We show an upper bound of sin(3 pi ( 3 pi/10 )/sin(2 pi/5)-sin (3 pi/10) < 5.70 . 70 on the spanning ratio of e 5-graphs, improving on the previous best known upper bound of 9.96 . 96 [Bose, Morin, van Renssen, and Verdonschot. The Theta-5-graph is a spanner. Computational Geometry, , 2015.]

The sparse affine regression problem can be cast as follows: Given a set S of n k points of S that is nearest to y. We describe an Od,epsilon(nk-1 logd-k +1 n)-time randomized (1 + epsilon)-approximation algorithm for this problem with d and epsilon constant. This is the first algorithm for this problem running in time o(nk). Its running time is similar to the query time of a data structure recently proposed by Har-Peled, Indyk, and Mahabadi (ICALP'18), while not requiring any preprocessing. Up to polylogarithmic factors, it matches a conditional lower bound relying on a conjecture about affine degeneracy testing. In the special case where k = d = O(1), we also provide a simple O(nd-1+delta)-time deterministic exact algorithm, for any delta > 0. Finally, we also show how to adapt the approximation algorithm for the sparse linear or convex regression problems with the same running time, up to polylogarithmic factors.

We show an upper bound of sin(3π/10) /sin(2π/5)-sin(3π/10) <5.70 on the spanning ratio of Θ_5-graphs, improving on the previous best known upper bound of 9.96 [Bose, Morin, van Renssen, and Verdonschot. The Theta-5-graph is a spanner. Computational Geometry, 2015.]

We show an upper bound of sin( 3π/10) /sin( 2π/5) -sin( 3π/10) <5.70 on the spanning ratio of _5 -graphs, improving on the previous best known upper bound of 9.96 [Bose, Morin, van Renssen, and Verdonschot. The Theta-5-graph is a spanner. Computational Geometry, 2015.]

We study the computation of the diameter and radius under the rectilinear link distance within a rectilinear polygonal domain of n vertices and h holes. We introduce a graph of oriented distances to encode the distance between pairs of points of the domain. This helps us transform the problem so that we can search through the candidates more efficiently. Our algorithm computes both the diameter and the radius in O(min⁡(nω,n2+nhlog⁡h+χ2)) time, where ω<2.373 denotes the matrix multiplication exponent and χ∈Ω(n)∩O(n2) is the number of edges of the graph of oriented distances. We also provide an alternative algorithm for computing the diameter that runs in O(n2log⁡n) time.

We show an upper bound of $\frac{ \sin\left(\frac{3\pi}{10}\right) }{ \sin\left(\frac{2\pi}{5}\right)-\sin\left(\frac{3\pi}{10}\right) }<5.70$ on the spanning ratio of $\Theta_5$-graphs, improving on the previous best known upper bound of $9.96$ [Bose, Morin, van Renssen, and Verdonschot. The Theta-5-graph is a spanner. Computational Geometry, 2015.]

The sparse regression problem, also known as best subset selection problem, can be cast as follows: Given a set $S$ of $n$ points in $\mathbb{R}^d$, a point $y\in \mathbb{R}^d$, and an integer $2 \leq k \leq d$, find an affine combination of at most $k$ points of $S$ that is nearest to $y$. We describe a $O(n^{k-1} \log^{d-k+2} n)$-time randomized $(1+\varepsilon)$-approximation algorithm for this problem with \(d\) and \(\varepsilon\) constant. This is the first algorithm for this problem running in time $o(n^k)$. Its running time is similar to the query time of a data structure recently proposed by Har-Peled, Indyk, and Mahabadi (ICALP'18), while not requiring any preprocessing. Up to polylogarithmic factors, it matches a conditional lower bound relying on a conjecture about affine degeneracy testing. In the special case where $k = d = O(1)$, we also provide a simple $O_\delta(n^{d-1+\delta})$-time deterministic exact algorithm, for any \(\delta > 0\). Finally, we show how to adapt the approximation algorithm for the sparse linear regression and sparse convex regression problems with the same running time, up to polylogarithmic factors.

We consider the following problem: given three sets of real numbers, output a word-RAM data structure from which we can efficiently recover the sign of the sum of any triple of numbers, one in each set. This is similar to a previous work by some of the authors to encode the order type of a finite set of points. While this previous work showed that it was possible to achieve slightly subquadratic space and logarithmic query time, we show here that for the simpler 3SUM problem, one can achieve an encoding that takes $\tilde{O}(N^{\frac 32})$ space for inputs sets of size $N$ and allows constant time queries in the word-RAM.

An edge guard set of a plane graph G is a subset $$\varGamma $$Γ of edges of G such that each face of G is incident to an endpoint of an edge in $$\varGamma $$Γ. Such a set is said to guardG. We improve the known upper bounds on the number of edges required to guard any n-vertex embedded planar graph G: (1) We present a simple inductive proof for a theorem of Everett and Rivera-Campo (Comput Geom Theory Appl 7:201–203, 1997) that G can be guarded with at most $$\frac{2n}{5}$$2n5 edges, then extend this approach with a deeper analysis to yield an improved bound of $$\frac{3n}{8}$$3n8 edges for any plane graph. (2) We prove that there exists an edge guard set of G with at most $$\frac{n}{3} + \frac{\alpha }{9}$$n3+α9 edges, where $$\alpha $$α is the number of quadrilateral faces in G. This improves the previous bound of $$\frac{n}{3} + \alpha $$n3+α by Bose et al. (Comput Geom Theory Appl 26(3):209–219, 2003). Moreover, if there is no short path between any two quadrilateral faces in G, we show that $$\frac{n}{3}$$n3 edges suffice, removing the dependence on $$\alpha $$α.

