Let S be a set of n points in Rd in general position. A set H of k-flats is called an mk-stabber of S if the relative interior of any m-simplex with vertices in S
is intersected by at least one element of H. In this paper we give lower and upper bounds on the size of minimum mk-stabbers of point sets in Rd. We study mainly mk-stabbers in the plane and in R3.Consejo Nacional de Ciencia y Tecnología (México)Ministerio de Economía y CompetitividadGeneralitat de CatalunyaEuropean Science FoundationMinisterio de Ciencia e Innovació

Cano Vila, Javier

Díaz Báñez, José Miguel (Coordinador)

Garijo Royo, Delia (Coordinador)

Hurtado Díaz, Ferran

Márquez Pérez, Alberto (Coordinador)

Urrutia Galicia, Jorge

Urrutia Galicia, Jorge (Coordinador)

English

idUS. Depósito de Investigación Universidad de Sevilla

XV Spanish Meeting on Computational Geometry, June 26-28, 2013Stabbing simplices of point sets with k-flatsJavier Cano∗1, Ferran Hurtado†2, and Jorge Urrutia‡11Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico City, Mexico.2Departament de Matemàtica Aplicada, Universitat Politécnica de Catalunya, Barcelona, Spain.AbstractLet S be a set of n points in Rd in general position.A set H of k-flats is called an mk-stabber of S if therelative interior of any m-simplex with vertices in Sis intersected by at least one element of H. In thispaper we give lower and upper bounds on the size ofminimum mk-stabbers of point sets in Rd. We studymainly mk-stabbers in the plane and in R3.IntroductionA set {x0, . . . , xk} ∈ Rd of k+1 points is called linearlyindependent if there is no linear combination λ0x0 +· · · + λkxk = 0 in which at least one λi 6= 0, k ≤ d.A set of points {x0, . . . , xk} ∈ Rd is called affinelyindependent if the set {x1−x0, . . . , xk−x0} is linearlyindependent, k ≥ 1. A set consisting of a single pointwill also be considered as affinely independent. Anaffine combination of a set of k+1 affinely independentpoints in Rd is a linear combination λ0x0 + · · ·+λkxkof x0, . . . , xk such that λ0 + · · · + λk = 1. The setof affine combinations of k + 1 affinely independentpoints of Rd is called a k-flat. In particular, a (d−1)-flat of Rd will be called a hyperplane. Following ourintuition, a 1-flat is a line, and a 2-flat is a plane. Ak-flat is isomorphic to the k-dimensional space Rk. Apoint set S in Rd is in general position if any subset ofS with at most d+1 elements is affinely independent.An m-simplex of Rd is the convex hull of a set ofm + 1 affinely independent points in Rd, m ≤ d. Forexample, in R2 a 0-simplex is a point, a 1-simplex isa segment and a 2-simplex is a triangle.Given a k-flat h and an m-simplex P of Rd, we saythat h stabs P if h intersects the relative interior of P.A set of k-flats H is called an mk-stabber of a set ofpoints S if every m-simplex induced by the elements∗Email: jcano@ciencias.unam.mx. Research supported byproject number 178379 Conacyt, México.†Email: ferran.hurtado@upc.edu. Research partially sup-ported by projects MINECO277 MTM2012-30951, Gen. Cat.DGR2009SGR1040, and ESF EUROCORES programme Eu-roGIGA, CRP ComPoSe IP04: MICINN Project EUI-EURC-2011-4306.‡Email: urrutia@matem.unam.mx. Research supported byproject number 178379 Conacyt, México..of S is stabbed by at least one element of H.For example, in the plane a set of points Q (respec-tively lines) is a 30-stabber (respectively a 31-stabber)of point set S if any triangle with vertices in S con-tains an element of Q in its interior (respectively isintersected by a line in Q).In this paper we study the following problem: Giventwo integers k < m, k < d, and m ≤ d+ 1, and a setS of n points in Rd in general position, how many k-flats are needed to stab all the m-simplices generatedby the elements of S?In