23,255 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
Optimal expansions in non-integer bases
For a given positive integer , let and . A sequence consisting of elements in is called
an expansion of if . It is known that
almost every belonging to the interval has uncountably many
expansions. In this paper we study the existence of expansions of
satisfying the inequalities , for each expansion of .Comment: 11 pages, 0 figures, to appear in Proc. Amer. Math. So
Optimality of the Width- Non-adjacent Form: General Characterisation and the Case of Imaginary Quadratic Bases
Efficient scalar multiplication in Abelian groups (which is an important
operation in public key cryptography) can be performed using digital
expansions. Apart from rational integer bases (double-and-add algorithm),
imaginary quadratic integer bases are of interest for elliptic curve
cryptography, because the Frobenius endomorphism fulfils a quadratic equation.
One strategy for improving the efficiency is to increase the digit set (at the
prize of additional precomputations). A common choice is the width\nbd-
non-adjacent form (\wNAF): each block of consecutive digits contains at
most one non-zero digit. Heuristically, this ensures a low weight, i.e.\ number
of non-zero digits, which translates in few costly curve operations. This paper
investigates the following question: Is the \wNAF{}-expansion optimal, where
optimality means minimising the weight over all possible expansions with the
same digit set?
The main characterisation of optimality of \wNAF{}s can be formulated in the
following more general setting: We consider an Abelian group together with an
endomorphism (e.g., multiplication by a base element in a ring) and a finite
digit set. We show that each group element has an optimal \wNAF{}-expansion if
and only if this is the case for each sum of two expansions of weight 1. This
leads both to an algorithmic criterion and to generic answers for various
cases.
Imaginary quadratic integers of trace at least 3 (in absolute value) have
optimal \wNAF{}s for . The same holds for the special case of base
and , which corresponds to Koblitz curves in
characteristic three. In the case of , optimality depends on
the parity of . Computational results for small trace are given
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