K-set (geometry)

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Points separated from others by a line

title: "K-set (geometry)" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["discrete-geometry", "matroid-theory"] description: "Points separated from others by a line" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/K-set_(geometry)" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Points separated from others by a line ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/d/db/k-sets.svg" caption="A set of six points (red), its six 2-sets (the sets of points contained in the blue ovals), and lines separating each k-set from the remaining points (dashed black)."] ::

In discrete geometry, a k-set of a finite point set S in the Euclidean plane is a subset of k elements of S that can be strictly separated from the remaining points by a line. More generally, in Euclidean space of higher dimensions, a k-set of a finite point set is a subset of k elements that can be separated from the remaining points by a hyperplane. In particular, when k=n/2 (where n is the size of S), the line or hyperplane that separates a k-set from the rest of S is a halving line or halving plane.

The k-sets of a set of points in the plane are related by projective duality to the k-levels in an arrangement of lines. The k-level in an arrangement of n lines in the plane is the curve consisting of the points that lie on one of the lines and have exactly k lines below them. Discrete and computational geometers have also studied levels in arrangements of more general kinds of curves and surfaces.

Combinatorial bounds

It is of importance in the analysis of geometric algorithms to bound the number of k-sets of a planar point set, or equivalently the number of k-levels of a planar line arrangement, a problem first studied by Lovász and Erdős et al. The best known upper bound for this problem is O(nk^{1/3}), as was shown by Tamal Dey using the crossing number inequality of Ajtai, Chvátal, Newborn, and Szemerédi. However, the best known lower bound is far from Dey's upper bound: it is \Omega(nc^{\sqrt{\log k}}) for some constant c, as shown by Tóth.

In three dimensions, the best upper bound known is O(nk^{3/2}), and the best lower bound known is \Omega(nkc^{\sqrt{\log k}}). For points in three dimensions that are in convex position, that is, are the vertices of some convex polytope, the number of k-sets is \Theta\bigl((n-k)k\bigr), which follows from arguments used for bounding the complexity of kth order Voronoi diagrams.

Bounds have also been proven on the number of \le k-sets, where a \le k-set is a j-set for some j\le k. In two dimensions, the maximum number of \le k-sets is exactly nk, while in d dimensions the bound is O(n^{\lfloor d/2\rfloor}k^{\lceil d/2\rceil}).

Halving lines

There are two separate but related ways in which halving lines are defined. The first simply separates half of the points of S from the other. The second is similar, but each line must pass through 2 points in S, and equally bisect the remaining points. The different definitions directly translate to each other in the dual line arrangement, where halving lines become dual to points. This is because, given an alternating sequence of cells and crossings in the dual arrangement, there will always be exactly one more cell than crossing.

For the case when k=n/2 (halving lines), the maximum number h_n of combinatorially distinct lines through two points of S that bisect the remaining points when k=1,2,\dots is For odd n, rather than exactly bisecting the points, one half should have \frac{n-1}{2} points and the other should have \frac{n-3}{2}.{{Cite journal | last1 = Bereg | first1 = Sergey | last2 = Haghpanah | first2 = Mohammadreza | title = New algorithms and bounds for halving pseudolines | journal = Discrete Applied Mathematics | volume = 319 | pages = 194–206 | date = 15 October 2022 | doi = 10.1016/j.dam.2021.05.029 | url = https://doi.org/10.1016/j.dam.2021.05.029 | access-date = 2025-07-07 | url-access = subscription | last1 = Alonso | first1 = E. | last2 = López | first2 = M. | last3 = Rodrigo | first3 = J. | title = An Improvement of the Upper Bound for the Number of Halving Lines of Planar Sets | journal = Symmetry | volume = 16 | issue = 7 | pages = 936 | date = 2024 | doi = 10.3390/sym16070936 | bibcode = 2024Symm...16..936A | doi-access = free

A weaker case of halving pseudolines in the projective plane is also studied, acting on an arrangement of pseudolines rather than a line arrangement. This problem relies more on combinatorics and less on geometry, but results from the weaker case carry over whenever the arrangement is stretchable, that is, can have its pseudolines straightened into lines without changing their combinatorial properties.

