Papers by adrian dumitrescu
Periodica Mathematica Hungarica, 2008
A set of points in the plane is said to be in general position if no three of them are collinear ... more A set of points in the plane is said to be in general position if no three of them are collinear and no four of them are cocircular. If a point set determines only distinct vectors, it is called parallelogram free. We show that there exist n-element point sets in the plane in general position, and parallelogram free, that determine only O(n 2 / √ log n) distinct distances. This answers a question of Erdős, Hickerson and Pach. We then revisit an old problem of Erdős : given any n points in the plane (or in d dimensions), how many of them can one select so that the distances which are determined are all distinct?-and provide (make explicit) some new bounds in one and two dimensions.
SIAM Journal on Discrete Mathematics, 2015
The problem of finding a collection of curves of minimum total length that meet all the lines int... more The problem of finding a collection of curves of minimum total length that meet all the lines intersecting a given polygon was initiated by Mazurkiewicz in 1916. Such a collection forms an opaque barrier for the polygon. In 1991 Shermer proposed an exponential-time algorithm that computes an interior-restricted barrier made of segments for any given convex n-gon. He conjectured that the barrier found by his algorithm is optimal, however this was refuted recently by Provan et al. Here we give a Shermer like algorithm that computes an interior polygonal barrier whose length is at most 1.7168 times the optimal and that runs in O(n) time. As a byproduct, we also deduce upper and lower bounds on the approximation ratio of Shermer's algorithm.
Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 2014
For two planar convex bodies, C and D, consider a packing S of n positive homothets of C containe... more For two planar convex bodies, C and D, consider a packing S of n positive homothets of C contained in D. We estimate the total perimeter of the bodies in S, denoted per(S), in terms of per(D) and n. When all homothets of C touch the boundary of the container D, we show that either per(S) = O(log n) or per(S) = O(1), depending on how C and D "fit together," and these bounds are the best possible apart from the constant factors. Specifically, we establish an optimal bound per(S) = O(log n) unless D is a convex polygon and every side of D is parallel to a corresponding segment on the boundary of C (for short, D is parallel to C). When D is parallel to C but the homothets of C may lie anywhere in D, we show that per(S) = O((1+esc(S)) log n/ log log n), where esc(S) denotes the total distance of the bodies in S from the boundary of D. Apart from the constant factor, this bound is also the best possible.
Lecture Notes in Computer Science, 2011
The problem of finding "small" sets that meet every straight-line which intersects a given convex... more The problem of finding "small" sets that meet every straight-line which intersects a given convex region was initiated by Mazurkiewicz in 1916. We call such a set an opaque set or a barrier for that region. We consider the problem of computing the shortest barrier for a given convex polygon with n vertices. No exact algorithm is currently known even for the simplest instances such as a square or an equilateral triangle. For general barriers, we present an approximation algorithm with ratio 1 2 + 2+ √ 2 π = 1.5867. . .. For connected barriers we achieve the approximation ratio 1.5716, while for single-arc barriers we achieve the approximation ratio π+5 π+2 = 1.5834. . .. All three algorithms run in O(n) time. We also show that if the barrier is restricted to the (interior and the boundary of the) input polygon, then the problem admits a fully polynomial-time approximation scheme for the connected case and a quadratic-time exact algorithm for the single-arc case.
Lecture Notes in Computer Science
Erdős, Purdy, and Straus conjectured that the number of distinct (nonzero) areas of the triangles... more Erdős, Purdy, and Straus conjectured that the number of distinct (nonzero) areas of the triangles determined by n noncollinear points in the plane is at least ⌊ n−1 2 ⌋, which is attained for ⌈n/2⌉ and respectively ⌊n/2⌋ equally spaced points lying on two parallel lines. We show that this number is at least 17 38 n − O(1) ≈ 0.4473n. The best previous bound, (√ 2 − 1)n − O(1) ≈ 0.4142n, which dates back to 1982, follows from the combination of a result of Burton and Purdy [5] and Ungar's theorem [23] on the number of distinct directions determined by n noncollinear points in the plane.
