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datovky
ds2-notes
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dbb01577
Commit
dbb01577
authored
Aug 28, 2021
by
Filip Stedronsky
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Succinct: merge definitions into a list
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fs-succinct/succinct.tex
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fs-succinct/succinct.tex
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4
View file @
dbb01577
...
@@ -24,14 +24,17 @@ The information-theoretical optimum is $OPT(n) := \lceil\log |X(n)|\rceil$
...
@@ -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
Now we can define three classes of data structures based on their fine-grained space
efficiency:
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.
A typical implicit data structure contains just its elements in some order and nothing more.
Examples include sorted arrays and heaps.
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
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
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
))
$
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.
...
@@ -45,4 +48,5 @@ fast operations on these space-efficient data structures.
\section
{
Succinct representation of strings
}
\section
{
Succinct representation of strings
}
\endchapter
\endchapter
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