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- Creator:
- Gerbner, Dániel, Toth, Csaba D., Keszegh, Balazs, and Dumitrescu, Adrian
- Description:
- Given n points in the plane, a covering path is a polygonal path that visits all the points. If no three points are collinear, every covering path requires at least n/2 segments, and n−1 straight line segments obviously suffice even if the covering path is required to be noncrossing. We show that every set of n points in the plane admits a (possibly self-crossing) covering path consisting of n/2+O(n/logn) straight line segments. If the path is required to be noncrossing, we prove that (1−ε)n straight line segments suffice for a small constant ε>0, and we exhibit n-element point sets that require at least 5n/9−O(1) segments in every such path. Further, the analogous question for noncrossing covering trees is considered and similar bounds are obtained. Finally, it is shown that computing a noncrossing covering path for n points in the plane requires Ω(nlogn) time in the worst case.
- Resource Type:
- Article
- Identifier:
- 0179-5376
- Campus Tesim:
- Northridge
- Creator:
- Dumitrescu, Adrian and Toth, Csaba D.
- Description:
- Let S be a set of n points in the unit square [,1]2, one of which is the origin. We construct n pairwise interior-disjoint axis-aligned empty rectangles such that the lower left corner of each rectangle is a point in S, and the rectangles jointly cover at least a positive constant area (about .9). This is a first step towards the solution of a longstanding conjecture that the rectangles in such a packing can jointly cover an area of at least 1/2.
- Resource Type:
- Article
- Identifier:
- 0209-9683
- Campus Tesim:
- Northridge
- Creator:
- Dumitrescu, Adrian, Balas, Kevin, and Toth, Csaba D.
- Description:
- For points in the unit square , an anchored rectangle packing consists of interior-disjoint axis-aligned empty rectangles such that point is a corner of the rectangle (that is, is anchored at ) for . We show that for every set of points in , there is an anchored rectangle packing of area at least , and for every , there are point sets for which the area of every anchored rectangle packing is at most . The maximum area of an anchored square packing is always at least and sometimes at most . The above constructive lower bounds immediately yield constant-factor approximations, of for rectangles and for squares, for computing anchored packings of maximum area in time. We prove that a simple greedy strategy achieves a -approximation for anchored square packings, and for lower-left anchored square packings. Reductions to maximum weight independent set (MWIS) yield a QPTAS for anchored rectangle packings in time and a PTAS for anchored square packings in time.
- Resource Type:
- Article
- Identifier:
- 1572-5286
- Campus Tesim:
- Northridge
- Creator:
- Dumitrescu, Adrian and Toth, Csaba D.
- Description:
- 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(logn) or per(S)=O(1) , depending on how C and D “fit together”. Apart from the constant factors, these bounds are the best possible. Specifically, we prove that per(S)=O(1) if 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 ) and per(S)=O(logn) otherwise. 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))logn/loglogn) , 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.
- Resource Type:
- Article
- Identifier:
- 2191-0383, 0138-4821
- Campus Tesim:
- Northridge
- Creator:
- Löffler, Maarten, Dumitrescu, Adrian, Toth, Csaba D., and Schulz, André
- Description:
- We give upper and lower bounds on the maximum and minimum number of geometric configurations of various kinds present (as subgraphs) in a triangulation of $n$ points in the plane. Configurations of interest include \emph{convex polygons}, \emph{star-shaped polygons} and \emph{monotone paths}. We also consider related problems for \emph{directed} planar straight-line graphs.
- Resource Type:
- Article
- Identifier:
- 0911-0119
- Campus Tesim:
- Northridge
- Creator:
- Schulz, André, Dumitrescu, Adrian, Sheffer, Adam, and Toth, Csaba D.
