On a two-player transversal game on a square grid
We give a short analysis of the \emph{transversal achievement game} on a square grid due to M. Erickson (2010).
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arXiv subjects
Adrian Dumitrescu
Publications and source records attributed to Adrian Dumitrescu.
Finding a Mediocre Player
Consider a totally ordered set $S$ of $n$ elements; as an example, a set of tennis players and their rankings. Further assume that their ranking is a total order and thus satisfies transitivity and anti-symmetry. Following Frances Yao (1974), an element (player) is said to be $(i,j)$-\emph{mediocre} if it is neither among the top $i$ nor among the bottom $j$ elements of $S$. Finding a mediocre element is closely related to finding the median element. More than $40$ years ago, Yao suggested a very simple and elegant algorithm for finding an $(i,j)$-mediocre element: Pick $i+j+1$ elements arbitrarily and select the $(i+1)$-th largest among them. She also asked: "Is this the best algorithm?" No one seems to have found a better algorithm ever since. We first provide a deterministic algorithm that beats the worst-case comparison bound in Yao's algorithm for a large range of values of $i$ (and corresponding suitable $j=j(i)$) even if the current best selection algorithm is used. We then repeat the exercise for randomized algorithms; the average number of comparisons of our algorithm beats the average comparison bound in Yao's algorithm for another large range of values of $i$ (and corresponding suitable $j=j(i)$) even if the best selection algorithm is used; the improvement is most notable in the symmetric case $i=j$. Moreover, the tight bound obtained in the analysis of Yao's algorithm allows us to give a definite answer for this class of algorithms. In summary, we answer Yao's question as follows: (i)~"Presently not" for deterministic algorithms and (ii)~"Definitely not" for randomized algorithms. (In fairness, it should be said however that Yao posed the question in the context of deterministic algorithms.)
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On the Longest Spanning Tree with Neighborhoods
We study a maximization problem for geometric network design. Given a set of $n$ compact neighborhoods in $\mathbb{R}^d$, select a point in each neighborhood, so that the longest spanning tree on these points (as vertices) has maximum length. Here we give an approximation algorithm with ratio $0.511$, which represents the first, albeit small, improvement beyond $1/2$. While we suspect that the problem is NP-hard already in the plane, this issue remains open.
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Multiparty Selection
Given a sequence $A$ of $n$ numbers and an integer (target) parameter $1\leq i\leq n$, the (exact) selection problem asks to find the $i$-th smallest element in $A$. An element is said to be $(i,j)$-mediocre if it is neither among the top $i$ nor among the bottom $j$ elements of $S$. The approximate selection problem asks to find a $(i,j)$-mediocre element for some given $i,j$; as such, this variant allows the algorithm to return any element in a prescribed range. In the first part, we revisit the selection problem in the two-party model introduced by Andrew Yao (1979) and then extend our study of exact selection to the multiparty model. In the second part, we deduce some communication complexity benefits that arise in approximate selection. In particular, we present a deterministic protocol for finding an approximate median among $k$ players.
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On the Shortest Separating Cycle
According to a result of Arkin~\etal~(2016), given $n$ point pairs in the plane, there exists a simple polygonal cycle that separates the two points in each pair to different sides; moreover, a $O(\sqrt{n})$-factor approximation with respect to the minimum length can be computed in polynomial time. Here the following results are obtained: (I)~We extend the problem to geometric hypergraphs and obtain the following characterization of feasibility. Given a geometric hypergraph on points in the plane with hyperedges of size at least $2$, there exists a simple polygonal cycle that separates each hyperedge if and only if the hypergraph is $2$-colorable. (II)~We extend the $O(\sqrt{n})$-factor approximation in the length measure as follows: Given a geometric graph $G=(V,E)$, a separating cycle (if it exists) can be computed in $O(m+ n\log{n})$ time, where $|V|=n$, $|E|=m$. Moreover, a $O(\sqrt{n})$-approximation of the shortest separating cycle can be found in polynomial time. Given a geometric graph $G=(V,E)$ in $\mathbb{R}^3$, a separating polyhedron (if it exists) can be found in $O(m+ n\log{n})$ time, where $|V|=n$, $|E|=m$. Moreover, a $O(n^{2/3})$-approximation of a separating polyhedron of minimum perimeter can be found in polynomial time. (III)~Given a set of $n$ point pairs in convex position in the plane, we show that a $(1+\varepsilon)$-approximation of a shortest separating cycle can be computed in time $n^{O(\varepsilon^{-1/2})}$. In this regard, we prove a lemma on convex polygon approximation that is of independent interest.
