arXiv subjects

Adrian Dumitrescu

Publications and source records attributed to Adrian Dumitrescu.

At least 37 records · Page 2Linked to original sources

arXiv2024

Geometric Variants of the Gale--Berlekamp Switching Game

The Gale-Berlekamp switching game is played on the following device: $G_n=\{1,2,\ldots,n\} \times \{1,2,\ldots,n\}$ is an $n \times n$ array of lights is controlled by $2n$ switches, one for each row or column. Given an (arbitrary) initial configuration of the board, the objective is to have as many lights on as possible. Denoting the maximum difference (discrepancy) between the number of lights that are on minus the number of lights that are off by $F(n)$, it is known (Brown and Spencer, 1971) that $F(n)= Θ(n^{3/2})$, and more precisely, that $F(n) \geq \left( 1+ o(1) \right) \sqrt{\frac{2}π} n^{3/2} \approx 0.797 \ldots n^{3/2}$. Here we extend the game to other playing boards. For example: (i)~For any constant $c>1$, if $c n$ switches are conveniently chosen, then the maximum discrepancy for the square board is $Ω(n^{3/2})$. From the other direction, suppose we fix any set of $a$ column switches, $b$ row switches, where $a \geq b$ and $a+b=n$. Then the maximum discrepancy is at most $-b (n-b)$. (ii) A board $H \subset \{1,\ldots,n\}^2$, with area $A=|H|$, is \emph{dense} if $A \geq c (u+v)^2$, for some constant $c>0$, where $u= |\{x \colon (x,y) \in H\}|$ and $v=|\{y \colon (x,y) \in H\}|$. For a dense board of area $A$, we show that the maximum discrepancy is $Θ(A^{3/4})$. This result is a generalization of the Brown and Spencer result for the original game. (iii) If $H$ consists of the elements of $G_n$ below the hyperbola $xy=n$, then its maximum discrepancy is $Ω(n)$ and $O(n (\log n)^{1/2})$.

math.CO↗

arXiv2024

A Stopping Game on Zero-Sum Sequences

We introduce and analyze a natural game formulated as follows. In this one-person game, the player is given a random permutation $A=(a_1,\dots, a_n)$ of a multiset $M$ of $n$ reals that sum up to $0$, where each of the $n!$ permutation sequences is equally likely. The player only knows the value of $n$ beforehand. The elements of the sequence are revealed one by one and the player can stop the game at any time. Once the process stops, say, after the $i$th element is revealed, the player collects the amount $\sum_{j=i+1}^{n} a_j$ as his/her payoff and the game is over (the payoff corresponds to the unrevealed part of the sequence). Three online algorithms are given for maximizing the expected payoff in the binary case when $M$ contains only $1$'s and $-1$'s. $\texttt{Algorithm 1}$ is slightly suboptimal, but is easier to analyze. Moreover, it can also be used when $n$ is only known with some approximation. $\texttt{Algorithm 2}$ is exactly optimal but not so easy to analyze on its own. $\texttt{Algorithm 3}$ is the simplest of all three. It turns out that the expected payoffs of the player are $Θ(\sqrt{n})$ for all three algorithms. In the end, we address the general problem and deal with an arbitrary zero-sum multiset, for which we show that our $\texttt{Algorithm 3}$ returns a payoff proportional to $\sqrt{n}$, which is worst case-optimal.

cs.DM↗

arXiv2024

Partitioning Complete Geometric Graphs on Dense Point Sets into Plane Subgraphs

A \emph{complete geometric graph} consists of a set $P$ of $n$ points in the plane, in general position, and all segments (edges) connecting them. It is a well known question of Bose, Hurtado, Rivera-Campo, and Wood, whether there exists a positive constant $c<1$, such that every complete geometric graph on $n$ points can be partitioned into at most $cn$ plane graphs (that is, noncrossing subgraphs). We answer this question in the affirmative in the special case where the underlying point set $P$ is \emph{dense}, which means that the ratio between the maximum and the minimum distances in $P$ is of the order of $Θ(\sqrt{n})$.

