Commit f74bb19c by Filip Stedronsky

### Succinct: init

parent 66411d7c
 ... ... @@ -16,7 +16,8 @@ CHAPTERS= \ 05-cache \ 06-hash \ 07-geom \ 08-string 08-string \ fs-succinct chapters: for ch in $(CHAPTERS) ; do$(MAKE) -C ch pics ; done ... ...
 TOP=.. include ../Makerules
 \ifx\chapter\undefined \input adsmac.tex \singlechapter{50} \fi \chapter[succinct]{Space-efficient data structures} In this chapter, we will explore space-efficient data structures. This may sound like a boring topic at first -- after all, many of the commonly-used data sctructures have linear space complexity, which is asymptotically optimal. However, in this chapter, we shall use a much more fine-grained notion of space efficiency and measure space requirements in bits. Imagine we have a universe $X$ and we want a data structure to hold a single element $x \in X$. For example, $X$ can be the universe of all length-$n$ sequences of integers from the range $[m]$ and we want a data structure to hold such a sequence (we shall limit ourselves to finite universes only). The information-theoretically optimal size of such a data structure is $OPT := \lceil\log |X|\rceil$ bits (which is essentilly the entropy of a uniform probability distribution over $X$). Now we can define three classes of data structures based on their fine-grained space efficiency: \defn{An {\I implicit data structure} is one that uses at most $OPT + \O(1)$ bits of space.} A typical implicit data structure contains just its elements in some order and nothing more. Examples include sorted arrays and heaps. \defn{A {\I succinct data structure} is one that uses at most $OPT + {\rm o}(OPT)$ bits of space.} \defn{A {\I compact data structure} is one that uses at most $\O(OPT)$ bits of space.} Note that some linear-space data structures are not even compact -- because we are counting bits now, not words. For example, a binary search tree representing a length-$n$ sequence of numbers from range $[m]$ needs $\O(n (\log n + \log m))$ bits, whereas $OPT$ is $n \log m$. For $n >> m$, this does not satisfy the requirements for a compact data structure. And of course, as with any data structure, we want to be able to perform reasonably fast operations on these space-efficient data structures. \section{Succinct representation of strings} \endchapter
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