For a finite set P in the plane, let b(P) be the smallest possible size of a set Q, Q∩P=∅, such that every segment with both endpoints in P contains at least one point of Q. We raise the problem of estimating b(n), the minimum of b(P) over all n-point sets P with no three points collinear. We review results providing bounds on b(n) and mention some additional observation

Matoušek, Jiří

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Discrete Comput Geom (2009) 42: 219–223DOI 10.1007/s00454-009-9185-zBlocking Visibility for Points in General PositionJirˇí MatoušekReceived: 12 July 2008 / Revised: 23 March 2009 / Accepted: 8 April 2009 /Published online: 2 May 2009© Springer Science+Business Media, LLC 2009Abstract For a finite set P in the plane, let b(P ) be the smallest possible size of aset Q, Q∩P = ∅, such that every segment with both endpoints in P contains at leastone point of Q. We raise the problem of estimating b(n), the minimum of b(P ) overall n-point sets P with no three points collinear. We review results providing boundson b(n) and mention some additional observations.Keywords Visibility · Visibility-blocking set · Behrend’s constructionLet P be an n-point set in the plane (or, more generally, in Rd). We define a visibility-blocking set for P as a set Q that is disjoint from P and such that every segment withendpoints in P contains at least one point of Q.If the points of P are all collinear, then there is a visibility blocking set withn − 1 points. The question raised in this note is what is the smallest possible size ofa visibility-blocking set for P having no three points collinear? That is, we letb(P ) := min{|Q| : Q a visibility-blocking set for P },b(n) := min{b(P ) : P ⊂ R2 with no three points collinear, |P | = n},and we would like to estimate the asymptotics of b(n) for large n.I arrived at this question when (unsuccessfully) attempting to prove the followingnice conjecture of Kára, Pór, and Wood [7, Conjecture 2]: For all integers k,  ≥ 2,J. Matoušek ()Department of Applied Mathematics and Institute of Theoretical Computer Science (ITI), CharlesUniversity, Malostranské nám. 25 118 00 Praha 1, Czech Republice-mail: matousek@kam.mff.cuni.czJ. MatoušekInstitute of Theoretical Computer Science, ETH Zurich, 8092 Zurich, Switzerland220 Discrete Comput Geom (2009) 42: 219–223there is an integer n such that every n-point set in the plane contains  collinearpoints or k pairwise visible points.I obtained Theorems 1 and 3 below, but after this note was accepted for publicationin the Klee Festschrift, it turned out that both of these results had been known earlierin somewhat different contexts. Moreover, I learned that other people considered theasymptotics of b(n) independently (a group of researchers at the Courant Institutein New York including Andreas Holmsen, János Pach, Radoš Radoicˇic´, and GáborTardos [6]) and also rediscovered some of these results. Thus, the present note mostlyreviews known results and adds some observations, which have not appeared in printas far as I know and which illustrate some of the difficulties inherent in the problem.I still believe that the inclusion of this note in the Festschrift is warranted by thebeauty of the problem—I think that Vic Klee would like it.Midpoints and an Upper Bound For a point set P , let μ(P ) be the cardinality ofthe set { 12 (p + q) : p,q ∈ P,p = q} of midpoints of all pairs of points of P . Clearly,b(P ) ≤ μ(P ). The problem of estimating μ(n) := minμ(P ), where the minimum isover all n-point planar sets P in general position, was raised, according to Pach [10],by F. Hurtado. Earlier Erdo˝s, Fishburn, and Füredi [4] studied the problem of esti-mating minμ(P ) over all n-point planar sets P in convex position and establishedthe (surprising) lower and upper bounds of 0.40n2 and 0.45n2, respectively.Now we recall an upper bound on μ(n) (and b(n)) due to Pach [10] (basedon Erdo˝s et al. [5]). For an integer N , let ν(N) be the maximum number of el-ements of a set A ⊆ {1,2, . . . ,N} that contains no three-term arithmetic progres-sion. Behrend [3], improving on a construction by Salem and Spencer, proved thatν(N) ≥ N1−O(1/√logN ). The following theorem, whose proof we recall, is based onBehrend’s construction.Theorem 1 (Pach [10]) There is a constant C such thatb(n) ≤ μ(n) ≤ neC√lognfor all sufficiently large n.Proof Let n be given and large, and let m and s be integer parameters to be