- Description:
- We obtain new lower and upper bounds for the maximum multiplicity of some weighted and, respectively, nonweighted common geometric graphs drawn on $n$ points in the plane in general position (with no three points collinear): perfect matchings, spanning trees, spanning cycles (tours), and triangulations. (i) We present a new lower bound construction for the maximum number of triangulations a set of $n$ points in general position can have. In particular, we show that a generalized double chain formed by two almost convex chains admits $\Omega (8.65^n)$ different triangulations. This improves the bound $\Omega (8.48^n)$ achieved by the previous best construction, the double zig-zag chain studied by Aichholzer et al. (ii) We obtain a new lower bound of $\Omega(12.00^n)$ for the number of noncrossing spanning trees of the double chain composed of two convex chains. The previous bound, $\Omega(10.42^n)$, stood unchanged for more than 10 years. (iii) Using a recent upper bound of $30^n$ for the number of triangulations, due to Sharir and Sheffer, we show that $n$ points in the plane in general position admit at most $O(68.62^n)$ noncrossing spanning cycles. (iv) We derive lower bounds for the number of maximum and minimum weighted geometric graphs (matchings, spanning trees, and tours). We show that the number of shortest tours can be exponential in $n$ for points in general position. These tours are automatically noncrossing. Likewise, we show that the number of longest noncrossing tours can be exponential in $n$. It was known that the number of shortest noncrossing perfect matchings can be exponential in $n$, and here we show that the number of longest noncrossing perfect matchings can be also exponential in $n$. It was known that the number of longest noncrossing spanning trees of a point set can be exponentially large, and here we show that this can be also realized with points in convex position. For points in convex position we re-derive tight bounds for the number of longest and shortest tours with some simpler arguments. We also give a combinatorial characterization of longest tours, which yields an $O(n\log n)$ time algorithm for computing them.
- Resource Type:
- Article
- Identifier:
- 0895-4801
- Campus Tesim:
- Northridge
- Creator:
- Dumitrescu, Adrian and Toth, Csaba D.
- Description:
- 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 also show how to compute efficiently the number of convex polygons in G.
- Resource Type:
- Article
- Identifier:
- 0963-5483
- Campus Tesim:
- Northridge
- Creator:
- Dumitrescu, Adrian, Balas, Kevin, and Toth, Csaba D.
- Description:
- For points p1,…,pn in the unit square [0,1]2, an anchored rectangle packing consists of interior-disjoint axis-aligned empty rectangles r1,…,rn⊆[0,1]2 such that point pi is a corner of the rectangle ri (that is, ri is anchored at pi) for i=1,…,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∈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 squarepacking 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−ε for rectangles and 5∕32 for squares, for computing anchored packings of maximum area in O(nlogn) 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 for anchored rectangle packings in exp(poly(ε−1logn)) time and a PTAS for anchored square packings in nO(1∕ε) time.
- Resource Type:
- Article
- Identifier:
- 1572-5286
- Campus Tesim:
- Northridge
- Creator:
- Dumitrescu, Adrian and Toth, Csaba D.
- Description:
- "We revisit the traveling salesman problem with neighborhoods (TSPN) and propose several new approximation algorithms. These constitute either first approximations (for hyperplanes, lines, and balls in Rd, for d ⩾ 3) or improvements over previous approximations achievable in comparable times (for unit disks in the plane). (I) Given a set of n hyperplanes in Rd, a traveling salesman problem (TSP) tour whose length is at most O(1) times the optimal can be computed in O(n) time when d is constant. (II) Given a set of n lines in Rd, a TSP tour whose length is at most O(log 3n) times the optimal can be computed in polynomial time for all d. (III) Given a set of n unit balls in Rd, a TSP tour whose length is at most O(1) times the optimal can be computed in polynomial time when d is constant."
- Resource Type:
- Article
- Identifier:
- 1549-6325
- Campus Tesim:
- Northridge
- Creator:
- Dumitrescu, Adrian, Minghu, Jiang, and Toth, Csaba D.
- Description:
- The problem of finding a collection of curves of minimum total length that meet all the lines intersecting a given planar convex body was initiated by Mazurkiewicz in 1916. Such a collection forms an opaque barrier for the convex body. 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, but 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.
- Resource Type:
- Article
- Identifier:
- 0895-4801
- Campus Tesim:
- Northridge