The 3SUM problem asks if an input n-set of real numbers contains a triple whose sum is zero. We qualify such a triple of degenerate because the probability of finding one in a random input is zero. We consider the 3POL problem, an algebraic generalization of 3SUM where we replace the sum function by a constant-degree polynomial in three variables. The motivations are threefold. Raz et al. gave an \(O(n^{11/6})\) upper bound on the number of degenerate triples for the 3POL problem. We give algorithms for the corresponding problem of counting them. Grønlund and Pettie designed subquadratic algorithms for 3SUM. We prove that 3POL admits bounded-degree algebraic decision trees of depth \(O(n^{12/7+\varepsilon })\), and we prove that 3POL can be solved in \(O(n^2 {(\log \log n)}^{3/2} / {(\log n)}^{1/2})\) time in the real-RAM model, generalizing their results. 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 show how to solve GPT in subquadratic time when the input points lie on a small number of constant-degree polynomial curves. Many other geometric degeneracy testing problems reduce to 3POL.

For most algorithms dealing with sets of points in the plane, the only relevant information carried by the input is the combinatorial configuration of the points: the orientation of each triple of points in the set (clockwise, counterclockwise, or collinear). This information is called the order type of the point set. In the dual, realizable order types and abstract order types are combinatorial analogues of line arrangements and pseudoline arrangements. Too often in the literature we analyze algorithms in the real-RAM model for simplicity, putting aside the fact that computers as we know them cannot handle arbitrary real numbers without some sort of encoding. Encoding an order type by the integer coordinates of some realizing point set is known to yield doubly exponential coordinates in some cases. Other known encodings can achieve quadratic space or fast orientation queries, but not both. In this contribution, we give a compact encoding for abstract order types that allows efficient query of the orientation of any triple: the encoding uses O(n^2) bits and an orientation query takes O(log n) time in the word-RAM model. This encoding is space-optimal for abstract order types. We show how to shorten the encoding to O(n^2 (loglog n)^2 / log n) bits for realizable order types, giving the first subquadratic encoding for those order types with fast orientation queries. We further refine our encoding to attain O(log n/loglog n) query time without blowing up the space requirement. In the realizable case, we show that all those encodings can be computed efficiently. Finally, we generalize our results to the encoding of point configurations in higher dimension.

We study the following family of problems: Given a set of $n$ points in convex position, what is the maximum number triangles one can create having these points as vertices while avoiding certain sets of forbidden configurations. As forbidden configurations we consider all 8 ways in which a pair of triangles in such a point set can interact. This leads to 256 extremal Turán-type questions. We give nearly tight (within a $\log n$ factor) bounds for 248 of these questions and show that the remaining 8 questions are all asymptotically equivalent to Stein's longstanding tripod packing problem.

We study the computation of the diameter and radius under the rectilinear link distance within a rectilinear polygonal domain of n vertices and h holes. We introduce a graph of oriented distances to encode the distance between pairs of points of the domain. This helps us transform the problem so that we can search through the candidates more efficiently. Our algorithm computes both the 1 Partially supported by the SNF Early Postdoc Mobility grant P2TIP2-168563, Switzerland, and F.R.S.-FNRS, Belgium. 2 Supported in part by ERC StG 757609. 3 Supported in part by KAKENHI No. 17K12635, Japan and NSF award CCF-1422311. 4 Supported by the Netherlands’ Organisation for Scientific Research (NWO) under project no. 024.002.003. 5 Partially supported by JSPS KAKENHI Grant Number 15K00009 and JST CREST Grant Number JPMJCR1402, and Kayamori Foundation of Informational Science Advancement. 6 Supported by the Fund for Research Training in Industry and Agriculture (FRIA). 7 Supported by JST ERATO Grant Number JPMJER1201, Japan. © Elena Arseneva, Man-Kwun Chiu, Matias Korman, Aleksandar Markovic, Yoshio Okamoto, Aurélien Ooms, André van Renssen, and Marcel Roeloffzen; licensed under Creative Commons License CC-BY 29th International Symposium on Algorithms and Computation (ISAAC 2018). Editors: Wen-Lian Hsu, Der-Tsai Lee, and Chung-Shou Liao; Article No. 58; pp. 58:1–58:13 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 58:2 Rectilinear Link Diameter and Radius in a Rectilinear Polygonal Domain diameter and the radius in O(min(n, n2 + nh log h + χ2)) time, where ω < 2.373 denotes the matrix multiplication exponent and χ ∈ Ω(n) ∩ O(n2) is the number of edges of the graph of oriented distances. We also provide an alternative algorithm for computing the diameter that runs in O(n2 logn) time. 2012 ACM Subject Classification Theory of computation → Computational geometry

The k-SUM problem is given n input real numbers to determine whether any k of them sum to zero. The problem is of tremendous importance in the emerging field of complexity theory within P, and it is in particular open whether it admits an algorithm of complexity O(n^c) with c

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. Gronlund 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.

In the $k$-SUM problem, we are given $n$ real numbers as input, and we are asked whether there exists a zero-sum $k$-subset. The problem is of tremendous importance in complexity theory, and it is in particular open whether it admits an algorithm of complexity $O(n^c)$ with $c<\lceil k / 2\rceil$. Revisiting a known algorithm due to Meiser (1993), we show that there exist linear decision trees of depth $O(n^3\log^3 n)$ solving this problem. Furthermore, we show that there exists a Las Vegas algorithm that runs in $O(n^{k+1})$ time performing exactly this number of linear queries on the input. We also consider a range of tradeoffs between the number of terms involved in the queries and the depth of the decision tree. In particular, we prove that there exists $o(n)$-linear decision trees of depth $O(n^c)$ for some constant $c$. The query complexities also hold for nonuniform real-RAM algorithms.