specifiedlater. Let G := {0,1, . . . , s − 1}m ⊂ Rm, and let Sk := {x ∈ G : ‖x‖2 = k}, where ‖ · ‖is the Euclidean norm.As in Behrend’s argument, G = ⋃m(s−1)2k=0 Sk , and thus by the pigeonhole principlethere exists k with |Sk| ≥ sm−2/m. We let P := Sk for some such k.Since the points of P lie on a sphere, no three of them are collinear. The midpointof every two points p,q ∈ G lies in G′ := {0, 12 ,1, 32 , . . . , s − 1}m, and thus Q :=G′ \ P is a visibility-blocking set for P .We let m := lnn, and let s be the smallest integer with sm−2/m ≥ n. Then|P | ≥ n, (s − 1)m−2 < mn, and|Q| ≤ |G′| = (2s − 1)m ≤ (3(s − 1))m ≤ 3m(s − 1)2mn ≤ neO(√logn),Discrete Comput Geom (2009) 42: 219–223 221as can easily be calculated (using (s − 1)2 ≤ (mn)2/(m−2) ≤ (n2)4/m, say). The im-plicit constant in the exponent can be improved by a more careful choice of m and bymore precise calculations.The bound in the theorem follows by projecting P and Q to a generic two-dimensional subspace of Rm. Lower Bounds I am aware only of the following rather trivial lower bound for b(n):Observation 2 If P is an n-point planar set with no three points collinear and withconv(P ) having p vertices, then b(P ) ≥ 3n − p − 3. In particular, b(n) ≥ 2n − 3.Proof A triangulation of P has 3n − p − 3 edges, and each point of a visibility-blocking set covers at most one of these. For point sets in convex position, a slightly superlinear lower bound can be given.The result is implicitly contained in Araujo et al. [1], and the argument goes back (atleast) to Kostochka and Kratochvíl [8].Theorem 3 For every n-point planar set P in (strictly) convex position, we haveb(P ) ≥{n∑mk=1 1/k for n = 2m + 1 odd,1 + n∑m−1k=1 1/k for n = 2m even.Thus, b(P ) = Ω(n logn).Proof Let p1,p2, . . . , pn be the points of P numbered along the circumference ofconv(P ). We define the length (pipj ) of the segment pipj , i < j , as the number ofconvex hull edges between pi and pj , i.e., min(j − i, n + i − j).Let us suppose that Q is a visibility-blocking set for P , and let q ∈ Q. The keyobservation is that if  is the smallest of the lengths of the segments pipj incidentto q , then q is incident to at most  segments.Thus, if we give the segment pipj weight 1/(pipj ), no q ∈ Q is incident tosegments of total weight more than 1. The theorem follows by summing the weightsof all segments. For the case where P is the vertex set of a regular convex n-gon, Poonen andRubinstein [13] show that, apart from the center, no point is the intersection of eightor more of the diagonals of the n-gon, and thus b(P ) = Ω(n2) in this case.Small Cases Now we return to arbitrary sets (not necessarily in convex position).The first few values of b(n) are b(2) = 1, b(3) = 3, b(4) = 5, b(5) = 8, b(6) = 10.The upper bounds are witnessed by Fig. 1. The lower bounds for n ≤ 4 are trivial(or follow from Observation 2). For n = 5,6, we distinguish two cases: If all ver-tices of P are on the convex hull, then the lower bound follows from Theorem 3,and otherwise, it is given by Observation 2. Determining the exact value gets morecomplicated for larger n, and I am not aware of a reasonably clean argument for any222 Discrete Comput Geom (2009) 42: 219–223Fig. 1 Upper bound examples for small nn ≥ 7. Interesting upper bound constructions for several small cases, to be reportedelsewhere, were given by Snoeyink and Speckmann (private communication).Additional Remarks1. A possibly easier version of the problem deals with pseudosegments instead ofstraight segments. That is, for a given point set P , we want to construct an arrange-ment A of pseudolines and a subset Q of its vertices such that P ∩ Q = ∅, eachp ∈ P is a vertex of A, no three points of P lie on a common pseudoline, andevery two points of P lie on a common pseudoline  and have a point of Q on thesegment of  between them.The lower bound of Theorem 3 still applies in this setting (with “convex posi-tion” interpreted appropriately in terms of the arrangement A), and here it is easyto provide an O(n logn) upper bound.Is there a linear upper bound for some P , not in “convex position”, in thepseudoline setting?Another variant of the problem, asked by Pach, is partitioning all the(n2)straight segments defined by the points into a small number of crossing families,i.e., families in which every two segments cross. Is a superlinear number of suchfamilies always necessary? (Also see Pach et al. [11] for some related