<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
a	O
fusion	B-Data_Structure
tree	I-Data_Structure
is	O
a	O
type	O
of	O
tree	B-Application
data	I-Application
structure	I-Application
that	O
implements	O
an	O
associative	B-Application
array	I-Application
on	O
-bit	O
integers	O
on	O
a	O
finite	O
universe	O
,	O
where	O
each	O
of	O
the	O
input	O
integer	O
has	O
size	O
less	O
than	O
2w	O
and	O
is	O
non-negative	O
.	O
</s>
<s>
When	O
operating	O
on	O
a	O
collection	O
of	O
key	B-Application
–	I-Application
value	I-Application
pairs	I-Application
,	O
it	O
uses	O
space	O
and	O
performs	O
searches	O
in	O
time	O
,	O
which	O
is	O
asymptotically	O
faster	O
than	O
a	O
traditional	O
self-balancing	B-Data_Structure
binary	I-Data_Structure
search	I-Data_Structure
tree	I-Data_Structure
,	O
and	O
also	O
better	O
than	O
the	O
van	B-Application
Emde	I-Application
Boas	I-Application
tree	I-Application
for	O
large	O
values	O
of	O
.	O
</s>
<s>
Fusion	B-Data_Structure
trees	I-Data_Structure
were	O
invented	O
in	O
1990	O
by	O
Michael	O
Fredman	O
and	O
Dan	O
Willard	O
.	O
</s>
<s>
In	O
1999	O
it	O
was	O
shown	O
how	O
to	O
implement	O
fusion	B-Data_Structure
trees	I-Data_Structure
under	O
a	O
model	O
of	O
computation	O
in	O
which	O
all	O
of	O
the	O
underlying	O
operations	O
of	O
the	O
algorithm	O
belong	O
to	O
AC0	O
,	O
a	O
model	O
of	O
circuit	O
complexity	O
that	O
allows	O
addition	O
and	O
bitwise	O
Boolean	O
operations	O
but	O
does	O
not	O
allow	O
the	O
multiplication	O
operations	O
used	O
in	O
the	O
original	O
fusion	B-Data_Structure
tree	I-Data_Structure
algorithm	O
.	O
</s>
<s>
A	O
dynamic	O
version	O
of	O
fusion	B-Data_Structure
trees	I-Data_Structure
using	O
hash	B-Algorithm
tables	I-Algorithm
was	O
proposed	O
in	O
1996	O
which	O
matched	O
the	O
original	O
structure	O
's	O
runtime	O
in	O
expectation	O
.	O
</s>
<s>
Another	O
dynamic	O
version	O
using	O
exponential	B-Data_Structure
tree	I-Data_Structure
was	O
proposed	O
in	O
2007	O
which	O
yields	O
worst-case	O
runtimes	O
of	O
per	O
operation	O
.	O
</s>
<s>
Finally	O
,	O
it	O
was	O
shown	O
that	O
dynamic	O
fusion	B-Data_Structure
trees	I-Data_Structure
can	O
perform	O
each	O
operation	O
in	O
time	O
deterministically	O
.	O
</s>
<s>
This	O
data	O
structure	O
is	O
helpful	O
for	O
predecessor	O
and	O
successor	O
search	O
operations	O
which	O
provide	O
an	O
output	O
as	O
the	O
greatest	O
key	B-Application
value	I-Application
smaller	O
than	O
the	O
queried	O
key	O
or	O
the	O
smallest	O
key	B-Application
value	I-Application
larger	O
than	O
the	O
queried	O
key	O
,	O
respectively	O
.	O
</s>
<s>
The	O
chief	O
idea	O
why	O
this	O
work	O
is	O
efficient	O
is	O
because	O
it	O
deploys	O
word-level	O
parallelism	B-Operating_System
,	O
