From adabac6babd5c6c52013f8ddf9e6a710ab5b5ba8 Mon Sep 17 00:00:00 2001 From: Filip Stedronsky <p@regnarg.cz> Date: Sat, 28 Aug 2021 15:06:47 +0200 Subject: [PATCH] Succinct: use new formalism in definitions --- fs-succinct/succinct.tex | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/fs-succinct/succinct.tex b/fs-succinct/succinct.tex index 151e8c8..90d790d 100644 --- a/fs-succinct/succinct.tex +++ b/fs-succinct/succinct.tex @@ -24,13 +24,13 @@ 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 that uses at most $OPT(n) + \O(1)$ bits of space.} +\defn{An {\I implicit data structure} is one with $s(n) \le OPT(n) + \O(1)$.} 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 $OPT(n) + {\rm o}(OPT(n))$ bits of space.} -\defn{A {\I compact data structure} is one that uses at most $\O(OPT(n))$ bits of space.} +\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 -- GitLab