math.CO↗

arXiv2024

On a Traveling Salesman Problem for Points in the Unit Cube

Let $X$ be an $n$-element point set in the $k$-dimensional unit cube $[0,1]^k$ where $k \geq 2$. According to an old result of Bollobás and Meir (1992), there exists a cycle (tour) $x_1, x_2, \ldots, x_n$ through the $n$ points, such that $\left(\sum_{i=1}^n |x_i - x_{i+1}|^k \right)^{1/k} \leq c_k$, where $|x-y|$ is the Euclidean distance between $x$ and $y$, and $c_k$ is an absolute constant that depends only on $k$, where $x_{n+1} \equiv x_1$. From the other direction, for every $k \geq 2$ and $n \geq 2$, there exist $n$ points in $[0,1]^k$, such that their shortest tour satisfies $\left(\sum_{i=1}^n |x_i - x_{i+1}|^k \right)^{1/k} = 2^{1/k} \cdot \sqrt{k}$. For the plane, the best constant is $c_2=2$ and this is the only exact value known. Bollob{á}s and Meir showed that one can take $c_k = 9 \left(\frac23 \right)^{1/k} \cdot \sqrt{k}$ for every $k \geq 3$ and conjectured that the best constant is $c_k = 2^{1/k} \cdot \sqrt{k}$, for every $k \geq 2$. Here we significantly improve the upper bound and show that one can take $c_k = 3 \sqrt5 \left(\frac23 \right)^{1/k} \cdot \sqrt{k}$ or $c_k = 2.91 \sqrt{k} \ (1+o_k(1))$. Our bounds are constructive. We also show that $c_3 \geq 2^{7/6}$, which disproves the conjecture for $k=3$. Connections to matching problems, power assignment problems, related problems, including algorithms, are discussed in this context. A slightly revised version of the Bollobás--Meir conjecture is proposed.

math.CO↗

arXiv2024

A Strongly Subcubic Combinatorial Algorithm for Triangle Detection with Applications

We revisit the algorithmic problem of finding a triangle in a graph: We give a randomized combinatorial algorithm for triangle detection in a given $n$-vertex graph with $m$ edges running in $O(n^{7/3})$ time, or alternatively in $O(m^{4/3})$ time. This may come as a surprise since it invalidates several conjectures in the literature. In particular, - the $O(n^{7/3})$ runtime surpasses the long-standing fastest algorithm for triangle detection based on matrix multiplication running in $O(n^ω) = O(n^{2.372})$ time, due to Itai and Rodeh (1978). - the $O(m^{4/3})$ runtime surpasses the long-standing fastest algorithm for triangle detection in sparse graphs based on matrix multiplication running in $O(m^{2ω/(ω+1)})= O(m^{1.407})$ time due to Alon, Yuster, and Zwick (1997). - the $O(n^{7/3})$ time algorithm for triangle detection leads to a $O(n^{25/9} \log{n})$ time combinatorial algorithm for $n \times n$ Boolean matrix multiplication, by a reduction of V. V. Williams and R.~R.~Williams (2018).This invalidates a conjecture of A.~Abboud and V. V. Williams (FOCS 2014). - the $O(m^{4/3})$ runtime invalidates a conjecture of A.~Abboud and V. V. Williams (FOCS 2014) that any combinatorial algorithm for triangle detection requires $m^{3/2 - o(1)}$ time. - as a direct application of the triangle detection algorithm, we obtain a faster exact algorithm for the $k$-clique problem, surpassing an almost $40$ years old algorithm of Ne{š}et{ř}il and Poljak (1985). This result strongly disproves the combinatorial $k$-clique conjecture. - as another direct application of the triangle detection algorithm, we obtain a faster exact algorithm for the \textsc{Max-Cut} problem, surpassing an almost $20$ years old algorithm of R.~R.~Williams (2005).

