Affiliation:
1. University of Leicester, Leicester, UK
2. Institute of Mathematical Sciences, Chennai, India
3. University of Waterloo, Waterloo, Canada
Abstract
We consider the
indexable dictionary
problem, which consists of storing a set
S
⊆ {0,…,
m
− 1} for some integer
m
while supporting the operations of rank(
x
), which returns the number of elements in
S
that are less than
x
if
x
∈
S
, and −1 otherwise; and select(
i
), which returns the
i
th smallest element in
S
. We give a data structure that supports both operations in
O
(1) time on the RAM model and requires B(
n, m
) +
o
(
n
) +
O
(lg lg
m
) bits to store a set of size
n
, where B(
n, m
) = ⌊lg (
m
/
n
)⌋ is the minimum number of bits required to store any
n
-element subset from a universe of size
m
. Previous dictionaries taking this space only supported (yes/no) membership queries in
O
(1) time. In the cell probe model we can remove the
O
(lg lg
m
) additive term in the space bound, answering a question raised by Fich and Miltersen [1995] and Pagh [2001].
We present extensions and applications of our indexable dictionary data structure, including:
—an information-theoretically optimal representation of a
k
-ary cardinal tree that supports standard operations in constant time;
—a representation of a multiset of size
n
from {0,…,
m
− 1} in B(
n, m
+
n
) +
o
(
n
) bits that supports (appropriate generalizations of) rank and select operations in constant time; and +
O
(lg lg
m
)
—a representation of a sequence of
n
nonnegative integers summing up to
m
in B(
n, m
+
n
) +
o
(
n
) bits that supports prefix sum queries in constant time.
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
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