From dbb015776ff28d3890518ab60ba22bf287850309 Mon Sep 17 00:00:00 2001
From: Filip Stedronsky <p@regnarg.cz>
Date: Sat, 28 Aug 2021 15:16:30 +0200
Subject: [PATCH] Succinct: merge definitions into a list
---
fs-succinct/succinct.tex | 12 ++++++++----
1 file changed, 8 insertions(+), 4 deletions(-)
diff --git a/fs-succinct/succinct.tex b/fs-succinct/succinct.tex
index 90d790d..b684710 100644
--- a/fs-succinct/succinct.tex
+++ b/fs-succinct/succinct.tex
@@ -24,14 +24,17 @@ The information-theoretical optimum is $OPT(n) := \lceil\log |X(n)|\rceil$
Now we can define three classes of data structures based on their fine-grained space
efficiency:
-\defn{An {\I implicit data structure} is one with $s(n) \le OPT(n) + \O(1)$.}
+\defn{A data structure is
+\tightlist{o}
+\:{\I implicit} when $s(n) \le OPT(n) + \O(1)$,
+\:{\I succinct} when $s(n) \le OPT(n) + {\rm o}(OPT(n))$,
+\:{\I compact} when $s(n) \le \O(OPT(n))$.
+\endlist
+}
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 with $s(n) \le OPT(n) + {\rm o}(OPT(n))$.}
-\defn{A {\I compact data structure} is one with $s(n) \le \O(OPT(n))$.}
-
Note that some linear-space data structures are not even compact -- because we
are counting bits now, not words. For example, a linked list representing a
length-$n$ sequence of numbers from range $[m]$ needs $\O(n (\log n + \log m))$
@@ -45,4 +48,5 @@ fast operations on these space-efficient data structures.
\section{Succinct representation of strings}
+
\endchapter
--
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