<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
the	O
predecessor	B-Data_Structure
problem	I-Data_Structure
involves	O
maintaining	O
a	O
set	O
of	O
items	O
to	O
,	O
given	O
an	O
element	O
,	O
efficiently	O
query	B-Library
which	O
element	O
precedes	O
or	O
succeeds	O
that	O
element	O
in	O
an	O
order	O
.	O
</s>
<s>
Data	B-General_Concept
structures	I-General_Concept
used	O
to	O
solve	O
the	O
problem	O
include	O
balanced	B-Data_Structure
binary	I-Data_Structure
search	I-Data_Structure
trees	I-Data_Structure
,	O
van	B-Application
Emde	I-Application
Boas	I-Application
trees	I-Application
,	O
and	O
fusion	B-Data_Structure
trees	I-Data_Structure
.	O
</s>
<s>
In	O
the	O
static	B-Data_Structure
predecessor	I-Data_Structure
problem	I-Data_Structure
,	O
the	O
set	O
of	O
elements	O
does	O
not	O
change	O
,	O
but	O
in	O
the	O
dynamic	B-Data_Structure
predecessor	I-Data_Structure
problem	I-Data_Structure
,	O
insertions	O
into	O
and	O
deletions	O
from	O
the	O
set	O
are	O
allowed	O
.	O
</s>
<s>
The	O
predecessor	B-Data_Structure
problem	I-Data_Structure
is	O
a	O
simple	O
case	O
of	O
the	O
nearest	B-Algorithm
neighbor	I-Algorithm
problem	I-Algorithm
,	O
and	O
data	B-General_Concept
structures	I-General_Concept
that	O
solve	O
it	O
have	O
applications	O
in	O
problems	O
like	O
integer	B-Algorithm
sorting	I-Algorithm
.	O
</s>
<s>
Data	B-General_Concept
structures	I-General_Concept
that	O
solve	O
the	O
problem	O
support	O
these	O
operations	O
:	O
</s>
<s>
In	O
addition	O
,	O
data	B-General_Concept
structures	I-General_Concept
which	O
solve	O
the	O
dynamic	B-Data_Structure
version	O
of	O
the	O
problem	O
also	O
support	O
these	O
operations	O
:	O
</s>
<s>
The	O
problem	O
is	O
typically	O
analyzed	O
in	O
a	O
transdichotomous	O
model	O
of	O
computation	O
such	O
as	O
word	B-Application
RAM	I-Application
.	O
</s>
<s>
One	O
simple	O
solution	O
to	O
this	O
problem	O
is	O
to	O
use	O
a	O
balanced	B-Data_Structure
binary	I-Data_Structure
search	I-Data_Structure
tree	I-Data_Structure
,	O
which	O
achieves	O
(	O
in	O
Big	O
O	O
notation	O
)	O
a	O
running	O
time	O
of	O
for	O
predecessor	O
queries	B-Library
.	O
</s>
<s>
The	O
Van	B-Application
Emde	I-Application
Boas	I-Application
tree	I-Application
achieves	O
a	O
query	B-Library
time	O
of	O
,	O
but	O
requires	O
space	O
.	O
</s>
<s>
Dan	O
Willard	O
proposed	O
an	O
improvement	O
on	O
this	O
space	O
usage	O
with	O
the	O
x-fast	B-Data_Structure
trie	I-Data_Structure
,	O
which	O
requires	O
space	O
and	O
the	O
same	O
query	B-Library
time	O
,	O
and	O
the	O
more	O
complicated	O
y-fast	B-Data_Structure
trie	I-Data_Structure
,	O
which	O
only	O
requires	O
space	O
.	O
</s>
<s>
Fusion	B-Data_Structure
trees	I-Data_Structure
,	O
introduced	O
by	O
Michael	O
Fredman	O
and	O
Willard	O
,	O
achieve	O
query	B-Library
time	O
and	O
for	O
predecessor	O
queries	B-Library
for	O
the	O
static	O
problem	O
.	O
</s>
<s>
The	O
dynamic	B-Data_Structure
problem	O
has	O
been	O
solved	O
using	O
exponential	B-Data_Structure
trees	I-Data_Structure
with	O
query	B-Library
time	O
,	O
and	O
with	O
expected	O
time	O
using	O
hashing	B-Algorithm
.	O
</s>
<s>
There	O
have	O
been	O
a	O
number	O
of	O
papers	O
proving	O
lower	O
bounds	O
on	O
the	O
predecessor	B-Data_Structure
problem	I-Data_Structure
,	O
or	O
identifying	O
what	O
the	O
running	O
time	O
of	O
asymptotically	B-General_Concept
optimal	I-General_Concept
solutions	O
would	O
be	O
.	O
</s>
<s>
For	O
example	O
,	O
Michael	O
Beame	O
and	O
Faith	O
Ellen	O
proved	O
that	O
for	O
all	O
values	O
of	O
,	O
there	B-Algorithm
exists	I-Algorithm
a	O
value	O
of	O
with	O
query	B-Library
time	O
(	O
in	O
Big	O
Theta	O
notation	O
)	O
,	O
and	O
similarly	O
,	O
for	O
all	O
values	O
of	O
,	O
there	B-Algorithm
exists	I-Algorithm
a	O
value	O
of	O
such	O
that	O
the	O
query	B-Library
time	O
is	O
.	O
</s>
<s>
For	O
the	O
static	B-Data_Structure
predecessor	I-Data_Structure
problem	I-Data_Structure
,	O
Mihai	O
Pătrașcu	O
and	O
Mikkel	O
Thorup	O
showed	O
the	O
following	O
lower	O
bound	O
for	O
the	O
optimal	O
search	O
time	O
,	O
in	O
the	O
cell-probe	B-Application
model	I-Application
:	O
</s>
<s>
and	O
the	O
van	B-Application
Emde	I-Application
Boas	I-Application
tree	I-Application
achieves	O
this	O
bound	O
.	O
</s>
