Online verification is hard.

diff --git a/PLAN b/PLAN
index 8495bd635e607489ae3ce2a01b42dab07a95e049..bef6665fdab792f1332dd97f105ded6a41e4d4f3 100644 (file)
--- a/PLAN
+++ b/PLAN
@@ -51,7 +51,6 @@ TODO:
 Spanning trees:
 
 - cite Eisner's tutorial \cite{eisner:tutorial}
-- \cite{pettie:onlineverify} online lower bound
 - move the remark on disconnected graphs? separate section?
 - mention graphs with non-unique weights? also in the separate section?
 - Some algorithms (most notably Fredman-Tarjan) do not need flattening

diff --git a/adv.tex b/adv.tex
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--- a/adv.tex
+++ b/adv.tex
@@ -1040,14 +1040,17 @@ Buchsbaum et al.~\cite{buchsbaum:verify} have recently shown that linear-time
 verification can be achieved even on the Pointer machine. They first solve the
 problem of finding the lowest common ancestors for a~set of pairs of vertices
 by batch processing: They combine an~algorithm of time complexity $\O(m\timesalpha(m,n))$
-based on the Union-Find data structure with the framework of topological graph
+based on the Disjoint Set Union data structure with the framework of topological graph
 computations developed in Section \ref{bucketsort}. Then they use a~similar
 technique for finding the peaks themselves.
 
 \rem
-The online version of this problem (build a~data structure for a~weighted tree
-in linear time and then answer queries for individual paths in constant time)
-is still open even for the RAM.
+The online version of this problem has turned out to be more difficult. It calls for an~algorithm
+that preprocesses the tree and then answers queries for peaks of paths presented online. Pettie
+\cite{pettie:onlineverify} has proven an~interesting lower bound based on the inverses of the
+Ackermann's function (see \ref{ackerinv}). If we want to answer queries within $t$~comparisons, we
+have to invest $\Omega(n\log\lambda_t(n))$ time into preprocessing. This implies that with
+preprocessing in linear time, the queries require $\Omega(\alpha(n))$ time.
 
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