cs.DS↗

arXiv2024

Finding Triangles or Independent Sets; and Other Dual Pair Approximations

We revisit the algorithmic problem of finding a triangle in a graph (\textsc{Triangle Detection}), and examine its relation to other problems such as \textsc{3Sum}, \textsc{Independent Set}, and \textsc{Graph Coloring}. We obtain several new algorithms: \smallskip (I) A simple randomized algorithm for finding a triangle in a graph. As an application, we study the range of a conjecture of Pǎtraşcu (2010) regarding the triangle detection problem. \smallskip (II) An algorithm which given a graph $G=(V,E)$ performs one of the following tasks in $O(m+n)$ (ie, linear) time: (i)~compute a $Ω(1/\sqrt{n})$-approximation of a maximum independent set in $G$ or (ii)~find a triangle in $G$. The run-time is faster than that for any previous method for each of these tasks. \smallskip (III) An algorithm which given a graph $G=(V,E)$ performs one of the following tasks in $O(m+n^{3/2})$ time: (i)~compute an $\sqrt{n}$-approximation for \textsc{Graph Coloring} of $G$ or (ii)~find a triangle in $G$. The run-time is faster than that for any previous method for each of these tasks on dense graphs, with $m =ω(n^{9/8})$. \smallskip (IV) The second and third results suggest the following broader research direction: if it is difficult to find (A) or (B) separately, can one find one of the two efficiently? This motivates the \emph{dual pair} concept we introduce. We discuss and provide several instances of dual-pair approximation.

cs.DS↗

arXiv2023

Two trees are better than one

We consider partitions of a point set into two parts, and the lengths of the minimum spanning trees of the original set and of the two parts. If $w(P)$ denotes the length of a minimum spanning tree of $P$, we show that every set $P$ of $n \geq 12$ points admits a bipartition $P= R \cup B$ for which the ratio $\frac{w(R)+w(B)}{w(P)}$ is strictly larger than $1$; and that $1$ is the largest number with this property. Furthermore, we provide a very fast algorithm that computes such a bipartition in $O(1)$ time and one that computes the corresponding ratio in $O(n \log{n})$ time. In certain settings, a ratio larger than $1$ can be expected and sometimes guaranteed. For example, if $P$ is a set of $n$ random points uniformly distributed in $[0,1]^2$ ($n \to \infty$), then for any $\eps>0$, the above ratio in a maximizing partition is at least $\sqrt2 -\eps$ with probability tending to $1$. As another example, if $P$ is a set of $n$ points with spread at most $α\sqrt{n}$, for some constant $α>0$, then the aforementioned ratio in a maximizing partition is $1 + Ω(α^{-2})$. All our results and techniques are extendable to higher dimensions.

cs.CG↗

arXiv2023

Observation Routes and External Watchman Routes

We introduce the Observation Route Problem ($\textsf{ORP}$) defined as follows: Given a set of $n$ pairwise disjoint compact regions in the plane, find a shortest tour (route) such that an observer walking along this tour can see (observe) some point in each region from some point of the tour. The observer does \emph{not} need to see the entire boundary of an object. The tour is \emph{not} allowed to intersect the interior of any region (i.e., the regions are obstacles and therefore out of bounds). The problem exhibits similarity to both the Traveling Salesman Problem with Neighborhoods ($\textsf{TSPN}$) and the External Watchman Route Problem ($\textsf{EWRP}$). We distinguish two variants: the range of visibility is either limited to a bounding rectangle, or unlimited. We obtain the following results: (I) Given a family of $n$ disjoint convex bodies in the plane, computing a shortest observation route does not admit a $(c\log n)$-approximation unless $\textsf{P} = \textsf{NP}$ for an absolute constant $c>0$. (This holds for both limited and unlimited vision.) (II) Given a family of disjoint convex bodies in the plane, computing a shortest external watchman route is $\textsf{NP}$-hard. (This holds for both limited and unlimited vision; and even for families of axis-aligned squares.) (III) Given a family of $n$ disjoint fat convex polygons, an observation tour whose length is at most $O(\log{n})$ times the optimal can be computed in polynomial time. (This holds for limited vision.) (IV) For every $n \geq 5$, there exists a convex polygon with $n$ sides and all angles obtuse such that its perimeter is \emph{not} a shortest external watchman route. This refutes a conjecture by Absar and Whitesides (2006).

