14,399 research outputs found
Improved Bounds for 3SUM, -SUM, and Linear Degeneracy
Given a set of real numbers, the 3SUM problem is to decide whether there
are three of them that sum to zero. Until a recent breakthrough by Gr{\o}nlund
and Pettie [FOCS'14], a simple -time deterministic algorithm for
this problem was conjectured to be optimal. Over the years many algorithmic
problems have been shown to be reducible from the 3SUM problem or its variants,
including the more generalized forms of the problem, such as -SUM and
-variate linear degeneracy testing (-LDT). The conjectured hardness of
these problems have become extremely popular for basing conditional lower
bounds for numerous algorithmic problems in P.
In this paper, we show that the randomized -linear decision tree
complexity of 3SUM is , and that the randomized -linear
decision tree complexity of -SUM and -LDT is , for any odd
. These bounds improve (albeit randomized) the corresponding
and decision tree bounds
obtained by Gr{\o}nlund and Pettie. Our technique includes a specialized
randomized variant of fractional cascading data structure. Additionally, we
give another deterministic algorithm for 3SUM that runs in time. The latter bound matches a recent independent bound by Freund
[Algorithmica 2017], but our algorithm is somewhat simpler, due to a better use
of word-RAM model
Linear relations of zeroes of the zeta-function
This article considers linear relations between the non-trivial zeroes of the
Riemann zeta-function. The main application is an alternative disproof to
Mertens' conjecture. We show that and that
.Comment: 12 pages, 2 figures, 2 tables. Version 2: some typos corrected. To
appear in Math. Com
Parallel Algorithms for Summing Floating-Point Numbers
The problem of exactly summing n floating-point numbers is a fundamental
problem that has many applications in large-scale simulations and computational
geometry. Unfortunately, due to the round-off error in standard floating-point
operations, this problem becomes very challenging. Moreover, all existing
solutions rely on sequential algorithms which cannot scale to the huge datasets
that need to be processed.
In this paper, we provide several efficient parallel algorithms for summing n
floating point numbers, so as to produce a faithfully rounded floating-point
representation of the sum. We present algorithms in PRAM, external-memory, and
MapReduce models, and we also provide an experimental analysis of our MapReduce
algorithms, due to their simplicity and practical efficiency.Comment: Conference version appears in SPAA 201
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