460 research outputs found

    Dominance Product and High-Dimensional Closest Pair under L∞L_\infty

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    Given a set SS of nn points in Rd\mathbb{R}^d, the Closest Pair problem is to find a pair of distinct points in SS at minimum distance. When dd is constant, there are efficient algorithms that solve this problem, and fast approximate solutions for general dd. However, obtaining an exact solution in very high dimensions seems to be much less understood. We consider the high-dimensional L∞L_\infty Closest Pair problem, where d=nrd=n^r for some r>0r > 0, and the underlying metric is L∞L_\infty. We improve and simplify previous results for L∞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)O(DP(n,d)\log n) time, and by a randomized algorithm that runs in O(DP(n,d))O(DP(n,d)) expected time, where DP(n,d)DP(n,d) is the time bound for computing the {\em dominance product} for nn points in Rd\mathbb{R}^d. That is a matrix DD, such that D[i,j]=∣{k∣pi[k]≀pj[k]}∣D[i,j] = \bigl| \{k \mid p_i[k] \leq p_j[k]\} \bigr|; this is the number of coordinates at which pjp_j dominates pip_i. For integer coordinates from some interval [βˆ’M,M][-M, M], we obtain an algorithm that runs in O~(min⁑{MnΟ‰(1,r,1), DP(n,d)})\tilde{O}\left(\min\{Mn^{\omega(1,r,1)},\, DP(n,d)\}\right) time, where Ο‰(1,r,1)\omega(1,r,1) is the exponent of multiplying an nΓ—nrn \times n^r matrix by an nrΓ—nn^r \times n matrix. We also give slightly better bounds for DP(n,d)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, kk-SUM, and Linear Degeneracy

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    Given a set of nn 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 Θ(n2)\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 kk-SUM and kk-variate linear degeneracy testing (kk-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 44-linear decision tree complexity of 3SUM is O(n3/2)O(n^{3/2}), and that the randomized (2kβˆ’2)(2k-2)-linear decision tree complexity of kk-SUM and kk-LDT is O(nk/2)O(n^{k/2}), for any odd kβ‰₯3k\ge 3. These bounds improve (albeit randomized) the corresponding O(n3/2log⁑n)O(n^{3/2}\sqrt{\log n}) and O(nk/2log⁑n)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(n2log⁑log⁑n/log⁑n)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

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    We give a fairly elementary and simple proof that shows that the number of incidences between mm points and nn lines in R3{\mathbb R}^3, so that no plane contains more than ss lines, is O(m1/2n3/4+m2/3n1/3s1/3+m+n) 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 mm and nn). 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
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