cs.CG↗

arXiv2023

Two-sided convexity testing with certificates

We revisit the problem of property testing for convex position for point sets in $\mathbb{R}^d$. Our results draw from previous ideas of Czumaj, Sohler, and Ziegler (ESA 2000). First, the algorithm is redesigned and its analysis is revised for correctness. Second, its functionality is expanded by (i)~exhibiting both negative and positive certificates along with the convexity determination, and (ii)~significantly extending the input range for moderate and higher dimensions. The behavior of the randomized tester is as follows: (i)~if $P$ is in convex position, it accepts; (ii)~if $P$ is far from convex position, with probability at least $2/3$, it rejects and outputs a $(d+2)$-point witness of non-convexity as a negative certificate; (iiii)~if $P$ is close to convex position, with probability at least $2/3$, it accepts and outputs an approximation of the largest subset in convex position. The algorithm examines a sublinear number of points and runs in subquadratic time for every dimension $d$.

cs.CG↗

arXiv2023

Almost Congruent Triangles

Almost $50$ years ago Erdős and Purdy asked the following question: Given $n$ points in the plane, how many triangles can be approximate congruent to equilateral triangles? They pointed out that by dividing the points evenly into three small clusters built around the three vertices of a fixed equilateral triangle, one gets at least $\left\lfloor \frac{n}{3} \right\rfloor \cdot \left\lfloor \frac{n+1}{3} \right\rfloor \cdot \left\lfloor \frac{n+2}{3} \right\rfloor$ such approximate copies. In this paper we provide a matching upper bound and thereby answer their question. More generally, for every triangle $T$ we determine the maximum number of approximate congruent triangles to $T$ in a point set of size $n$. Parts of our proof are based on hypergraph Turán theory: for each point set in the plane and a triangle $T$, we construct a $3$-uniform hypergraph $\mathcal{H}=\mathcal{H}(T)$, which contains no hypergraph as a subgraph from a family of forbidden hypergraphs $\mathcal{F}=\mathcal{F}(T)$. Our upper bound on the number of edges of $\mathcal{H}$ will determine the maximum number of triangles that are approximate congruent to $T$.

math.CO↗

arXiv2023

Peeling Sequences

Given a set of $n$ labeled points in general position in the plane, we remove all of its points one by one. At each step, one point from the convex hull of the remaining set is erased. In how many ways can the process be carried out? The answer obviously depends on the point set. If the points are in convex position, there are exactly $n!$ ways, which is the maximum number of ways for $n$ points. But what is the minimum number? It is shown that this number is (roughly) at least $3^n$ and at most $12.29^n$.

