From 8ef36d9c3827f4f9c4ad864f93e68f35e7223170 Mon Sep 17 00:00:00 2001
From: Martin Mares <mj@ucw.cz>
Date: Tue, 19 Feb 2008 13:38:19 +0100
Subject: [PATCH] Start with ranks.

---
 notation.tex |  2 ++
 ram.tex      |  3 ++-
 rank.tex     | 35 ++++++++++++++++++++++++++++++++++-
 3 files changed, 38 insertions(+), 2 deletions(-)

diff --git a/notation.tex b/notation.tex
index 5517660..a9f1fa2 100644
--- a/notation.tex
+++ b/notation.tex
@@ -57,6 +57,8 @@
 \n{$\bxor$}{bitwise non-equivalence: $(x\bxor y)[i]=1$ iff $x[i]\ne y[i]$}
 \n{$x \shl n$}{bitwise shift of~$x$ by $n$~positions to the left: $x\shl n = x\cdot 2^n$}
 \n{$x \shr n$}{bitwise shift of~$x$ by $n$~positions to the right: $x\shr n = \lfloor x/2^n \rfloor$}
+\n{$R_{C,\prec}(x)$}{the rank of~$x$ in a~set~$C$ ordered by~$\prec$ \[rankdef]}
+\n{$R^{-1}_{C,\prec}(i)$}{the unrank of~$i$: the $i$-th smallest element of a~set~$C$ ordered by~$\prec$ \[rankdef]}
 }
 
 %--------------------------------------------------------------------------------
diff --git a/ram.tex b/ram.tex
index f576e83..2442df3 100644
--- a/ram.tex
+++ b/ram.tex
@@ -3,6 +3,7 @@
 \fi
 
 \chapter{Fine Details of Computation}
+\id{ramchap}
 
 \section{Models and machines}
 
@@ -342,7 +343,7 @@ and Willard. It will also form a~basis for the rest of this chapter.
 
 %--------------------------------------------------------------------------------
 
-\section{Bits and vectors}
+\section{Bits and vectors}\id{bitsect}
 
 In this rather technical section, we will show how the RAM can be used as a~vector
 computer to operate in parallel on multiple elements, as long as these elements
diff --git a/rank.tex b/rank.tex
index a0823ea..365fa94 100644
--- a/rank.tex
+++ b/rank.tex
@@ -4,7 +4,40 @@
 
 \chapter{Ranking Combinatorial Structures}
 
-\section{Ranking of permutations}
+\section{Ranking and unranking}
+
+The techniques for building efficient data structures on the RAM described
+in Chapter~\ref{ramchap} can be also used for a~variety of problems related
+to ranking of combinatorial structures. Generally, the problems are stated
+in the following way:
+
+\defn\id{rankdef}%
+Let~$C$ be a~set of objects and~$\prec$ a~linear order on~$C$. The \df{rank}
+$R_{C,\prec}(x)$ of an~element $x\in C$ is the number of elements $y\in C$ such that $y\prec x$.
+When~$\prec$ is defined on some superset of~$C$, we naturally extend the rank
+to elements outside~$C$.
+We will call the function $R_{C,\prec}$ the \df{ranking function} for $C$ and~$\prec$
+and its inverse $R^{-1}_{C,\prec}$ the \df{unranking function}. When the set
+and the order are clear from the context, we will use just~$R(x)$ and $R^{-1}(x)$.
+
+\example
+Let us consider the set $C_k=\{\0,\1\}^k$ of all binary strings of length~$k$ ordered
+lexicographically. Then $R^{-1}(i)$ is the $i$-th smallest element of this set, that
+is the number~$i$ written in binary and padded to~$k$ digits (i.e., $\(i)_k$ in the
+notation of Section~\ref{bitsect}). Obviously, $R(x)$ is the integer whose binary
+representation is~$x$.
 
+\FIXME{Uses of ranks.}
+
+\para
+In this chapter, we will investigate how to compute the ranking and unranking
+functions on different sets. Usually, we will make use of the fact that the ranks
+(and hence the input and output of our algorithm) are large numbers, so we can
+use the integers of a~similar magnitude to represent non-trivial data structures.
+This will allow us to design efficient algorithms on the RAM.
+
+\FIXME{In the whole chapter, assume RAM.}
+
+\section{Ranking of permutations}
 
 \endpart
-- 
2.39.5