Ackermann's function.

diff --git a/PLAN b/PLAN
index 174f3a601158b3b67fbcdf0dfb2cd8922f140f34..8fc44e12f493ff446a2966fb6679e164d94d64d6 100644 (file)
--- a/PLAN
+++ b/PLAN
@@ -97,7 +97,8 @@ Notation:
 - introduce \widehat\O early
 - unify { x ; ... }, { x | ...} and { x : ... }
 - capitalize Pointer Machine
-- define \alpha(m,n)
+- Ackermann: which of the Tarjan's set union papers should we cite?
+- Ackermann function vs. Ackermann's function
 
 Varia:
 

diff --git a/biblio.bib b/biblio.bib
index cef5f704cdb85ba70bec2d51de1e01fe96a129c6..d0b97845b3cb55569b8b5dcca8a5f17295d5d0c7 100644 (file)
--- a/biblio.bib
+++ b/biblio.bib
@@ -1261,3 +1261,25 @@
   pages={743--750},
   year={2002}
 }
+
+@article{ ackermann:function,
+  title={{Zum Hilbertschen Aufbau der reellen Zahlen}},
+  author={Ackermann, W.},
+  journal={Mathematische Annalen},
+  volume={99},
+  number={1},
+  pages={118--133},
+  year={1928},
+  publisher={Springer},
+  notes={In German}
+}
+
+@article{ klazar:gdss,
+  title={{Generalized Davenport--Schinzel Sequences: Results, problems, and applications}},
+  author={Klazar, M.},
+  journal={INTEGERS: The Electronic Journal of Combinatorial Number Theory},
+  volume={2},
+  number={A11},
+  pages={2},
+  year={2002}
+}

diff --git a/macros.tex b/macros.tex
index cc1a0e941ac6b1f43a540559b57720ffaa8bd9a1..b350a789dbf78c04d1c6c8a9b51384a74b56006e 100644 (file)
--- a/macros.tex
+++ b/macros.tex
@@ -46,7 +46,7 @@
 \def\deg{\mathop{\rm deg}}
 \def\timesalpha{\mskip2mu\alpha}
 \def\timesbeta{\mskip2mu\beta}
-\def\tower{\mathop\uparrow}
+\def\tower{\mathbin\uparrow}
 \def\Forb{{\rm Forb}}
 \def\minorof{\preccurlyeq}
 \def\per{\mathop{\rm per}}

diff --git a/mst.tex b/mst.tex
index 591d70cd051144d3bf86a5f36869f1cbf7d37443..5fef32f8046510d0325a115aa73568a0ca782915 100644 (file)
--- a/mst.tex
+++ b/mst.tex
@@ -494,7 +494,7 @@ with surprising efficiency:
 \thmn{Disjoint Set Union, Tarjan and van Leeuwen \cite{tarjan:setunion}}%
 Starting with a~trivial equivalence with single-element classes, a~sequence of operations
 comprising of $n$~\<Union>s intermixed with $m\ge n$~\<Find>s can be processed in time
-$\O(m\timesalpha(m,n))$, where $\alpha(m,n)$ is the inverse Ackermann function.\foot{See Section \ref{ackersec} for a~precise definition.}
+$\O(m\timesalpha(m,n))$, where $\alpha(m,n)$ is the inverse Ackermann's function.\foot{See Section \ref{ackersec} for a~precise definition.}
 
 \proof
 See \cite{tarjan:setunion}.

diff --git a/notation.tex b/notation.tex
index caaf93eb23d558d7c04930b5ef632bfb50beddc3..b63e59588a1e816c96e8a802273fea5280dd4a57 100644 (file)
--- a/notation.tex
+++ b/notation.tex
@@ -142,6 +142,62 @@ where $U^*$ is the complete graph on~$U$. Similarly for $G.F$ and $G.U$.
 
 %--------------------------------------------------------------------------------
 
-\section{Ackermann function and its inverse}\id{ackersec}%
+\section{Ackermann's function and its inverse}\id{ackersec}%
+
+The Ackermann's function is an~extremely quickly growing function that has been
+originally introduced by Ackermann \cite{ackermann:function} in the context of
+computability theory. Its original purpose was to demonstrate that a~function
+can be recursive, but not primitive recursive. At the first sight, it does not
+seem related to efficient algorithms at all. Its inverse however occurs in
+analysis of various algorithms and mathematical structures surprisingly often:
+We meet it in Section \ref{classalg} in the time complexity of the Disjoint Set Union
+data structure and also in the best known upper bound on the decision tree
+complexity of minimum spanning trees in Section \ref{optalgsect}. Another
+important application is the complexity of Davenport-Schinzel sequences (see
+Klazar's survey \cite{klazar:gdss}), but as far as we know, it is not otherwise
+related to the topic of our study.
+
+Various sources tend to differ in the exact definition of both the Ackermann's
+function and its inverse, but most of the definitions differ in factors that
+are negligible when compared with the asymptotic growth of the function.
+We will use the definition by double induction given by Tarjan \cite{tarjan:setunion},
+which are predominant in the literature on graph algorithms:
+
+\defn
+The \df{Ackermann's function} $A(x,y)$ is a~function on non-negative integers defined by:
+$$\eqalign{
+A(0,y) &:= 2y, \cr
+A(x,0) &:= 0, \cr
+A(x,1) &:= 2 \quad \hbox{for $x\ge 1$}, \cr
+A(x,y) &:= A(x-1, A(x,y-1)) \quad \hbox{for $x\ge 1$, $y\ge 2$}. \cr
+}$$
+Sometimes, a~single-parameter version of this function is also used. It is defined
+as the diagonal of the previous function, i.e., $A(n):=A(n,n)$.
+
+\example
+We can try evaluating $A(x,y)$ in some points:
+$$\eqalign{
+A(x,2) &= A(x-1, A(x,1)) = A(x-1,2) = A(0,2) = 4, \cr
+A(1,y) &= A(0, A(1,y-1)) = 2A(1,y-1) = 2^{y-1}A(1,1) = 2^y, \cr
+A(2,y) &= A(1, A(2,y-1)) = 2^{A(2,y-1)} = 2\tower y. \cr
+A(3,y) &= \hbox{the tower function iterated $y$~times \dots} \cr
+A(4,3) &= A(3,A(4,2)) = A(3,4) = A(2,A(3,3)) = A(2,A(2,A(3,2))) = \cr
+       &= A(2,A(2,4)) = 2\tower(2\tower 4) = 2\tower 65536. \cr
+}$$
+
+\defn
+The \df{Inverse Ackermann's function} $\alpha(x,y)$ is defined by:
+$$
+\alpha(x,n) := \min\{ y \mid A(x,y) > \log n \}.
+$$
+Again, a~single-parameter version is occasionally considered:
+$$
+\alpha(n) = \min\{ x \mid A(x,x) > \log n \}.
+$$
+
+\example
+$\alpha(1,n) = \O(\log\log n)$, $\alpha(2,n) = \O(\log^* n)$, $\alpha(3,n)$ grows even slower.
+
+\FIXME{Relationship between single- and double-parameter versions.}
 
 \endpart