math.CO↗

arXiv2023

On the Stretch Factor of Polygonal Chains

Let $P=(p_1, p_2, \dots, p_n)$ be a polygonal chain in $\mathbb{R}^d$. The stretch factor of $P$ is the ratio between the total length of $P$ and the distance of its endpoints, $\sum_{i = 1}^{n-1} |p_i p_{i+1}|/|p_1 p_n|$. For a parameter $c \geq 1$, we call $P$ a $c$-chain if $|p_ip_j|+|p_jp_k| \leq c|p_ip_k|$, for every triple $(i,j,k)$, $1 \leq i 0$, there is a noncrossing $c$-chain that has stretch factor $Ω(n^{1/2-\varepsilon})$, for sufficiently large constant $c=c(\varepsilon)$; (ii) on the other hand, the stretch factor of a $c$-chain $P$ is $O\left(n^{1/2}\right)$, for every constant $c\geq 1$, regardless of whether $P$ is crossing or noncrossing; and (iii) we give a randomized algorithm that can determine, for a polygonal chain $P$ in $\mathbb{R}^2$ with $n$ vertices, the minimum $c\geq 1$ for which $P$ is a $c$-chain in $O\left(n^{2.5}\ \mathrm{polylog}\ n\right)$ expected time and $O(n\log n)$ space. These results generalize to $\mathbb{R}^d$. For every dimension $d\geq 2$ and every $\varepsilon>0$, we construct a noncrossing $c$-chain that has stretch factor $Ω\left(n^{(1-\varepsilon)(d-1)/d}\right)$; on the other hand, the stretch factor of any $c$-chain is $O\left((n-1)^{(d-1)/d}\right)$; for every $c>1$, we can test whether an $n$-vertex chain in $\mathbb{R}^d$ is a $c$-chain in $O\left(n^{3-1/d}\ \mathrm{polylog}\ n\right)$ expected time and $O(n\log n)$ space.

cs.CG↗

arXiv2022

Finding Points in Convex Position in Density-Restricted Sets

For a finite set $A\subset \mathbb{R}^d$, let $Δ(A)$ denote the spread of $A$, which is the ratio of the maximum pairwise distance to the minimum pairwise distance. For a positive integer $n$, let $γ_d(n)$ denote the largest integer such that any set $A$ of $n$ points in general position in $\mathbb{R}^d$, satisfying $Δ(A) \leq αn^{1/d}$ for a fixed $α>0$, contains at least $γ_d(n)$ points in convex position. About $30$ years ago, Valtr proved that $γ_2(n)=Θ(n^{1/3})$. Since then no further results have been obtained in higher dimensions. Here we continue this line of research in three dimensions and prove that $γ_3(n) =Θ(n^{1/2})$. The lower bound implies the following approximation: Given any $n$-element point set $A\subset \mathbb{R}^3$ in general position, satisfying $Δ(A) \leq αn^{1/3}$ for a fixed $α$, a $Ω(n^{-1/6})$-factor approximation of the maximum-size convex subset of points can be computed by a randomized algorithm in $O(n \log{n})$ expected time.

math.CO↗

arXiv2022

The Dirac--Goodman--Pollack Conjecture

In one of their seminal articles on allowable sequences, Goodman and Pollack gave combinatorial generalizations for three problems in discrete geometry, one of which being the Dirac conjecture. According to this conjecture, any set of $n$ noncollinear points in the plane has a point incident to at least $c n$ connecting lines determined by the set. The notion of allowable sequences of permutations provides a natural combinatorial setting for analyzing these problems. Within this formalism, the conjectured generalization reads as follows: \emph{Any nontrivial allowable $n$-sequence $Σ$ has a local sequence $Λ_i$ whose half-period is at least $c n$.} The conjecture is confirmed here with a concrete bound $c=1/845$. Several related problems are discussed.

math.CO↗

arXiv2021

Convex Polygons in Cartesian Products

We study several problems concerning convex polygons whose vertices lie in a Cartesian product of two sets of $n$ real numbers (for short, \emph{grid}). First, we prove that every such grid contains $Ω(\log n)$ points in convex position and that this bound is tight up to a constant factor. We generalize this result to $d$ dimensions (for a fixed $d\in \mathbb{N}$), and obtain a tight lower bound of $Ω(\log^{d-1}n)$ for the maximum number of points in convex position in a $d$-dimensional grid. Second, we present polynomial-time algorithms for computing the longest $x$- or $y$-monotone convex polygonal chain in a grid that contains no two points with the same $x$- or $y$-coordinate. We show that the maximum size of a convex polygon with such unique coordinates can be efficiently approximated up to a factor of $2$. Finally, we present exponential bounds on the maximum number of point sets in convex position in such grids, and for some restricted variants. These bounds are tight up to polynomial factors.