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Selection Algorithms with Small Groups
We revisit the selection problem, namely that of computing the $i$th order statistic of $n$ given elements, in particular the classic deterministic algorithm by grouping and partition due to Blum, Floyd, Pratt, Rivest, and Tarjan (1973). Whereas the original algorithm uses groups of odd size at least $5$ and runs in linear time, it has been perpetuated in the literature that using smaller group sizes will force the worst-case running time to become superlinear, namely $Ω(n \log{n})$. We first point out that the usual arguments found in the literature justifying the superlinear worst-case running time fall short of proving this claim. We further prove that it is possible to use group size smaller than $5$ while maintaining the worst case linear running time. To this end we introduce three simple variants of the classic algorithm, the repeated step algorithm, the shifting target algorithm, and the hyperpair algorithm, all running in linear time.
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New Lower Bounds for the Number of Pseudoline Arrangements
Arrangements of lines and pseudolines are fundamental objects in discrete and computational geometry. They also appear in other areas of computer science, such as the study of sorting networks. Let $B_n$ be the number of nonisomorphic arrangements of $n$ pseudolines and let $b_n=\log_2{B_n}$. The problem of estimating $B_n$ was posed by Knuth in 1992. Knuth conjectured that $b_n \leq {n \choose 2} + o(n^2)$ and also derived the first upper and lower bounds: $b_n \leq 0.7924 (n^2 +n)$ and $b_n \geq n^2/6 -O(n)$. The upper bound underwent several improvements, $b_n \leq 0.6988\, n^2$ (Felsner, 1997), and $b_n \leq 0.6571\, n^2$ (Felsner and Valtr, 2011), for large $n$. Here we show that $b_n \geq cn^2 -O(n \log{n})$ for some constant $c>0.2083$. In particular, $b_n \geq 0.2083\, n^2$ for large $n$. This improves the previous best lower bound, $b_n \geq 0.1887\, n^2$, due to Felsner and Valtr (2011). Our arguments are elementary and geometric in nature. Further, our constructions are likely to spur new developments and improved lower bounds for related problems, such as in topological graph drawings.
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Online Unit Covering in Euclidean Space
We revisit the online Unit Covering problem in higher dimensions: Given a set of $n$ points in $\mathbb{R}^d$, that arrive one by one, cover the points by balls of unit radius, so as to minimize the number of balls used. In this paper, we work in $\mathbb{R}^d$ using Euclidean distance. The current best competitive ratio of an online algorithm, $O(2^d d \log{d})$, is due to Charikar et al. (2004); their algorithm is deterministic. (I) We give an online deterministic algorithm with competitive ratio $O(1.321^d)$, thereby sharply improving on the earlier record by a large exponential factor. In particular, the competitive ratios are $5$ for the plane and $12$ for $3$-space (the previous ratios were $7$ and $21$, respectively). For $d=3$, the ratio of our online algorithm matches the ratio of the current best offline algorithm for the same problem due to Biniaz et al. (2017), which is remarkable (and rather unusual). (II) We show that the competitive ratio of every deterministic online algorithm (with an adaptive deterministic adversary) for Unit Covering in $\mathbb{R}^d$ under the $L_{2}$ norm is at least $d+1$ for every $d \geq 1$. This greatly improves upon the previous best lower bound, $Ω(\log{d} / \log{\log{\log{d}}})$, due to Charikar et al. (2004). (III) We obtain lower bounds of $4$ and $5$ for the competitive ratio of any deterministic algorithm for online Unit Covering in $\mathbb{R}^2$ and respectively $\mathbb{R}^3$; the previous best lower bounds were both $3$. (IV) When the input points are taken from the square or hexagonal lattices in $\mathbb{R}^2$, we give deterministic online algorithms for Unit Covering with an optimal competitive ratio of $3$.