blocking-type questions.)2. Here is another variation of the problem: Instead of requiring that each segmentdetermined by two points of P contains a point of Q, it now suffices that each linedetermined by two points of P contains a point of Q. Perhaps surprisingly, thereare examples showing that O(n) points suffice to block all lines.One such example can be constructed using integer points on the curve y = x3(a similar construction was used, e.g., for the orchard problem; see [9]). A simplerexample, inspired by a relation of the problem to Ungar’s theorem (see [9] again),was communicated to me by Pinchasi [12]: Take the vertex set of a regular (2n)-gon centered at the origin and apply a projective transform that sends the lineat infinity to the x-axis, obtaining a (2n)-point set P (the points of P lie on ahyperbola). Since the vertices of the regular (2n)-gon determine only 2n distinctdirections, there is a (2n)-point set Q on the x-axis that intersects all the linesdetermined by P .3. The set P in the example just mentioned can also be partitioned into two n-pointsubsets P1 and P2 (namely, the points above and below the x-axis) in such a waythat every segment p1p2, p1 ∈ P1, p2 ∈ P2, contains a point of Q. This shows thata natural bipartite version of the original visibility-blocking problem has a linearupper bound.Discrete Comput Geom (2009) 42: 219–223 223Pach [10] proves a superlinear lower bound for μ(n), the minimum cardinalityof the set of all midpoints for n points in general position, using Freiman’s theoremon set addition. This argument can be adapted to give a superlinear lower bound forthe midpoints in the bipartite setting as well. Indeed, let P1,P2 be disjoint n-pointsets with P := P1 ∪ P2 in general position, and suppose that the set P1 + P2 ={p1 + p2 : p1 ∈ P1,p2 ∈ P2} has cardinality O(n). Then P has Ω(n3) additivefour-tuples, i.e., four-tuples (p1,p2,p3,p4) with p1 + p2 = p3 + p4, and by theBalog–Szemerédi theorem [2] there is a subset P ′ ⊆ P of size Ω(n) with |P ′ +P ′| = O(|P ′|). Then, as in Pach’s argument, Freiman’s theorem implies that for asufficiently large n, the set P ′ contains three collinear points, a contradiction. Thisshows that Pach’s lower bound idea for μ(n) is not directly applicable to b(n).Acknowledgements I would like to thank Bettina Speckmann and Jack Snoeyink for discussions andideas concerning small cases, Rom Pinchasi for the simple line-blocking example mentioned above, ImreBárány for discussions concerning the conjecture of [7], Andreas Holmsen for telling me about the pa-pers [1] and [13], János Pach for comments and additional references, and David Wood and Jean Cardinalfor bringing Pach’s paper [10] to my attention.References1. Araujo, G., Dumitrescu, A., Hurtado, F., Noy, M., Urrutia, J.: Comput. Geom. 32(1), 59–69 (2005)2. Balog, A., Szemerédi, E.: A statistical theorem of set addition. Combinatorica 14(3), 263–268 (1994)3. Behrend, F.A.: On sets of integers which contain no three terms in arithmetical progression. Proc.Natl. Acad. Sci. USA 32, 331–332 (1946)4. Erdo˝s, P., Fishburn, P., Füredi, Z.: Midpoints of diagonals of convex n-gons. SIAM J. Discrete Math.4(3), 329–341 (1991)5. Erdo˝s, P., Füredi, Z., Pach, J., Ruzsa, I.: The grid revisited. Discrete Math. 111(1–3), 189–196 (1993)6. Holmsen, A.: e-mail communication, October 20087. Kára, J., Pór, A., Wood, D.R.: On the chromatic number of the visibility graph of a set of points in theplane. Discrete Comput. Geom. 34(3), 497–506 (2005)8. Kostochka, A., Kratochvíl, J.: Covering and coloring polygon-circle graphs. Discrete Math. 163, 299–305 (1997)9. Pach, J.: Finite point configurations. In: Goodman, J.E., et al. (eds.) Handbook of Discrete and Com-putational Geometry, pp. 3–18. CRC Press, Boca Raton (1997)10. Pach, J.: Midpoints of segments induced by a point set. Geombinatorics 13(2), 98–105 (2003)11. Pach, J., Pinchasi, R., Sharir, M.: On the number of directions determined by a three-dimensionalpoints set. J. Comb. Theory Ser. A 108(1), 1–16 (2004)12. Pinchasi, R.: Private e-mail communication (2006)13. Poonen, B., Rubinstein, M.: The number of intersection points made by the diagonals of a regularpolygon. SIAM J. Discrete Math. 11(1), 135–156 (1998)

Blocking Visibility for Points in General Position

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Blocking Visibility for Points in General Position

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