cs.CG↗

arXiv2021

Online unit clustering in higher dimensions

We revisit the online Unit Clustering and Unit Covering problems in higher dimensions: Given a set of $n$ points in a metric space, that arrive one by one, Unit Clustering asks to partition the points into the minimum number of clusters (subsets) of diameter at most one; while Unit Covering asks to cover all points by the minimum number of balls of unit radius. In this paper, we work in $\mathbb{R}^d$ using the $L_\infty$ norm. We show that the competitive ratio of any online algorithm (deterministic or randomized) for Unit Clustering must depend on the dimension $d$. We also give a randomized online algorithm with competitive ratio $O(d^2)$ for Unit Clustering of integer points (i.e., points in $\mathbb{Z}^d$, $d\in \mathbb{N}$, under $L_{\infty}$ norm). We show that the competitive ratio of any deterministic online algorithm for Unit Covering is at least $2^d$. This ratio is the best possible, as it can be attained by a simple deterministic algorithm that assigns points to a predefined set of unit cubes. We complement these results with some additional lower bounds for related problems in higher dimensions.

cs.CG↗

arXiv2021

Piercing All Translates of a Set of Axis-Parallel Rectangles

For a given shape $S$ in the plane, one can ask what is the lowest possible density of a point set $P$ that pierces ("intersects", "hits") all translates of $S$. This is equivalent to determining the covering density of $S$ and as such is well studied. Here we study the analogous question for families of shapes where the connection to covering no longer exists. That is, we require that a single point set $P$ simultaneously pierces each translate of each shape from some family $\mathcal F$. We denote the lowest possible density of such an $\mathcal F$-piercing point set by $π_T(\mathcal F)$. Specifically, we focus on families $\mathcal F$ consisting of axis-parallel rectangles. When $|\mathcal F|=2$ we exactly solve the case when one rectangle is more squarish than $2\times 1$, and give bounds (within $10\,\%$ of each other) for the remaining case when one rectangle is wide and the other one is tall. When $|\mathcal F|\ge 2$ we present a linear-time constant-factor approximation algorithm for computing $π_T(\mathcal F)$ (with ratio $1.895$).

cs.CG↗

arXiv2021

Sparse Hop Spanners for Unit Disk Graphs

A unit disk graph $G$ on a given set $P$ of points in the plane is a geometric graph where an edge exists between two points $p,q \in P$ if and only if $|pq| \leq 1$. A spanning subgraph $G'$ of $G$ is a $k$-hop spanner if and only if for every edge $pq\in G$, there is a path between $p,q$ in $G'$ with at most $k$ edges. We obtain the following results for unit disk graphs in the plane. (I) Every $n$-vertex unit disk graph has a $5$-hop spanner with at most $5.5n$ edges. We analyze the family of spanners constructed by Biniaz (2020) and improve the upper bound on the number of edges from $9n$ to $5.5n$. (II) Using a new construction, we show that every $n$-vertex unit disk graph has a $3$-hop spanner with at most $11n$ edges. (III) Every $n$-vertex unit disk graph has a $2$-hop spanner with $O(n\log n)$ edges. This is the first nontrivial construction of $2$-hop spanners. (IV) For every sufficiently large positive integer $n$, there exists a set $P$ of $n$ points on a circle, such that every plane hop spanner on $P$ has hop stretch factor at least $4$. Previously, no lower bound greater than $2$ was known. (V) For every finite point set on a circle, there exists a plane (i.e., crossing-free) $4$-hop spanner. As such, this provides a tight bound for points on a circle. (VI) The maximum degree of $k$-hop spanners cannot be bounded from above by a function of $k$ for any positive integer $k$.

cs.CG↗