SIAM Journal on Discrete Mathematics, 2008
Let G be a connected graph, and let V and V ′ two n-element subsets of its vertex set V (G). Imag... more Let G be a connected graph, and let V and V ′ two n-element subsets of its vertex set V (G). Imagine that we place a chip at each element of V and we want to move them into the positions of V ′ (V and V ′ may have common elements). A move is defined as shifting a chip from v 1 to v 2 (v 1 , v 2 ∈ V (G)) on a path formed by edges of G so that no intermediate vertices are occupied. We give upper and lower bounds on the number of moves that are necessary, and analyze the computational complexity of this problem under various assumptions: labeled versus unlabeled chips, arbitrary graphs versus the case when the graph is the rectangular (infinite) planar grid, etc. We prove hardness and inapproximability results for several variants of the problem. We also give a linear-time algorithm which performs an optimal (minimum) number of moves for the unlabeled version in a tree, and a constant-ratio approximation algorithm for the unlabeled version in a graph. The graph algorithm uses the tree algorithm as a subroutine.
Journal of Discrete Algorithms, 2009
An orthogonal spanner network for a given set of n points in the plane is a plane straight line g... more An orthogonal spanner network for a given set of n points in the plane is a plane straight line graph with axis-aligned edges that connects all input points. We show that for any set of n points in the plane, there is an orthogonal spanner network that (i) is short having a total edge length at most a constant times the length of a Euclidean minimum spanning tree for the point set; (ii) is small having O (n) vertices and edges; and (iii) has constant geometric dilation, which means that for any two points u and v in the network, the shortest path in the network between u and v is at most a constant times longer than the Euclidean distance between u and v. Such a network can be constructed in O (n log n) time.
International Journal of Computational Geometry & Applications, 2010
Maximum Independent Set (MIS) and its relative Maximum Weight Independent Set (MWIS) are well-kno... more Maximum Independent Set (MIS) and its relative Maximum Weight Independent Set (MWIS) are well-known problems in combinatorial optimization; they are NP-hard even in the geometric setting of unit disk graphs. In this paper, we study the Maximum Area Independent Set (MAIS) problem, a natural restricted version of MWIS in disk intersection graphs where the weight equals the disk area. We obtain: (i) Quantitative bounds on the maximum total area of an independent set relative to the union area; (ii) Practical constant-ratio approximation algorithms for finding an independent set with a large total area relative to the union area.
International Journal of Computational Geometry & Applications, 2008
Given a pair of start and target configurations, each consisting of n pairwise disjoint disks in ... more Given a pair of start and target configurations, each consisting of n pairwise disjoint disks in the plane, what is the minimum number of moves that suffice for transforming the start configuration into the target configuration? In one move a disk slides in the plane without intersecting any other disk, so that its center moves along an arbitrary (open) continuous curve. We discuss efficient algorithms for this task and estimate their number of moves under different assumptions on disk radii. For example, with n congruent disks, [Formula: see text] moves always suffice for transforming the start configuration into the target configuration; on the other hand, [Formula: see text] moves are sometimes necessary.
Discrete Optimization, 2011
The Fermat-Weber center of a planar body Q is a point in the plane from which the average distanc... more The Fermat-Weber center of a planar body Q is a point in the plane from which the average distance to the points in Q is minimal. We first show that for any convex body Q in the plane, the average distance from the Fermat-Weber center of Q to the points of Q is larger than 1 6 • ∆(Q), where ∆(Q) is the diameter of Q. This proves a conjecture of Carmi, Har-Peled and Katz. From the other direction, we prove that the same average distance is at most 2(4− √ 3) 13 • ∆(Q) < 0.3490 • ∆(Q). The new bound substantially improves the previous bound of 2 3 √ 3 • ∆(Q) ≈ 0.3849 • ∆(Q) due to Abu-Affash and Katz, and brings us closer to the conjectured value of 1 3 • ∆(Q). We also confirm the upper bound conjecture for centrally symmetric planar convex bodies.
Discrete & Computational Geometry, 2010
Let Ω be a disk of radius R in the plane. A set F of unit disks contained in Ω forms a maximal pa... more Let Ω be a disk of radius R in the plane. A set F of unit disks contained in Ω forms a maximal packing if the unit disks are pairwise interior-disjoint and the set is maximal, i.e., it is not possible to add another disk to F while maintaining the packing property. A point p is hidden within the "forest" defined by F if any ray with apex p intersects some disk of F , that is, a person standing at p can hide without being seen from outside the forest. We show that if the radius R of Ω is large enough, one can find a hidden point for any maximal packing of unit disks in Ω. This proves a conjecture of Joseph Mitchell. We also present an O(n 5/2 log n)-time algorithm that, given a forest with n (not necessarily congruent) disks, computes the boundary illumination map of all disks in the forest.