A conjecture exists for the relationship between halving lines and the rectilinear crossing number \overline{cr}(n) for a set of n points, which states that, for each arrangement of points in which \overline{cr}(n) is minimized, the number of halving lines is maximized.{{cite journal | last1 = Aichholzer | first1 = Oswin | last2 = García | first2 = Jesús | last3 = Orden | first3 = David | last4 = Ramos | first4 = Pedro | title = New Lower Bounds for the Number of (≤ k)-Edges and the Rectilinear Crossing Number of Kn | journal = Discrete & Computational Geometry | volume = 38 | issue = 1 | pages = 1–14 | date = 2007 | doi = 10.1007/s00454-007-1325-8 | url = https://link.springer.com/content/pdf/10.1007/s00454-007-1325-8.pdf | access-date = 22 July 2025

Construction algorithms

Edelsbrunner and Welzl first studied the problem of constructing all k-sets of an input point set, or dually of constructing the k-level of an arrangement. The k-level version of their algorithm can be viewed as a plane sweep algorithm that constructs the level in left-to-right order. Viewed in terms of k-sets of point sets, their algorithm maintains a dynamic convex hull for the points on each side of a separating line, repeatedly finds a bitangent of these two hulls, and moves each of the two points of tangency to the opposite hull. Chan surveys subsequent results on this problem, and shows that it can be solved in time proportional to Dey's O(nk^{1/3}) bound on the complexity of the k-level.

Agarwal and Matoušek describe algorithms for efficiently constructing an approximate level; that is, a curve that passes between the (k-\delta)-level and the (k+\delta)-level for some small approximation parameter \delta. They show that such an approximation can be found, consisting of a number of line segments that depends only on n/\delta and not on n or k.

Matroid generalizations

The planar k-level problem can be generalized to one of parametric optimization in a matroid: one is given a matroid in which each element is weighted by a linear function of a parameter \lambda, and must find the minimum weight basis of the matroid for each possible value of \lambda. If one graphs the weight functions as lines in a plane, the k-level of the arrangement of these lines graphs as a function of \lambda the weight of the largest element in an optimal basis in a uniform matroid, and Dey showed that his O(nk^{1/3}) bound on the complexity of the k-level could be generalized to count the number of distinct optimal bases of any matroid with n elements and rank k.

For instance, the same O(nk^{1/3}) upper bound holds for counting the number of different minimum spanning trees formed in a graph with n edges and k vertices, when the edges have weights that vary linearly with a parameter \lambda. This parametric minimum spanning tree problem has been studied by various authors and can be used to solve other bicriterion spanning tree optimization problems.

However, the best known lower bound for the parametric minimum spanning tree problem is \Omega(n\log k), a weaker bound than that for the k-set problem. For more general matroids, Dey's O(nk^{1/3}) upper bound has a matching lower bound.

Examples of halving lines

Image:Halving lines for 2 points, no intersect.svg|2 points, 1 line Image:Halving lines for 4 points, no intersect.svg|4 points, 3 lines Image:Halving lines for 6 points, no intersect.svg|6 points, 6 lines Image:Halving lines for 8 points, no intersect.svg|8 points, 9 lines

