7,181 research outputs found
Succinct Representations of Permutations and Functions
We investigate the problem of succinctly representing an arbitrary
permutation, \pi, on {0,...,n-1} so that \pi^k(i) can be computed quickly for
any i and any (positive or negative) integer power k. A representation taking
(1+\epsilon) n lg n + O(1) bits suffices to compute arbitrary powers in
constant time, for any positive constant \epsilon <= 1. A representation taking
the optimal \ceil{\lg n!} + o(n) bits can be used to compute arbitrary powers
in O(lg n / lg lg n) time.
We then consider the more general problem of succinctly representing an
arbitrary function, f: [n] \rightarrow [n] so that f^k(i) can be computed
quickly for any i and any integer power k. We give a representation that takes
(1+\epsilon) n lg n + O(1) bits, for any positive constant \epsilon <= 1, and
computes arbitrary positive powers in constant time. It can also be used to
compute f^k(i), for any negative integer k, in optimal O(1+|f^k(i)|) time.
We place emphasis on the redundancy, or the space beyond the
information-theoretic lower bound that the data structure uses in order to
support operations efficiently. A number of lower bounds have recently been
shown on the redundancy of data structures. These lower bounds confirm the
space-time optimality of some of our solutions. Furthermore, the redundancy of
one of our structures "surpasses" a recent lower bound by Golynski [Golynski,
SODA 2009], thus demonstrating the limitations of this lower bound.Comment: Preliminary versions of these results have appeared in the
Proceedings of ICALP 2003 and 2004. However, all results in this version are
improved over the earlier conference versio
Tight Cell Probe Bounds for Succinct Boolean Matrix-Vector Multiplication
The conjectured hardness of Boolean matrix-vector multiplication has been
used with great success to prove conditional lower bounds for numerous
important data structure problems, see Henzinger et al. [STOC'15]. In recent
work, Larsen and Williams [SODA'17] attacked the problem from the upper bound
side and gave a surprising cell probe data structure (that is, we only charge
for memory accesses, while computation is free). Their cell probe data
structure answers queries in time and is succinct in the
sense that it stores the input matrix in read-only memory, plus an additional
bits on the side. In this paper, we essentially settle the
cell probe complexity of succinct Boolean matrix-vector multiplication. We
present a new cell probe data structure with query time
storing just bits on the side. We then complement our data
structure with a lower bound showing that any data structure storing bits
on the side, with must have query time satisfying . For , any data structure must have . Since lower bounds in the cell probe model also apply to
classic word-RAM data structures, the lower bounds naturally carry over. We
also prove similar lower bounds for matrix-vector multiplication over
LRM-Trees: Compressed Indices, Adaptive Sorting, and Compressed Permutations
LRM-Trees are an elegant way to partition a sequence of values into sorted
consecutive blocks, and to express the relative position of the first element
of each block within a previous block. They were used to encode ordinal trees
and to index integer arrays in order to support range minimum queries on them.
We describe how they yield many other convenient results in a variety of areas,
from data structures to algorithms: some compressed succinct indices for range
minimum queries; a new adaptive sorting algorithm; and a compressed succinct
data structure for permutations supporting direct and indirect application in
time all the shortest as the permutation is compressible.Comment: 13 pages, 1 figur
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