -96,9 +96,10 @@ value or we set $\<ckey>(v)=+\infty$. The \<ckey>s obey the standard \df{heap or
 
 Each vertex is also assigned its \df{rank,} which is a~non-negative integer.
 The ranks of leaves are always zero, the rank of every internal vertex can be
-arbitrary, but it must be strictly greater than the ranks of its sons. We
-define the rank of the whole queue to be equal to the rank of its root vertex and
-similarly for its \<ckey>.
+arbitrary, but it must be strictly greater than the ranks of its sons. Also,
+we will always maintain the ranks in such a~way that both sons will have the
+equal ranks. We define the rank of the whole queue to be equal to the rank of
+its root vertex and similarly for its \<ckey>.
 
 A~queue is called \df{complete} if every two vertices joined by an~edge have rank
 difference exactly one. In other words, it is a~complete binary tree and the