Notes

References

  • {{cite journal | last = Agarwal | first = P. K. | author-link = Pankaj K. Agarwal | title = Partitioning arrangements of lines I: An efficient deterministic algorithm | journal = Discrete & Computational Geometry | volume = 5 | issue = 1 | year = 1990 | pages = 449–483 | doi = 10.1007/BF02187805 | doi-access = free
  • {{cite conference | last1 = Agarwal | first1 = P. K. | author1-link = Pankaj K. Agarwal | last2 = Aronov | first2 = B. | author2-link = Boris Aronov | last3 = Sharir | first3 = M. | author3-link = Micha Sharir | title = On levels in arrangements of lines, segments, planes, and triangles | book-title = Proc. 13th Annual Symposium on Computational Geometry | year = 1997 | pages = 30–38}}
  • {{cite journal | last1 = Alon | first1 = N. | author1-link = Noga Alon | last2 = Győri | first2 = E. | title = The number of small semi-spaces of a finite set of points in the plane | journal = Journal of Combinatorial Theory | series=Series A | volume = 41 | pages = 154–157 | year = 1986 | doi = 10.1016/0097-3165(86)90122-6| doi-access = free}}
  • {{cite web |last = Chan |first = T. M. |author-link = Timothy M. Chan |title = Remarks on k-level algorithms in the plane |year = 1999 |url = http://www.cs.uwaterloo.ca/~tmchan/lev2d_7_7_99.ps.gz |url-status = dead |archive-url = https://web.archive.org/web/20101104182509/http://www.cs.uwaterloo.ca/~tmchan/lev2d_7_7_99.ps.gz |archive-date = 2010-11-04
  • {{cite journal | last = Chan | first = T. M. | author-link = Timothy M. Chan | title = On levels in arrangements of curves | journal = Discrete & Computational Geometry | volume = 29 | pages = 375–393 | year = 2003 | doi = 10.1007/s00454-002-2840-2 | issue = 3 | doi-access = free
  • {{cite journal | last = Chan | first = T. M. | author-link = Timothy M. Chan | title = On levels in arrangements of curves, II: a simple inequality and its consequence | journal = Discrete & Computational Geometry | volume = 34 | pages = 11–24 | year = 2005a | doi = 10.1007/s00454-005-1165-3 | doi-access = free
  • {{cite conference | last = Chan | first = T. M. | author-link = Timothy M. Chan | title = On levels in arrangements of surfaces in three dimensions | book-title = Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms | pages = 232–240 | year = 2005b | url = http://www.cs.uwaterloo.ca/~tmchan/surf_soda.ps}}
  • {{cite conference | last = Chan | first = T. M. | author-link = Timothy M. Chan | title = Finding the shortest bottleneck edge in a parametric minimum spanning tree | book-title = Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms | pages = 232–240 | year = 2005c | url = http://www.cs.uwaterloo.ca/~tmchan/bottle_soda.ps}}
  • {{cite journal | last1 = Chazelle | first1 = B. | author1-link = Bernard Chazelle | last2 = Preparata | first2 = F. P. | author2-link = Franco Preparata | title = Halfspace range search: an algorithmic application of k-sets | journal = Discrete & Computational Geometry | volume = 1 | issue = 1 | year = 1986 | pages = 83–93 |mr=0824110 | doi = 10.1007/BF02187685 | doi-access = free
  • {{cite journal | last1 = Clarkson | first1 = K. L. | author1-link = Kenneth L. Clarkson | last2 = Shor | first2 = P. | author2-link = Peter Shor | title = Applications of random sampling, II | journal = Discrete & Computational Geometry | volume = 4 | pages = 387–421 | year = 1989 | doi = 10.1007/BF02187740 | doi-access = free
  • {{cite journal | last1 = Cole | first1 = R. | last2 = Sharir | first2 = M. | author2-link = Micha Sharir | last3 = Yap | first3 = C. K. | title = On k-hulls and related problems | journal = SIAM Journal on Computing | volume = 16 | issue = 1 | year = 1987 | pages = 61–77 |mr=0873250 | doi = 10.1137/0216005}}
  • {{cite journal | last = Dey | first = T. K. | author-link = Tamal Dey | title = Improved bounds for planar k-sets and related problems | journal = Discrete & Computational Geometry | volume = 19 | issue = 3 | year = 1998 | pages = 373–382 | doi = 10.1007/PL00009354 | mr=1608878 | doi-access = free
  • {{cite journal | last1 = Edelsbrunner | first1 = H. | author1-link = Herbert Edelsbrunner | last2 = Welzl | first2 = E. | author2-link = Emo Welzl | title = Constructing belts in two-dimensional arrangements with applications | journal = SIAM Journal on Computing | volume = 15 | issue = 1 | year = 1986 | pages = 271–284 | doi = 10.1137/0215019}}
  • {{cite journal | title = Geometric lower bounds for parametric matroid optimization | last = Eppstein | first = D. | author-link = David Eppstein | journal = Discrete & Computational Geometry | volume = 20 | pages = 463–476 | year = 1998 | url = http://www.ics.uci.edu/~eppstein/pubs/Epp-DCG-98.pdf | doi = 10.1007/PL00009396 | doi-access = free | issue = 4}}