this paper we focus mainly on stabbing all ther-simplices of point sets on the plane or in R3, andgive some results for higher dimensions.The problem of finding sets of points that staball the triangles of a point set has been studied byKatchalsky and Meir [4], and independently by Czy-zowicz, Kranakis, and Urrutia [2]. They prove thatfor any point set S in the plane in general positionwith n elements, such that its convex hull has c ele-ments, the set of triangles of S can be stabbed withexactly 2n − c − 2 points, and that such a bound istight, as any triangulation of S has 2n−c−2 triangles.Stabbers for other convex holes, such as quadrilateralsand pentagons, have also been studied [1].Given a point set S in Rd, we define fmk (S) as thesize of the smallest mk-stabber of S. We define fmk (n)as the largest value that a mk-stabber can have overall the point sets S of Rd with n elements. With thisterminology, the preceding result in the plane trans-lates to f20 (S) = 2n− c− 2.In this paper we show upper and lower bounds forfmk (n), for point sets on the plane and R3 that aretight up to a constant.1 Well-separated sets and gener-alized ham-sandwich cutsA family of convex sets C0, . . . , Ck of Rd, k ≤ d, iswell separated if for any choice of xi ∈ Ci, the set ofpoints {x0, . . . , xk} is affinely independent. A familyP0, . . . , Pk of finite point sets in Rd is well separatedif their convex hulls Conv(P0), . . . ,Conv(Pk) are wellseparated, k ≤ d.91Stabbing simplices with k-flatsNotice that in particular, this implies that for anypair of different indexes i, j ≤ k, the convex hulls ofPi and Pj do not intersect.Let P0, . . . , Pk be a family of pairwise disjoint finitepoint sets in Rd, k ≤ d. Given positive integers ai ≤|Pi|, an (a0, . . . , ak)-cut is a hyperplane h for whichh ∩ Pi 6= ∅ and |h+ ∩ Pi| = ai, 0 ≤ i ≤ d, where h+ isthe half-space bounded below by h.The following result by Steiger and Zao will be use-ful to us:Theorem 1 ([8]) Let P0, . . . , Pd be well-separatedpoint sets in Rd such that P0 ∪ · · · ∪ Pd is in generalposition, and let a0, . . . , ad be positive integers suchthat ai ≤ |Pi|. Then there is a unique (a0, . . . , ad)-cutof P0, . . . , Pd.Finally, we recall the almost folklore result aboutham-sandwich cuts:Theorem 2 (Ham-sandwich theorem, [9])Every d finite sets in Rd can be simultaneouslybisected by a hyperplane. A hyperplane h bisects afinite set P if each of the open half-spaces defined byh contains at most b |P |2 c points of P .Now we have all the tools that we will use to givean algorithm for constructing a worst-case optimalmk-stabber for point sets in the plane and in the 3-dimensional space.2 Stabbers in the planeWe start by studying the following problem: Howmany lines are necessary to stab the set of line tri-angles (segments) determined by triples (pairs) of el-ements of a point set S in the plane? In our previousterminology, determine lower and upper bounds forf11 (n) and f21 (n) for point sets on the plane. We firstprove:Theorem 3 For any set S of n points in the planer ≤ f21 (S) ≤ dn4 e, where r is the smallest number suchthat n2 ≤ 2 + 2 + 3 + · · ·+ r. Our bounds are tight.Proof. The lower bound follows from the fact that rlines, no three of which intersect at a point, divide theplane into 2 + 2 + 3 + · · ·+ r convex regions, and if aset of r lines stabs all of the triangles with vertices inS, then S has at most two elements in each of theseregions.We now prove that f21 (S) ≤ dn4 e. To this end, wenow show how to obtain a set H with dn4 e straight-lines that is a 21-stabber of S. To prove our result, itis sufficient to find a set H with dn4 e lines such thatany cell of the arrangement generated by H containsat most two elements of S.First, we put in H one straight