230,609 research outputs found
Succinct Indexable Dictionaries with Applications to Encoding -ary Trees, Prefix Sums and Multisets
We consider the {\it indexable dictionary} problem, which consists of storing
a set for some integer , while supporting the
operations of \Rank(x), which returns the number of elements in that are
less than if , and -1 otherwise; and \Select(i) which returns
the -th smallest element in . We give a data structure that supports both
operations in O(1) time on the RAM model and requires bits to store a set of size , where {\cal B}(n,m) = \ceil{\lg
{m \choose n}} is the minimum number of bits required to store any -element
subset from a universe of size . Previous dictionaries taking this space
only supported (yes/no) membership queries in O(1) time. In the cell probe
model we can remove the additive term in the space bound,
answering a question raised by Fich and Miltersen, and Pagh.
We present extensions and applications of our indexable dictionary data
structure, including:
An information-theoretically optimal representation of a -ary cardinal
tree that supports standard operations in constant time,
A representation of a multiset of size from in bits that supports (appropriate generalizations of) \Rank
and \Select operations in constant time, and
A representation of a sequence of non-negative integers summing up to
in bits that supports prefix sum queries in constant
time.Comment: Final version of SODA 2002 paper; supersedes Leicester Tech report
2002/1
Nearly Optimal Static Las Vegas Succinct Dictionary
Given a set of (distinct) keys from key space , each associated
with a value from , the \emph{static dictionary} problem asks to
preprocess these (key, value) pairs into a data structure, supporting
value-retrieval queries: for any given , must
return the value associated with if , or return if . The special case where is called the \emph{membership}
problem. The "textbook" solution is to use a hash table, which occupies linear
space and answers each query in constant time. On the other hand, the minimum
possible space to encode all (key, value) pairs is only bits, which could be much less.
In this paper, we design a randomized dictionary data structure using
bits of space, and it
has \emph{expected constant} query time, assuming the query algorithm can
access an external lookup table of size . The lookup table depends
only on , and , and not the input. Previously, even for
membership queries and , the best known data structure with
constant query time requires bits of space
(Pagh [Pag01] and P\v{a}tra\c{s}cu [Pat08]); the best-known using
space has query time ; the only known
non-trivial data structure with space has
query time and requires a lookup table of size (!). Our new
data structure answers open questions by P\v{a}tra\c{s}cu and Thorup
[Pat08,Tho13].
We also present a scheme that compresses a sequence to its
zeroth order (empirical) entropy up to extra
bits, supporting decoding each in expected time.Comment: preliminary version appeared in STOC'2
A probabilistic interpretation of set-membership filtering: application to polynomial systems through polytopic bounding
Set-membership estimation is usually formulated in the context of set-valued
calculus and no probabilistic calculations are necessary. In this paper, we
show that set-membership estimation can be equivalently formulated in the
probabilistic setting by employing sets of probability measures. Inference in
set-membership estimation is thus carried out by computing expectations with
respect to the updated set of probability measures P as in the probabilistic
case. In particular, it is shown that inference can be performed by solving a
particular semi-infinite linear programming problem, which is a special case of
the truncated moment problem in which only the zero-th order moment is known
(i.e., the support). By writing the dual of the above semi-infinite linear
programming problem, it is shown that, if the nonlinearities in the measurement
and process equations are polynomial and if the bounding sets for initial
state, process and measurement noises are described by polynomial inequalities,
then an approximation of this semi-infinite linear programming problem can
efficiently be obtained by using the theory of sum-of-squares polynomial
optimization. We then derive a smart greedy procedure to compute a polytopic
outer-approximation of the true membership-set, by computing the minimum-volume
polytope that outer-bounds the set that includes all the means computed with
respect to P
A modular CMOS analog fuzzy controller
The low/medium precision required for many fuzzy applications makes analog circuits natural candidates to design fuzzy chips with optimum speed/power figures. This paper presents a sixteen rules-two inputs analog fuzzy controller in a CMOS 1 /spl mu/m single-poly technology based on building blocks implementations previously proposed by the authors (1995). However, such building blocks are rearranged here to get a highly modular architecture organized from two high level blocks: the label block and the rule block. In addition, sharing of membership function circuits allows a compact design with low area and power consumption and its highly modular architecture will permit to increase the number of inputs and rules in future chips with hardly design effort. The paper includes measurements from a silicon prototype of the controller
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