  • {{cite journal | last = Eppstein | first = David | date = August 2022 | doi = 10.1007/s00453-022-01024-9 | journal = Algorithmica | title = A stronger lower bound on parametric minimum spanning trees| volume = 85 | issue = 6 | pages = 1738–1753 | doi-access = free | arxiv = 2105.05371
  • {{cite conference | last1=Erdős | first1=P. | authorlink1=Paul Erdős | last2=Lovász | first2=L. | authorlink2=László Lovász | last3=Simmons | first3=A. | last4=Straus | first4=E. G. | authorlink4=Ernst G. Straus | title = Dissection graphs of planar point sets | book-title = A Survey of Combinatorial Theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) | publisher = North-Holland | location = Amsterdam | date = 1973 | pages = 139–149 |mr=0363986}}
  • {{cite journal | title = Using sparsification for parametric minimum spanning tree problems | last1=Fernández-Baca | first1=D. | last2=Slutzki | first2=G. | last3=Eppstein | first3=D. | authorlink3=David Eppstein | journal = Nordic Journal of Computing | volume = 3 | issue = 4 | pages = 352–366 | year = 1996}}
  • {{cite book | last1 = Gusfield | first1 = D. | title = Sensitivity analysis for combinatorial optimization | series = Tech. Rep. UCB/ERL M80/22 | publisher = University of California, Berkeley | date = 1980}}
  • {{cite journal | last1=Hassin | first1=R. | last2=Tamir | first2=A. | title=Maximizing classes of two-parametric objectives over matroids | journal=Mathematics of Operations Research | volume=14 | pages=362–375 | year=1989 | doi=10.1287/moor.14.2.362 | issue=2}}
  • {{cite journal | last1=Ishii | first1=H. | last2=Shiode | first2=S. | last3=Nishida | first3=T. | title = Stochastic spanning tree problem | journal = Discrete Applied Mathematics | volume = 3 | pages = 263–273 | year = 1981 | doi = 10.1016/0166-218X(81)90004-4 | issue = 4| doi-access = free
  • {{cite book | last1=Katoh | first1=N. | last2=Ibaraki | first2=T.|author2-link=Toshihide Ibaraki | title = On the total number of pivots required for certain parametric combinatorial optimization problems | series = Working Paper 71 | publisher = Inst. Econ. Res., Kobe Univ. of Commerce | date = 1983}}
  • {{cite journal | last = Lee | first = Der-Tsai | author-link = Der-Tsai Lee | title = On k-nearest neighbor Voronoi diagrams in the plane | journal = IEEE Transactions on Computers | volume = 31 | year = 1982 | pages = 478–487 | doi = 10.1109/TC.1982.1676031 | issue = 6}}
  • {{cite journal | last1=Lovász | first1 =L. | authorlink1=László Lovász | title = On the number of halving lines | journal = Annales Universitatis Scientiarum Budapestinensis de Rolando Eőtvős Nominatae Sectio Mathematica | volume = 14 | year = 1971 | pages = 107–108}}
  • {{cite journal | last = Matoušek | first = J. | author-link = Jiří Matoušek (mathematician) | title = Construction of ε-nets | journal = Discrete & Computational Geometry | volume = 5 | issue = 5 | year = 1990 | pages = 427–448 |mr=1064574 | doi = 10.1007/BF02187804| doi-access = free
  • {{cite journal | last = Matoušek | first = J. | author-link = Jiří Matoušek (mathematician) | title = Approximate levels in line arrangements | journal = SIAM Journal on Computing | volume = 20 | issue = 2 | pages = 222–227 | year = 1991 | doi = 10.1137/0220013}}
  • {{cite journal | last1=Sharir | first1=M. | authorlink1=Micha Sharir | last2=Smorodinsky | first2=S. | last3=Tardos | first3=G. | title = An improved bound for k-sets in three dimensions | journal = Discrete & Computational Geometry | volume = 26 | year = 2001 | issue=2 | pages = 195–204 | doi = 10.1007/s00454-001-0005-3| doi-access = free
  • {{cite journal | last1 = Tóth | first1 = G. | title = Point sets with many k-sets | journal = Discrete & Computational Geometry | volume = 26 | issue = 2 | pages = 187–194 | year = 2001 | doi = 10.1007/s004540010022| doi-access = free

References

  1. {{harvtxt. Agarwal. Aronov. Sharir. 1997; {{harvtxt. Chan. 2003; {{harvtxt. Chan. 2005a; {{harvtxt. Chan. 2005b.
  2. {{harvtxt. Chazelle. Preparata. 1986; {{harvtxt. Cole. Sharir. Yap. 1987; {{harvtxt. Edelsbrunner. Welzl. 1986.
  3. {{harvtxt. Lee. 1982; {{harvtxt. Clarkson. Shor. 1989.
  4. [https://oeis.org/A076523/a076523_1.svg Triangular Convex Hull]
  5. {{harvtxt. Agarwal. 1990; {{harvtxt. Matoušek. 1990; {{harvtxt. Matoušek. 1991.
  6. {{harvtxt. Gusfield. 1980; {{harvtxt. Ishii. Shiode. Nishida. 1981; {{harvtxt. Katoh. Ibaraki. 1983; {{harvtxt. Hassin. Tamir. 1989; {{harvtxt. Fernández-Baca. Slutzki. Eppstein. 1996; {{harvtxt. Chan. 2005c.

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