FITTING FLATS TO POINTS WITH OUTLIERS

2011 ◽  
Vol 21 (05) ◽  
pp. 559-569
Author(s):  
GUILHERME D. DA FONSECA
Keyword(s):  
Lower Bound ◽  
Computer Science ◽  
Upper Bound ◽  
Arbitrary Constant ◽  
Fundamental Problem ◽  
Constant Factor ◽  
Infinite Cylinder ◽  
Running Time ◽  
Point Set ◽  
Set Of Points

Determining the best shape to fit a set of points is a fundamental problem in many areas of computer science. We present an algorithm to approximate the k-flat that best fits a set of n points with n - m outliers. This problem generalizes the smallest m-enclosing ball, infinite cylinder, and slab. Our algorithm gives an arbitrary constant factor approximation in O(nk+2/m) time, regardless of the dimension of the point set. While our upper bound nearly matches the lower bound, the algorithm may not be feasible for large values of k. Fortunately, for some practical sets of inliers, we reduce the running time to O(nk+2/mk+1), which is linear when m = Ω(n).

scholarly journals Finding Points in General Position

2017 ◽  
Vol 27 (04) ◽  
pp. 277-296 ◽  
Author(s):  
Vincent Froese ◽  
Iyad Kanj ◽  
André Nichterlein ◽  
Rolf Niedermeier
Keyword(s):  
Lower Bound ◽  
General Position ◽  
Subset Selection ◽  
Selection Problem ◽  
Fixed Parameter Tractability ◽  
Maximum Cardinality ◽  
Exponential Time ◽  
Running Time ◽  
Fixed Parameter ◽  
Set Of Points

We study the General Position Subset Selection problem: Given a set of points in the plane, find a maximum-cardinality subset of points in general position. We prove that General Position Subset Selection is NP-hard, APX-hard, and present several fixed-parameter tractability results for the problem as well as a subexponential running time lower bound based on the Exponential Time Hypothesis.

scholarly journals Optimal Time-Space Trade-Offs for Sorting

1998 ◽  
Vol 5 (10) ◽  
Author(s):  
Jakob Pagter ◽  
Theis Rauhe
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Fundamental Problem ◽  
Full Range ◽  
Optimal Time ◽  
Sorting Algorithm ◽  
Logarithmic Factor ◽  
Time Space ◽  
Trade Offs ◽  
Space Product

We study the fundamental problem of sorting in a sequential model of computation and in particular consider the time-space trade-off (product of time and space) for this problem.<br />Beame has shown a lower bound of  Omega(n^2) for this product leaving a gap of a logarithmic factor up to the previously best known upper bound of O(n^2 log n) due to Frederickson. Since then, no progress has been made towards tightening this gap.<br />The main contribution of this paper is a comparison based sorting algorithm which closes this gap by meeting the lower bound of Beame. The time-space product O(n^2) upper bound holds for the full range of space bounds between log n and n/log n. Hence in this range our algorithm is optimal for comparison based models as well as for the very powerful general models considered by Beame.

PARAMETRIC POLYMATROID OPTIMIZATION AND ITS GEOMETRIC APPLICATIONS

2002 ◽  
Vol 12 (05) ◽  
pp. 429-443 ◽  
Author(s):  
NAOKI KATOH ◽  
HISAO TAMAKI ◽  
TAKESHI TOKUYAMA
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Transportation Problem ◽  
Minimum Weight ◽  
Planar Point ◽  
Line Arrangement ◽  
Optimal Bound ◽  
Straight Line ◽  
Point Set ◽  
Multiple Levels

We give an optimal bound on the number of transitions of the minimum weight base of an integer valued parametric polymatroid. This generalizes and unifies Tamal Dey's O(k1/3 n) upper bound on the number of k-sets (and the complexity of the k-level of a straight-line arrangement), David Eppstein's lower bound on the number of transitions of the minimum weight base of a parametric matroid, and also the Θ(kn) bound on the complexity of the at-most-k level (the union of i-levels for i = 1,2,…,k) of a straight-line arrangement. As applications, we improve Welzl's upper bound on the sum of the complexities of multiple levels, and apply this bound to the number of different equal-sized-bucketings of a planar point set with parallel partition lines. We also consider an application to a special parametric transportation problem.

scholarly journals On Cartesian Products which Determine Few Distinct Distances

10.37236/7736 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Cosmin Pohoata
Keyword(s):  
Lower Bound ◽  
Cartesian Products ◽  
Point Set ◽  
Set Of Points

Every set of points $\mathcal{P}$ determines $\Omega(|\mathcal{P}| / \log |\mathcal{P}|)$ distances. A close version of this was initially conjectured by Erdős in 1946 and rather recently proved by Guth and Katz. We show that when near this lower bound, a point set $\mathcal{P}$ of the form $A \times A$ must satisfy $|A - A| \ll |A|^{2-\frac{2}{7}} \log^{\frac{1}{7}} |A|$. This improves recent results of Hanson and Roche-Newton. 

scholarly journals Counting Plane Graphs: Cross-Graph Charging Schemes

2013 ◽  
Vol 22 (6) ◽  
pp. 935-954 ◽  
Author(s):  
MICHA SHARIR ◽  
ADAM SHEFFER
Keyword(s):  
Upper Bound ◽  
Planar Graphs ◽  
Upper Bounds ◽  
Straight Edge ◽  
Specific Point ◽  
Plane Graphs ◽  
Fixed Set ◽  
Point Set ◽  
Vertex Degrees ◽  
Set Of Points