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Problems on Track Runners
Consider the circle $C$ of length 1 and a circular arc $A$ of length $\ell\in (0,1)$. It is shown that there exists $k=k(\ell) \in \mathbb{N}$, and a schedule for $k$ runners along the circle with $k$ constant but distinct positive speeds so that at any time $t \geq 0$, at least one of the $k$ runners is not in $A$. On the other hand, we show the following: Assume that $k$ runners $1,2,\ldots,k$, with constant rationally independent (thus distinct) speeds $ξ_1,ξ_2,\ldots,ξ_k$, run clockwise along a circle of length $1$, starting from arbitrary points. For every circular arc $A\subset C$ and for every $T>0$, there exists $t>T$ such that all runners are in $A$ at time $t$. Several other problems of a similar nature are investigated.
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A Selectable Sloppy Heap
We study the selection problem, namely that of computing the $i$th order statistic of $n$ given elements. Here we offer a data structure called \emph{selectable sloppy heap} handling a dynamic version in which upon request: (i)~a new element is inserted or (ii)~an element of a prescribed quantile group is deleted from the data structure. Each operation is executed in (ideal!) constant time---and is thus independent of $n$ (the number of elements stored in the data structure)---provided that the number of quantile groups is fixed. This is the first result of this kind accommodating both insertion and deletion in constant time. As such, our data structure outperforms the soft heap data structure of Chazelle (which only offers constant amortized complexity for a fixed error rate $0<\varepsilon \leq 1/2$) in applications such as dynamic percentile maintenance. The design demonstrates how slowing down a certain computation can speed up the data structure.
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Approximation algorithms for TSP with neighborhoods in the plane
In the Euclidean TSP with neighborhoods (TSPN), we are given a collection of n regions (neighborhoods) and we seek a shortest tour that visits each region. As a generalization of the classical Euclidean TSP, TSPN is also NP-hard. In this paper, we present new approximation results for the TSPN, including (1) a constant-factor approximation algorithm for the case of arbitrary connected neighborhoods having comparable diameters; and (2) a PTAS for the important special case of disjoint unit disk neighborhoods (or nearly disjoint, nearly-unit disks). Our methods also yield improved approximation ratios for various special classes of neighborhoods, which have previously been studied. Further, we give a linear-time O(1)-approximation algorithm for the case of neighborhoods that are (infinite) straight lines.
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Convex polygons in geometric triangulations
We show that the maximum number of convex polygons in a triangulation of $n$ points in the plane is $O(1.5029^n)$. This improves an earlier bound of $O(1.6181^n)$ established by van Kreveld, Löffler, and Pach (2012) and almost matches the current best lower bound of $Ω(1.5028^n)$ due to the same authors. Given a planar straight-line graph $G$ with $n$ vertices, we show how to compute efficiently the number of convex polygons in $G$.
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Perfect vector sets, properly overlapping partitions, and largest empty box
We revisit the following problem (along with its higher dimensional variant): Given a set $S$ of $n$ points inside an axis-parallel rectangle $U$ in the plane, find a maximum-area axis-parallel sub-rectangle that is contained in $U$ but contains no points of $S$. (I) We present an algorithm that finds a large empty box amidst $n$ points in $[0,1]^d$: a box whose volume is at least $\frac{\log{d}}{4(n + \log{d})}$ can be computed in $O(n+d \log{d})$ time. (II) To better analyze the above approach, we introduce the concepts of perfect vector sets and properly overlapping partitions, in connection to the minimum volume of a maximum empty box amidst $n$ points in the unit hypercube $[0,1]^d$, and derive bounds on their sizes.
