More on the Ackermann's function and its inverses.

diff --git a/mst.tex b/mst.tex
index 5fef32f8046510d0325a115aa73568a0ca782915..9f8a7414bc097bba7a59f91e324daf9408d3e1f2 100644 (file)
--- a/mst.tex
+++ b/mst.tex
@@ -494,7 +494,8 @@ 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's function.\foot{See Section \ref{ackersec} for a~precise definition.}
+$\O(m\timesalpha(m,n))$, where $\alpha(m,n)$ is a~certain inverse of the Ackermann's function
+(see Definition \ref{ackerinv}).
 
 \proof
 See \cite{tarjan:setunion}.

diff --git a/notation.tex b/notation.tex
index efc401473c2207a2cf5cf3ecd5462d5de641f90c..5d6933cb874d96c625ed3bec5740552387947ae1 100644 (file)
--- a/notation.tex
+++ b/notation.tex
@@ -37,7 +37,6 @@
 \n{$G/e$}{multigraph contraction \[contract]}
 \n{$G.e$}{simple graph contraction \[simpcont]}
 \n{$G/X$, $G.X$}{contraction by a~set $X$ of vertices or edges \[setcont]}
-\n{$\alpha(n)$}{the inverse Ackermann's function}
 \n{$f[X]$}{function applied to a set: $f[X]:=\{ f(x) ; x\in X \}$}
 \n{$f[e]$}{as edges are two-element sets, $f[e]$ maps both endpoints of an edge~$e$}
 \n{$\varrho({\cal C})$}{edge density of a graph class~$\cal C$ \[density]}
@@ -88,6 +87,11 @@
 \n{${\cal D}(G)$}{The optimal MSF decision tree for a~graph~$G$ \[decdef]}
 \n{$D(G)$}{The depth of ${\cal D}(G)$ \[decdef]}
 \n{$D(m,n)$}{Decision tree complexity of MSF \[decdef]}
+\n{$A(x,y)$}{The Ackermann's function \[ackerdef]}
+\n{$A(x)$}{The diagonal Ackermann's function \[ackerdef]}
+\n{$a(x,n)$}{The inverse of the $x$-th row of the Ackermann's function \[ackerinv]}
+\n{$a(n)$}{The diagonal inverse of the Ackermann's function \[ackerinv]}
+\n{$\alpha(m,n)$}{$\alpha(m,n) := \min\{ x\ge 1 \mid A(x,4\lceil m/n\rceil) > \log n \}$ \[ackerinv]}
 }
 
 %--------------------------------------------------------------------------------
@@ -148,13 +152,13 @@ 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
+seem related to efficient algorithms at all. Its various inverses however occur 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
+We meet them 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
+Klazar's survey \cite{klazar:gdss}), but as far as we know, there are not otherwise
 related to the topic of our study.
 
 Various sources tend to differ in the exact definition of both the Ackermann's
@@ -163,7 +167,7 @@ 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 is predominant in the literature on graph algorithms:
 
-\defn
+\defn\id{ackerdef}%
 The \df{Ackermann's function} $A(x,y)$ is a~function on non-negative integers defined as:
 $$\eqalign{
 A(0,y) &:= 2y, \cr
@@ -185,20 +189,46 @@ A(4,3) &= A(3,A(4,2)) = A(3,4) = A(2,A(3,3)) = A(2,A(2,A(3,2))) = \cr
        &= A(2,A(2,4)) = 2\tower(2\tower 4) = 2\tower 65536. \cr
 }$$
 
-\defn
-The \df{Inverse Ackermann's function} $\alpha(x,y)$ is defined as:
+\para
+Three functions related to the inverse of the function~$A$ are usually considered:
+
+\defn\id{ackerinv}%
+The \df{row inverse} $a(x,y)$ of the Ackermann's function is defined as:
+$$
+a(x,n) := \min\{ y \mid A(x,y) > \log n \}.
 $$
-\alpha(x,n) := \min\{ y \mid A(x,y) > \log n \}.
+The \df{diagonal inverse} $a(n)$ is defined as:
 $$
-Again, a~single-parameter version is occasionally considered:
+a(n) := \min\{ x \mid A(x,x) > \log n \}.
 $$
-\alpha(n) := \min\{ x \mid A(x,x) > \log n \}.
+The \df{alpha function} $\alpha(m,n)$ is defined for $m\ge n$ as:
+$$
+\alpha(m,n) :=  \min\{ x\ge 1 \mid A(x,4\lceil m/n\rceil) > \log n \}.
 $$
 
 \example
-$\alpha(1,n) = \O(\log\log n)$, $\alpha(2,n) = \O(\log^* n)$, $\alpha(3,n)$ grows even slower.
+$a(1,n) = \O(\log\log n)$, $a(2,n) = \O(\log^* n)$, $a(3,n)$ grows even slower
+and $a(n)$ is asymptotically smaller than $a(x,n)$ for any fixed~$x$.
 
 \obs
-As the rows of the function~$A$ are increasing, we have $A(x,y) \ge A(x,x) = A(x)$ for $y\ge x$
+The rows of $A(x,y)$ are non-decreasing and so are the columns, so $\alpha(m,n)$
+is maximized when $m=n$. Thus $\alpha(m,n) \le 3$ whenever $\log n < A(3,4)$,
+which happens for all ``practical'' values of~$m$.
+
+\lemma
+$\alpha(m,n) \le a(n)+1$.
+
+\proof
+$A(x,4\lceil m/n\rceil) \ge A(x,4) = A(x-1,A(x,3)) \ge A(x-1,x-1)$, so $A(x,4\lceil m/n\rceil)$
+rises above $\log n$ no later than $A(x-1,x-1)$ does so.
+\qed
+
+\lemma\id{alphaconst}%
+When $k$~is a~fixed constant and $m\ge n\cdot a(k,n)$, then $\alpha(m,n) \le k$.
+
+\proof
+The choice of~$m$ guarantees that $A(x,4\lceil m/n\rceil) \ge A(x,a(k,n))$, which
+is greater than $\log n$ for all $x \ge k$.
+\qed
 
 \endpart

diff --git a/opt.tex b/opt.tex
index 9de4d64a617cfd05943cdd07d4f3c0c29663d661..15b97d510b3796933ef1c5b137381fa208038cad 100644 (file)
--- a/opt.tex
+++ b/opt.tex
@@ -1123,6 +1123,16 @@ We bound $D(m,n)$ by the number of comparisons performed by the algorithm of Cha
 or Pettie \cite{pettie:ackermann}.
 \qed
 
+Similarly to the Iterated Jarn\'\i{}k's algorithm, this bound is actually linear for classes of graphs
+that do not have density extremely close to constant:
+
+\cor
+The Optimal MST algorithm runs in linear time whenever $m\ge n\cdot a(k,n)$ for any fixed~$k$.
+
+\proof
+Combine the previous theorem with Lemma \ref{alphaconst}.
+\qed
+
 \rem
 It is also known from \cite{pettie:optimal} that when we run the Optimal algorithm on a~random
 graph drawn from either $G_{n,p}$ (random graphs on~$n$ vertices, each edge is included with probability~$p$