We study cross-graph charging schemes for graphs drawn in the plane. These are charging schemes where charge is moved across vertices of different graphs. Such methods have recently been used to obtain various properties of triangulations that are embedded in a fixed set of points in the plane. We generalize this method to obtain results for various other types of graphs that are embedded in the plane. Specifically, we obtain a new bound ofO*(187.53N) (where theO*(⋅) notation hides polynomial factors) for the maximum number of crossing-free straight-edge graphs that can be embedded in any specific set ofNpoints in the plane (improving upon the previous best upper bound 207.85Nin Hoffmann, Schulz, Sharir, Sheffer, Tóth and Welzl [14]). We also derive upper bounds for numbers of several other types of plane graphs (such as connected and bi-connected plane graphs), and obtain various bounds on the expected vertex-degrees in graphs that are uniformly chosen from the set of all crossing-free straight-edge graphs that can be embedded in a specific point set.We then apply the cross-graph charging-scheme method to graphs that allow certain types of crossings. Specifically, we consider graphs with no set ofkpairwise crossing edges (more commonly known ask-quasi-planar graphs). Fork=3 andk=4, we prove that, for any setSofNpoints in the plane, the number of graphs that have a straight-edgek-quasi-planar embedding overSis only exponential inN.

scholarly journals Induced Ramsey Number for a Star Versus a Fixed Graph

10.37236/9358 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Maria Axenovich ◽  
Izolda Gorgol
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Upper Bound ◽  
Constant Factor ◽  
Ramsey Number ◽  
Large N ◽  
Star Versus

We write $F{\buildrel {\text{ind}} \over \longrightarrow}(H,G)$ for graphs $F, G,$ and $H$, if for any coloring of the edges of $F$ in red and blue, there is either a red induced copy of $H$ or a blue induced copy of $G$. For graphs $G$ and $H$, let $\mathrm{IR}(H,G)$ be the smallest number of vertices in a graph $F$ such that $F{\buildrel {\text{ind}} \over \longrightarrow}(H,G)$. In this note we consider the case when $G$ is a star on $n$ edges, for large $n$ and $H$ is a fixed graph. We prove that  $$ (\chi(H)-1) n \leq \mathrm{IR}(H, K_{1,n}) \leq (\chi(H)-1)^2n + \epsilon n,$$ for any $\epsilon>0$,  sufficiently large $n$, and $\chi(H)$ denoting the chromatic number of $H$. The lower bound is asymptotically tight  for any fixed bipartite $H$. The upper bound is attained up to a constant factor, for example when $H$ is a clique.

State Complexity of k-Union and k-Intersection for Prefix-Free Regular Languages

2015 ◽  
Vol 26 (02) ◽  
pp. 211-227 ◽  
Author(s):  
Hae-Sung Eom ◽  
Yo-Sub Han ◽  
Kai Salomaa
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Finite Automata ◽  
Constant Factor ◽  
The State ◽  
Regular Languages ◽  
Deterministic Finite Automata ◽  
State Complexity ◽  
Binary Alphabet ◽  
Multiple Intersections

We investigate the state complexity of multiple unions and of multiple intersections for prefix-free regular languages. Prefix-free deterministic finite automata have their own unique structural properties that are crucial for obtaining state complexity upper bounds that are improved from those for general regular languages. We present a tight lower bound construction for k-union using an alphabet of size k + 1 and for k-intersection using a binary alphabet. We prove that the state complexity upper bound for k-union cannot be reached by languages over an alphabet with less than k symbols. We also give a lower bound construction for k-union using a binary alphabet that is within a constant factor of the upper bound.

scholarly journals A Lower Bound on the Average Degree Forcing a Minor

10.37236/8847 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Sergey Norin ◽  
Bruce Reed ◽  
Andrew Thomason ◽  
David R. Wood
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Constant Factor ◽  
Average Degree ◽  
Complete Graphs ◽  
Sparse Graphs ◽  
Dense Graphs ◽  
A Minor

We show that for sufficiently large $d$ and for $t\geq d+1$,  there is a graph $G$ with average degree $(1-\varepsilon)\lambda  t \sqrt{\ln d}$ such that almost every graph $H$ with $t$ vertices and average degree $d$ is not a minor of $G$, where $\lambda=0.63817\dots$ is an explicitly defined constant. This generalises analogous results for complete graphs by Thomason (2001) and for general dense graphs by Myers and Thomason (2005). It also shows that an upper bound for sparse graphs by Reed and Wood (2016) is best possible up to a constant factor.

scholarly journals A Bound on a Convexity Measure for Point Sets

2019 ◽  
Vol 29 (04) ◽  
pp. 301-306
Author(s):  
Danny Rorabaugh
Keyword(s):  
Upper Bound ◽  
Interior Angle ◽  
Point Sets ◽  
Planar Point ◽  
Angle Measure ◽  
Convex Position ◽  
Point Set ◽  
Set Of Points ◽  
Convexity Measure

A planar point set is in convex position precisely when it has a convex polygonization, that is, a polygonization with maximum interior angle measure at most [Formula: see text]. We can thus talk about the convexity of a set of points in terms of its min-max interior angle measure. The main result presented here is a nontrivial upper bound of the min-max value in terms of the number of points in the set. Motivated by a particular construction, we also pose a natural conjecture for the best upper bound.

IMPROVED BOUNDS FOR THE CROSSING NUMBER OF THE MESH OF TREES

2003 ◽  
Vol 04 (01) ◽  
pp. 17-35 ◽  
Author(s):  
ROBERT CIMIKOWSKI ◽  
IMRICH VRT'O
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Crossing Number ◽  
Constant Factor ◽  
Mesh Of Trees

Improved bounds for the crossing number of the mesh of trees graph, Mn, are derived. In particular, we derive a new lower bound of [Formula: see text] which improves on the previous bound of Leighton [11] by a constant factor, and an upper bound of [Formula: see text]. In addition, we construct drawings of Mn which achieve the upper bound number of crossings. We also prove that the crossing number of M4 is 4.

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