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Monotone Paths in Geometric Triangulations
(I) We prove that the (maximum) number of monotone paths in a geometric triangulation of $n$ points in the plane is $O(1.7864^n)$. This improves an earlier upper bound of $O(1.8393^n)$; the current best lower bound is $Ω(1.7003^n)$. (II) Given a planar geometric graph $G$ with $n$ vertices, we show that the number of monotone paths in $G$ can be computed in $O(n^2)$ time.
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Constant-Factor Approximation for TSP with Disks
We revisit the traveling salesman problem with neighborhoods (TSPN) and present the first constant-ratio approximation for disks in the plane: Given a set of $n$ disks in the plane, a TSP tour whose length is at most $O(1)$ times the optimal can be computed in time that is polynomial in $n$. Our result is the first constant-ratio approximation for a class of planar convex bodies of arbitrary size and arbitrary intersections. In order to achieve a $O(1)$-approximation, we reduce the traveling salesman problem with disks, up to constant factors, to a minimum weight hitting set problem in a geometric hypergraph. The connection between TSPN and hitting sets in geometric hypergraphs, established here, is likely to have future applications.
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Lower bounds on the dilation of plane spanners
(I) We exhibit a set of 23 points in the plane that has dilation at least $1.4308$, improving the previously best lower bound of $1.4161$ for the worst-case dilation of plane spanners. (II) For every integer $n\geq13$, there exists an $n$-element point set $S$ such that the degree 3 dilation of $S$ denoted by $δ_0(S,3) \text{ equals } 1+\sqrt{3}=2.7321\ldots$ in the domain of plane geometric spanners. In the same domain, we show that for every integer $n\geq6$, there exists a an $n$-element point set $S$ such that the degree 4 dilation of $S$ denoted by $δ_0(S,4) \text{ equals } 1 + \sqrt{(5-\sqrt{5})/2}=2.1755\ldots$ The previous best lower bound of $1.4161$ holds for any degree. (III) For every integer $n\geq6 $, there exists an $n$-element point set $S$ such that the stretch factor of the greedy triangulation of $S$ is at least $2.0268$.
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Lattice spanners of low degree
Let $δ_0(P,k)$ denote the degree $k$ dilation of a point set $P$ in the domain of plane geometric spanners. If $Λ$ is the infinite square lattice, it is shown that $1+\sqrt{2} \leq δ_0(Λ,3) \leq (3+2\sqrt2) \, 5^{-1/2} = 2.6065\ldots$ and $δ_0(Λ,4) = \sqrt{2}$. If $Λ$ is the infinite hexagonal lattice, it is shown that $δ_0(Λ,3) = 1+\sqrt{3}$ and $δ_0(Λ,4) = 2$. All our constructions are planar lattice tilings constrained to degree $3$ or $4$.
math.MG↗
Anchored Rectangle and Square Packings
For points $p_1,\ldots , p_n$ in the unit square $[0,1]^2$, an \emph{anchored rectangle packing} consists of interior-disjoint axis-aligned empty rectangles $r_1,\ldots , r_n\subseteq [0,1]^2$ such that point $p_i$ is a corner of the rectangle $r_i$ (that is, $r_i$ is \emph{anchored} at $p_i$) for $i=1,\ldots, n$. We show that for every set of $n$ points in $[0,1]^2$, there is an anchored rectangle packing of area at least $7/12-O(1/n)$, and for every $n\in \mathbf{N}$, there are point sets for which the area of every anchored rectangle packing is at most $2/3$. The maximum area of an anchored \emph{square} packing is always at least $5/32$ and sometimes at most $7/27$. The above constructive lower bounds immediately yield constant-factor approximations, of $7/12 -\varepsilon$ for rectangles and $5/32$ for squares, for computing anchored packings of maximum area in $O(n\log n)$ time. We prove that a simple greedy strategy achieves a $9/47$-approximation for anchored square packings, and $1/3$ for lower-left anchored square packings. Reductions to maximum weight independent set (MWIS) yield a QPTAS and a PTAS for anchored rectangle and square packings in $n^{O(1/\varepsilon)}$ and $\exp({\rm poly}(\log (n/\varepsilon)))$ time, respectively.
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