Computational Geometry, 2009
This paper studies non-crossing geometric perfect matchings. Two such perfect matchings are compa... more This paper studies non-crossing geometric perfect matchings. Two such perfect matchings are compatible if they have the same vertex set and their union is also non-crossing. Our first result states that for any two perfect matchings M and M of the same set of n points, for some k ∈ O(log n), there is a sequence of perfect matchings M = M 0 , M 1 ,. .. , M k = M , such that each M i is compatible with M i+1. This improves the previous best bound of k n − 2. We then study the conjecture: every perfect matching with an even number of edges has an edge-disjoint compatible perfect matching. We introduce a sequence of stronger conjectures that imply this conjecture, and prove the strongest of these conjectures in the case of perfect matchings that consist of vertical and horizontal segments. Finally, we prove that every perfect matching with n edges has an edge-disjoint compatible matching with approximately 4n/5 edges.
Combinatorics, Probability and Computing, 2008
Given a set of s points and a set of n2 lines in three-dimensional Euclidean space such that each... more Given a set of s points and a set of n2 lines in three-dimensional Euclidean space such that each line is incident to n points but no n lines are coplanar, we show that s = Ω(n11/4). This is the first non-trivial answer to a question recently posed by Jean Bourgain.
Let S be a set of n points in the plane and consider a family of (nondegenerate) pairwise congrue... more Let S be a set of n points in the plane and consider a family of (nondegenerate) pairwise congruent triangles whose vertices belong to S. While the number of such triangles can grow superlinearly in n-as it happens in lattice sections of the integer grid-it has been conjectured by Brass that the number of pairwise congruent empty triangles is only at most linear. We disprove this conjecture by constructing point sets with Ω(n log n) empty congruent triangles.
Algorithmica, 2009
New tight bounds are presented on the minimum length of planar straight line graphs connecting n ... more New tight bounds are presented on the minimum length of planar straight line graphs connecting n given points in the plane and having convex faces. Specifically, we show that the minimum length of a convex Steiner partition for n points in the plane is at most O(log n/ log log n) times longer than a Euclidean minimum spanning tree (EMST), and this bound is the best possible. Without Steiner points, the corresponding bound is known to be Θ(log n), attained for n vertices of a pseudo-triangle. We also show that the minimum length convex Steiner partition of n points along a pseudo-triangle is at most O(log log n) times longer than an EMST, and this bound is also the best possible. Our methods are constructive and lead to O(n log n) time algorithms for computing convex Steiner partitions having O(n) Steiner points and weight within the above worst-case bounds in both cases.
Algorithmica, 2010
According to a classical result of Grünbaum, the transversal number τ (F) of any family F of pair... more According to a classical result of Grünbaum, the transversal number τ (F) of any family F of pairwise-intersecting translates or homothets of a convex body C in R d is bounded by a function of d. Denote by α(C) (resp. β(C)) the supremum of the ratio of the transversal number τ (F) to the packing number ν(F) over all finite families F of translates (resp. homothets) of a convex body C in R d. Kim et al. recently showed that α(C) is bounded by a function of d for any convex body C in R d , and gave the first bounds on α(C) for convex bodies C in R d and on β(C) for convex bodies C in the plane. Here we show that β(C) is also bounded by a function of d for any convex body C in R d , and present new or improved bounds on both α(C) and β(C) for various convex bodies C in R d for all dimensions d. Our techniques explore interesting inequalities linking the covering and packing densities of a convex body. Our methods for obtaining upper bounds are constructive and lead to efficient constant-factor approximation algorithms for finding a minimum-cardinality point set that pierces a set of translates or homothets of a convex body.
Proceedings of the twenty-third annual symposium on Computational geometry - SCG '07, 2007
Abstract Consider the following survival problem: Given a set of k trajectories (paths) with maxi... more Abstract Consider the following survival problem: Given a set of k trajectories (paths) with maximum unit speed in a boundedregion over a (long) time interval [0, T], find another trajectory (if itexists) subject to the same maximum unit speed limit, that avoids (that is, ...
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Papers by adrian dumitrescu