line ` that separatesS into two subsets of size dn2 e and bn2 c respectively.Refer to these sets as S1 and S2. Clearly, these sets arewell separated, and by Theorem 1 we can find a (2, 2)-cut for S1 and S2. This cut is a line `1 that leavestwo elements x1, x2 of S1 and two elements x′1, x′2 ofS2 above it. These points are separated from the re-maining points of S by ` and `1, if `1 contains someelement of S1 or S2 move it slightly. Thus any trian-gle containing at least one vertex in {x1, x2, x′1, x′2} isintersected by ` or `1.We repeat this recursively on S1 \{x1, x2} and S2 \{x′1, x′2}, dn4 e − 2 times, obtaining a set of dn4 e lines(including ` and `1) that is a 21-stabber of S.To show that our bound is tight, let S be aset of n points in the plane in convex position la-beled p0, p1, . . . , pn−1 in counterclockwise order, withn even. Consider the set of n2 triangles pipi+1pi+2, ieven, and i ≤ n2 for every even value of i. Observethat any two of these triangles intersect at exactly onevertex, for n ≥ 6.We claim that any straight line stabs at most 2 ofthese triangles. This is true since every such trianglehas exactly two edges of the convex hull of Pn; thenany straight line stabbing one such triangle stabs atleast one edge of the convex hull. Any straight linestabs at most two edges of the convex hull, so then itcan stab at most two of such triangles. Thus to staball of those triangles we need at least one straightline for every two triangles. Observe next that if theelements of S are in convex position, then to stab theedges of the convex hull of S we need at last dn4 e lines.Thus our upper bound is tight. The key property of H in the proof of this result isthe fact that any cell of the arrangement induced byH contains at most two elements of S. Observe thatif instead of finding a (2, 2)-cut for S1 and S2, we use(m− 1,m− 1)-cuts, then we obtain a set of lines suchthat any cell of the arrangement generated by themcontains at most m − 1 elements of S. An m islandof S is a subset S′ of S with exactly m elements andsuch that the convex hull of S′ contains no elementof S \ S′ in its convex hull. The next result followseasily from the proof of Theorem 3.Theorem 4 For any set S of n points in the plane,d n2(m−1)e lines are always sufficient and sometimesnecessary to stab all of the m islands of S.In particular, for m = 2, this proves that f11 (S) ≤dn2 e; that is, to stab the segments of S we need at mostdn2 e lines. It is easy to see that the bound dn2(m−1)ein Theorem 4 is achieved for point sets in convex po-sition.We close this section by observing that the problemof calculating f11 (S) is equivalent to calculating theminimum number of lines such that in each face of92XV Spanish Meeting on Computational Geometry, June 26-28, 2013the arrangement defined by these lines, there is atmost one element of S. This problem is known as theshattering problem and its decision version is knownto be NP -complete [5].3 Stabbers in the spaceWe consider now the problem of stabbing the tetra-hedra induced by an n point set S in R3 using sets ofplanes. Our approach is similar to that of the previoussection.For the lower bound we will use the following setof n points in the plane. Let f : R → R3, f(a) =(a, a2, a3). f(R) is known as the momentum curve inR3. Clearly, any plane in R3 intersects f(R) in at most3 points. Let Cn = {p0, . . . , pn−1} with pi = f(i).Since any plane intersects f(R) in at most 3 points,Cn is in general position. Given any segment pipj ,let f([i, j]) be the curve containing the points f(x),with i ≤ x ≤ j. We call f([i, j]) the shadow of thesegment pipj in the momentum curve. The followingobservation will be useful.Observation 1 Let pi, pj ∈ Cn. Then any plane in-tersecting the relative interior