460 research outputs found

### Dominance Product and High-Dimensional Closest Pair under $L_\infty$

Given a set $S$ of $n$ points in $\mathbb{R}^d$, the Closest Pair problem is
to find a pair of distinct points in $S$ at minimum distance. When $d$ is
constant, there are efficient algorithms that solve this problem, and fast
approximate solutions for general $d$. However, obtaining an exact solution in
very high dimensions seems to be much less understood. We consider the
high-dimensional $L_\infty$ Closest Pair problem, where $d=n^r$ for some $r >
0$, and the underlying metric is $L_\infty$.
We improve and simplify previous results for $L_\infty$ Closest Pair, showing
that it can be solved by a deterministic strongly-polynomial algorithm that
runs in $O(DP(n,d)\log n)$ time, and by a randomized algorithm that runs in
$O(DP(n,d))$ expected time, where $DP(n,d)$ is the time bound for computing the
{\em dominance product} for $n$ points in $\mathbb{R}^d$. That is a matrix $D$,
such that $D[i,j] = \bigl| \{k \mid p_i[k] \leq p_j[k]\} \bigr|$; this is the
number of coordinates at which $p_j$ dominates $p_i$. For integer coordinates
from some interval $[-M, M]$, we obtain an algorithm that runs in
$\tilde{O}\left(\min\{Mn^{\omega(1,r,1)},\, DP(n,d)\}\right)$ time, where
$\omega(1,r,1)$ is the exponent of multiplying an $n \times n^r$ matrix by an
$n^r \times n$ matrix.
We also give slightly better bounds for $DP(n,d)$, by using more recent
rectangular matrix multiplication bounds. Computing the dominance product
itself is an important task, since it is applied in many algorithms as a major
black-box ingredient, such as algorithms for APBP (all pairs bottleneck paths),
and variants of APSP (all pairs shortest paths)

### Improved Bounds for 3SUM, $k$-SUM, and Linear Degeneracy

Given a set of $n$ 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 $\Theta(n^2)$-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 $k$-SUM and
$k$-variate linear degeneracy testing ($k$-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 $4$-linear decision tree
complexity of 3SUM is $O(n^{3/2})$, and that the randomized $(2k-2)$-linear
decision tree complexity of $k$-SUM and $k$-LDT is $O(n^{k/2})$, for any odd
$k\ge 3$. These bounds improve (albeit randomized) the corresponding
$O(n^{3/2}\sqrt{\log n})$ and $O(n^{k/2}\sqrt{\log n})$ 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 $O(n^2 \log\log n /
\log n )$ 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

### Incidences between points and lines in three dimensions

We give a fairly elementary and simple proof that shows that the number of
incidences between $m$ points and $n$ lines in ${\mathbb R}^3$, so that no
plane contains more than $s$ lines, is $O\left(m^{1/2}n^{3/4}+
m^{2/3}n^{1/3}s^{1/3} + m + n\right)$ (in the precise statement, the constant
of proportionality of the first and third terms depends, in a rather weak
manner, on the relation between $m$ and $n$).
This bound, originally obtained by Guth and Katz~\cite{GK2} as a major step
in their solution of Erd{\H o}s's distinct distances problem, is also a major
new result in incidence geometry, an area that has picked up considerable
momentum in the past six years. Its original proof uses fairly involved
machinery from algebraic and differential geometry, so it is highly desirable
to simplify the proof, in the interest of better understanding the geometric
structure of the problem, and providing new tools for tackling similar
problems. This has recently been undertaken by Guth~\cite{Gu14}. The present
paper presents a different and simpler derivation, with better bounds than
those in \cite{Gu14}, and without the restrictive assumptions made there. Our
result has a potential for applications to other incidence problems in higher
dimensions

- β¦