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Register a new userSubquadratic Algorithms for Some textsc{3Sum}-Hard Geometric Problems in the Algebraic Decision Tree Model
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Added by
Micha Sharir
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We present subquadratic algorithms in the algebraic decision-tree model for several textsc{3Sum}-hard geometric problems, all of which can be reduced to the following question: Given two sets $A$, $B$, each consisting of $n$ pairwise disjoint segments in the plane, and a set $C$ of $n$ triangles in the plane, we want to count, for each triangle $Deltain C$, the number of intersection points between the segments of $A$ and those of $B$ that lie in $Delta$. The problems considered in this paper have been studied by Chan~(2020), who gave algorithms that solve them, in the standard real-RAM model, in $O((n^2/log^2n)log^{O(1)}log n)$ time. We present solutions in the algebraic decision-tree model whose cost is $O(n^{60/31+varepsilon})$, for any $varepsilon>0$. Our approach is based on a primal-dual range searching mechanism, which exploits the multi-level polynomial partitioning machinery recently developed by Agarwal, Aronov, Ezra, and Zahl~(2020). A key step in the procedure is a variant of point location in arrangements, say of lines in the plane, which is based solely on the emph{order type} of the lines, a handicap that turns out to be beneficial for speeding up our algorithm.
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The 3SUM problem asks if an input $n$-set of real numbers contains a triple whose sum is zero. We consider the 3POL problem, a natural generalization of 3SUM where we replace the sum function by a constant-degree polynomial in three variables. The motivations are threefold. Raz, Sharir, and de Zeeuw gave a $O(n^{11/6})$ upper bound on the number of solutions of trivariate polynomial equations when the solutions are taken from the cartesian product of three $n$-sets of real numbers. We give algorithms for the corresponding problem of counting such solutions. Gro nlund and Pettie recently designed subquadratic algorithms for 3SUM. We generalize their results to 3POL. Finally, we shed light on the General Position Testing (GPT) problem: Given $n$ points in the plane, do three of them lie on a line?, a key problem in computational geometry. We prove that there exist bounded-degree algebraic decision trees of depth $O(n^{frac{12}{7}+varepsilon})$ that solve 3POL, and that 3POL can be solved in $O(n^2 {(log log n)}^frac{3}{2} / {(log n)}^frac{1}{2})$ time in the real-RAM model. Among the possible applications of those results, we show how to solve GPT in subquadratic time when the input points lie on $o({(log n)}^frac{1}{6}/{(log log n)}^frac{1}{2})$ constant-degree polynomial curves. This constitutes a first step towards closing the major open question of whether GPT can be solved in subquadratic time. To obtain these results, we generalize important tools --- such as batch range searching and dominance reporting --- to a polynomial setting. We expect these new tools to be useful in other applications.
Quantum computers can execute algorithms that dramatically outperform classical computation. As the best-known example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article reviews the current state of quantum algorithms, focusing on algorithms with superpolynomial speedup over classical computation, and in particular, on problems with an algebraic flavor.
In this article, we consider a collection of geometric problems involving points colored by two colors (red and blue), referred to as bichromatic problems. The motivation behind studying these problems is two fold; (i) these problems appear naturally and frequently in the fields like Machine learning, Data mining, and so on, and (ii) we are interested in extending the algorithms and techniques for single point set (monochromatic) problems to bichromatic case. For all the problems considered in this paper, we design low polynomial time exact algorithms. These algorithms are based on novel techniques which might be of independent interest.
We improve the running times of $O(1)$-approximation algorithms for the set cover problem in geometric settings, specifically, covering points by disks in the plane, or covering points by halfspaces in three dimensions. In the unweighted case, Agarwal and Pan [SoCG 2014] gave a randomized $O(nlog^4 n)$-time, $O(1)$-approximation algorithm, by using variants of the multiplicative weight update (MWU) method combined with geometric data structures. We simplify the data structure requirement in one of their methods and obtain a deterministic $O(nlog^3 nloglog n)$-time algorithm. With further new ideas, we obtain a still faster randomized $O(nlog n(loglog n)^{O(1)})$-time algorithm. For the weighted problem, we also give a randomized $O(nlog^4nloglog n)$-time, $O(1)$-approximation algorithm, by simple modifications to the MWU method and the quasi-uniform sampling technique.
We present algorithms for length-constrained maximum sum segment and maximum density segment problems, in particular, and the problem of finding length-constrained heaviest segments, in general, for a sequence of real numbers. Given a sequence of n real numbers and two real parameters L and U (L <= U), the maximum sum segment problem is to find a consecutive subsequence, called a segment, of length at least L and at most U such that the sum of the numbers in the subsequence is maximum. The maximum density segment problem is to find a segment of length at least L and at most U such that the density of the numbers in the subsequence is the maximum. For the first problem with non-uniform width there is an algorithm with time and space complexities in O(n). We present an algorithm with time complexity in O(n) and space complexity in O(U). For the second problem with non-uniform width there is a combinatorial solution with time complexity in O(n) and space complexity in O(U). We present a simple geometric algorithm with the same time and space complexities. We extend our algorithms to respectively solve the length-constrained k maximum sum segments problem in O(n+k) time and O(max{U, k}) space, and the length-constrained $k$ maximum density segments problem in O(n min{k, U-L}) time and O(U+k) space. We present extensions of our algorithms to find all the length-constrained segments having user specified sum and density in O(n+m) and O(nlog (U-L)+m) times respectively, where m is the number of output. Previously, there was no known algorithm with non-trivial result for these problems. We indicate the extensions of our algorithms to higher dimensions. All the algorithms can be extended in a straight forward way to solve the problems with non-uniform width and non-uniform weight.
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