of the segment pipj in-tersects its shadow.Consider the set {t0, . . . , tbn−13 c} of tetrahedra, suchthat the vertex set of ti is {p3i, p3i+1, p3i+2, p3i+3}.Observe that the interiors of these tetrahedra are pair-wise disjoint, and that ti and ti+1 share exactly onevertex, namely p3i+3.Observation 2 If a plane intersects the interior of atetrahedron, then it intersects the relative interior ofone of its edges.We define the shadow of ti as the shadow of the seg-ment p3ip3i+3 which is in fact the union of the shadowsof the edges of ti. Then the relative interiors of theshadows of {t0, . . . , tbn−13 c} are pairwise disjoint. Thenext lemma follows now directly from Observations 1and 2:Lemma 5 If a plane stabs ti, then it intersects therelative interior of its shadow.Since any plane intersects the momentum curve atmost three times, it follows that any plane intersectsat most three elements of {t0, . . . , tbn−13 c}.By Observation 1, any plane stabbing ti must in-tersect its shadow; thus any any plane stabs at mostthree elements of {t0, . . . , tbn−13 c}.Thus we have:Lemma 6 f32 (n) ≥ dn−19 e.We will now prove that f32 (n) ≤ dn9 e + 2. To thisend, we are going to give an algorithm that splits anypoint set S into a a family of well-separated familiesof subsets of S.First find two parallel planes π1 and π2 such thatthey split S into three subsets S1, S2 and S3 such that||Si| − |Sj || ≤ 1, i 6= j ≤ 3. That is, each Si containsbn3 c or dn3 e elements.We now apply the ham-sandwich theorem in R3to find a plane π3 that simultaneously bisects S1, S2and S3. Let S+i be the subset of Si that lies aboveπ3 and S−i the one that lies below π3, with i = 1, 2, 3.This gives us two disjoint well-separated families F ={S+1 , S−2 , S+3 } and F ′ = {S−1 , S+2 , S−3 }. Similarly tothe algorithm in the previous section, iteratively find(3, 3, 3)-cuts for F , removing the nine points abovethe cut until each set in F contains at most threepoints, and adding the cut planes to the 32-stabber.Then repeat the same process for F ′.It is now easy to see that the set of planes we obtainhas at most dn9 e + 2 elements, and that they form astabber of all the tetrahedra with vertices in S. Thuswe have proved:Theorem 7 dn−19 e ≤ f32 (n) ≤ dn9 e+ 2.Observe that using similar arguments we can provethat f22 (n) ≈ n/6 (stabbing triangles with planes) andf12 (n) ≈ n/3 (stabbing segments with planes).3.1 Stabbing 2-simplices of point setswith linesFor the problem of stabbing triangles with lines, it iseasy to see that in some instances, we need at least2n−42 lines. Indeed, consider any set of points S inR3 with n points in convex position. Observe thatthe convex hull of S contains exactly 2n−4 triangles,and that any line intersects the interior of at most twoof these triangles. Thus we have proved that in R3,f21 (n) ≥ n− 2.We now prove that 2n−5 lines are always sufficientto stab all of the triangles of S. To prove this, firstproject the elements of S into the real plane, thusobtaining a point set S′ in R2. It is known that if theconvex hull of S′ has c elements, it is always possibleto find a set of points Q with at most 2n − c − 2points that stabs all of the triangles of S′; see [2, 4].Each point q ∈ Q generates a vertical line `q passingthrough q. Since each triangle T of S projects to atriangle T ′ of S′, there is a line `q, q ∈ Q that stabsT ′ and thus T . Since in the worst case, the convexhull of S′ has three points, we have proved:Theorem 8 In R3, f21 (n) ≤ 2n− 5.93Stabbing simplices with k-flats3.2 Stabbing the 3-simplices of a pointset in R3 with pointsWe now turn our attention to the problem of stabbingthe set of tetrahedra generated by a point set S inR3 using sets of points. We will assume that no twoelements of S lie on a