which	O
means	O
that	O
the	O
computation	O
on	O
multiple	O
words	O
takes	O
place	O
simultaneously	O
,	O
making	O
the	O
number	O
of	O
overall	O
operations	O
way	O
less	O
than	O
they	O
should	O
be	O
.	O
</s>
<s>
A	O
fusion	B-Data_Structure
tree	I-Data_Structure
is	O
essentially	O
a	O
B-tree	B-Architecture
with	O
branching	O
factor	O
of	O
(	O
any	O
small	O
exponent	O
is	O
also	O
possible	O
as	O
it	O
will	O
not	O
have	O
a	O
great	O
impact	O
on	O
the	O
height	O
of	O
the	O
tree	O
)	O
,	O
which	O
gives	O
it	O
a	O
height	O
of	O
.	O
</s>
<s>
To	O
achieve	O
the	O
desired	O
runtimes	O
for	O
updates	O
and	O
queries	O
,	O
the	O
fusion	B-Data_Structure
tree	I-Data_Structure
must	O
be	O
able	O
to	O
search	O
a	O
node	O
containing	O
up	O
to	O
keys	O
in	O
constant	O
time	O
.	O
</s>
<s>
With	O
only	O
standard	O
word	O
operations	O
,	O
such	O
as	O
those	O
of	O
the	O
C	B-Language
programming	I-Language
language	I-Language
,	O
it	O
is	O
difficult	O
to	O
directly	O
compute	O
the	O
perfect	O
sketch	O
of	O
a	O
key	O
in	O
constant	O
time	O
.	O
</s>
<s>
Let	O
z	O
=	O
(	O
0y	O
)	O
k	O
,	O
which	O
can	O
be	O
computed	O
in	O
constant	O
time	O
(	O
multiply	O
y	O
by	O
the	O
constant	O
(	O
0b1	O
)	O
k	O
)	O
,	O
to	O
make	O
it	O
as	O
long	O
as	O
the	O
node	O
sketch	O
such	O
that	O
each	O
word	O
in	O
the	O
node	O
sketch	O
can	O
be	O
compared	O
with	O
the	O
query	O
integer	O
y	O
in	O
one	O
operation	O
,	O
demonstrating	O
word-level	O
parallelism	B-Operating_System
.	O
</s>
<s>
If	O
the	O
next	O
bit	O
of	O
q	O
is	O
0	O
,	O
then	O
the	O
successor	O
of	O
q	O
is	O
contained	O
in	O
the	O
p1	O
subtree	B-Application
,	O
and	O
if	O
the	O
next	O
bit	O
of	O
q	O
is	O
1	O
,	O
then	O
the	O
predecessor	O
of	O
q	O
is	O
contained	O
in	O
the	O
p0	O
subtree	B-Application
.	O
</s>
<s>
An	O
application	O
of	O
fusion	B-Data_Structure
trees	I-Data_Structure
to	O
hash	B-Algorithm
tables	I-Algorithm
was	O
given	O
by	O
Willard	O
,	O
who	O
describes	O
a	O
data	O
structure	O
for	O
hashing	O
in	O
which	O
an	O
outer-level	O
hash	B-Algorithm
table	I-Algorithm
with	O
hash	O
chaining	O
is	O
combined	O
with	O
a	O
fusion	B-Data_Structure
tree	I-Data_Structure
representing	O
each	O
hash	O
chain	O
.	O
</s>
<s>
In	O
hash	O
chaining	O
,	O
in	O
a	O
hash	B-Algorithm
table	I-Algorithm
with	O
a	O
constant	O
load	O
factor	O
,	O
the	O
average	O
size	O
of	O
a	O
chain	O
is	O
constant	O
,	O
but	O
additionally	O
with	O
high	O
probability	O
all	O
chains	O
have	O
size	O
,	O
where	O
is	O
the	O
number	O
of	O
hashed	O
items	O
.	O
</s>
<s>
This	O
chain	O
size	O
is	O
small	O
enough	O
that	O
a	O
fusion	B-Data_Structure
tree	I-Data_Structure
can	O
handle	O
searches	O
and	O
updates	O
within	O
it	O
in	O
constant	O
time	O
per	O
operation	O
.	O
</s>