line vertical to the real plane.Consider again the momentum curve f(a) =(a, a2, a3), and the set of points Cn = {p0, . . . , pn}as defined above. Let 0 ≤ i, j ≤ n such thati + 1 < j. Let Ti,j be the tetrahedra with vertex set{pi, pi+1, pj , pj+1}. It is well known [3] that the set oftetrahedra Ti,j , with i and j as above, have disjointinteriors, and form a tetrahedralization of the convexhull of Cn = {p0, . . . , pn}. Since any set of points thatstabs the tetrahedra of S contains at least one pointin each of these Ti,j , it follows that f30 (n + 1) is atleast(n−12).We now prove an upper bound on f30 (n + 1). Weproceed as follows: For every two elements xi, xj ∈ S,place a point pi,j slightly above the midpoint of theline segment joining xi to xj . For every xi ∈ S, placea point pi slightly below xi.We now prove that the set of points containing allthe p′i,js and the pi’s stabs all the tetrahedra of S.Consider any four points xi, xj , xk, x` of S. Let Hbe the convex hull of {xi, xj , xk, x`}. Two cases arise.In the first case, H projects to a convex quadrilateralin the plane. In this case, one edge of the convex hullof {xi, xj , xk, x`} is not visible from below. Supposew.l.o.g. that it is the edge joining xk to x`. Then thepoint pi,j stabs H.Suppose then that H projects to a triangle T onthe plane. Then one vertex of H, say xi, belongs tothe interior of T . Two sub-cases arise. In the firstsub-case, when we see H from below, xi is not visi-ble. In this case, the point pi stabs the convex hull of{xi, xj , xk, x`}. In the remaining sub-case, pi,j stabsH.Thus we have that we can always choose(n+12)+ npoints that stab the tetrahedra of S. It is not hardto see that we can reduce the above number by 9 byobserving that if the line segment joining xi to xj isnot visible from below, and this line segment belongsto the convex hull of S; then the point pi,j is redun-dant. Also if a point xi belongs to the convex hull ofS and is visible from below, then pi is redundant.Thus we have proved:Theorem 9(n−12)≤ f30 (n+ 1) ≤(n+12)+ n− 9.4 Some thoughts on higher di-mensionsSome of the results in this paper; e.g. Lemma 5and Lemma 6, generalise easily to higher dimensions,yielding the following result:Lemma 10 In Rd, dn−1d2 e ≤ fdd−1(n).And in general:Lemma 11 In Rd, dn−1dm e ≤ fmd−1(n).To finish this paper, we conjecture:Conjecture 1 fdm(n) ≈ nm·d .References[1] J. Cano, A. García, F. Hurtado, T. Sakai, J. Tejel,J. Urrutia, Blocking the k-holes of point sets in theplane, submitted.[2] J. Czyzowicz, E. Kranakis, J. Urrutia, Guarding theconvex subsets of a point set, in: 12th CanadianConference on Computational Geometry, Frederic-ton, New Brunswick, Canada, 2000, 47–50.[3] B. Grünbaum, Convex Polytopes, Interscience Mono-graphs in Pure and Applied Mathematics, Vol. XVI,John Wiley & Sons, London-New York-Sydney.[4] M. Katchalski, A. Meir, On empty triangles deter-mined by points in the plane, Acta Math. Hungar 51(1988), 323–328.[5] R. Freimer, J. S. B. Mitchell, C. D. Piatko, On theComplexity of Shattering Using Arrangements, in:Proceedings of the 2nd Canadian Conference on Com-putational Geometry, 1990, 218–222.[6] A. Pór, D. R. Wood, On visibility and blockers, J.Computational Geometry 1(1) (2010), 29–40.[7] T. Sakai, J. Urrutia, Covering the convex quadri-laterals of point sets, Graphs and Combinatorics 38(2007), 343–358.[8] W. Steiger, J. Zhao, Generalized Ham-SandwichCuts, Discrete and Computational Geometry 44(3)(2010), 535–545.[9] H. Steinhaus, A note on the ham sandwich theorem,Mathesis Polska 9 (1938), 26–28.94

Stabbing simplices of point sets with k-flats

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Stabbing simplices of point sets with k-flats

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