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Mutual witness proximity graphs
Information Processing Letters 114 (2014) 519–523
Contents lists available at ScienceDirect
Information Processing Letters www.elsevier.com/locate/ipl
Mutual witness proximity graphs Boris Aronov a,∗ , Muriel Dulieu a , Ferran Hurtado b a b
Dept. of Computer Science and Engineering, Polytechnic School of Engineering, New York University, Brooklyn, NY, USA Dept. de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Barcelona, Spain
a r t i c l e
i n f o
Article history: Received 18 August 2013 Received in revised form 8 March 2014 Accepted 2 April 2014 Available online 16 April 2014 Communicated by R. Uehara Keywords: Computational geometry Proximity graphs Witness graphs Delaunay graphs Gabriel graphs Rectangle of influence graphs
a b s t r a c t This paper describes one variation on witness proximity graphs called mutual witness proximity graphs. Two witness proximity graphs are said to be mutual when, given two sets of points A and B, A is the vertex set of the first graph and the witness set of the second one, while B is the witness set of the first graph and the vertex set of the second one. We show that in the union of two mutual witness Delaunay graphs, there are 2 edges, where n = | A | + | B |, which is tight in the worst case. We also always at least n− 2 show that if two mutual witness Delaunay graphs are complete, then the sets A and B are circularly separable; if two mutual witness Gabriel graphs are complete, then the sets A and B are linearly separable; but two mutual witness rectangle graphs might be complete, with A and B not linearly separable. © 2014 Elsevier B.V. All rights reserved.
1. Introduction A proximity graph is a graph whose vertices correspond to some geometric objects, and an edge connects two of them when the objects are considered to be neighbors according to some proximity criterion. These graphs have been widely used for spatial data analysis [1,2], pattern recognition (see [3,4] and references therein), data mining [5], and in many disciplines where classification or interpolation is required [2,4,6]. On the other hand, as connecting neighbors naturally yields an aesthetically appealing layout of a graph, proximity graphs have also been a topic thoroughly studied in the area of graph drawing [7,8]. We consider proximity graphs on point sets on the plane. Given a point set S, the Delaunay graph DG( S ), the Gabriel graph GG( S ), and the rectangle of influence graph RIG( S ) of S all have vertex set S. Given p , q ∈ S, there is an edge connecting p and q in DG( S ) if there is some disk with p and q on its boundary, whose interior is empty of
*
Corresponding author. E-mail addresses: [email protected] (B. Aronov), [email protected] (M. Dulieu), [email protected] (F. Hurtado). http://dx.doi.org/10.1016/j.ipl.2014.04.001 0020-0190/© 2014 Elsevier B.V. All rights reserved.
points from S. The vertices p and q are adjacent in GG( S ) if the closed disk with diameter pq covers no other points from S. Finally, there is an edge connecting p and q in RIG( S ) if the closed axis-aligned rectangle B ( p , q) with p and q as opposite corners contains no other points from S. These graphs belong to the Delaunay family of graphs, and define neighbor relationships that describe how the points interact; see [3] for a survey on the topic and its applications. The rectangle of influence graph was introduced by Ichino and Sklansky in [9]. In the same work they also defined the mutual neighborhood graph of a point set A against a second point set B, denoted MNG( A | B ): its vertex set is A, and there is an edge connecting p , q ∈ A when the rectangle B ( p , q) contains no points from B (with the exception of p and q themselves, should they belong to B). The motivation behind their definition was to consider the union of the graphs MNG( A | B ) and MNG( B | A ) as a descriptor of the interaction between the sets A and B. A systematic study of witness proximity graphs, which are proximity graphs on a point set with respect to a second point set, was undertaken in [10–13] and also studied later in other works such as [14] and [15]. For the sake of
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conciseness we don’t give here the general definition but concentrate on the two graphs we study in the following sections. We define the witness Delaunay graph of a point set A of vertices in the plane, with respect to a point set B of witnesses, denoted DG− ( A , B ), as the graph with vertex set A in which two points x, y ∈ A are adjacent when there is an open disk whose boundary passes through x and y and that does not contain any witness w ∈ B. Notice that we don’t care about the presence of elements from A \ B inside the disk. DG− ( A , B ) is a negative-witness graph because the witnesses from B “prevent” adjacencies. DG− ( A , ∅) is simply the complete graph K | A | . DG− ( A , A ) is precisely the Delaunay graph DG( A ), which under standard non-degeneracy assumptions is a triangulation, denoted by DT( A ). For a pair of points p, q in the plane, we denote by D pq the closed disk with diameter pq. The witness Gabriel graph GG− ( A , B ) is defined by two sets of points A and B; A is the set of vertices of the graph and B is the set of witnesses. There is an edge xy in GG− ( A , B ) if, and only if, there is no point of B in D xy \ {x, y }. It is a negative witness graph. GG− ( A , A ) is precisely the original Gabriel graph GG( A ). Following the rationale of Ichino and Sklansky [9], it is natural to wonder what would be the characteristics of the two interdependent witness graphs that we call mutual witness graphs, corresponding to the witness Delaunay graphs and to the witness Gabriel graphs. In both cases the two graphs would be mutual in the following sense: Given two sets of points A and B, A is the vertex set of the first graph and the witness set of the second one, while B is the witness set of the first graph and the vertex set of the second one. In Sections 2 and 3 we consider the properties of the mutual witness Delaunay graphs and the mutual witness Gabriel graphs, respectively, the first one being the main subject of this work. We conclude in Section 4 with an observation on the mutual neighborhood graphs originally introduced by Ichino and Sklansky. We assume hereafter strong general position of the point sets A and B: no three collinear points and no four concyclic points. Removing this assumption involves complicating some arguments and invalidating other theorems below; for several of the objects there are no agreed upon standard definitions in presence of degeneracies. 2. Mutual witness Delaunay graphs We first consider the joint size of the mutual witness Delaunay graphs DG− ( A , B ) and DG− ( B , A ): Theorem 1. Given disjoint sets of points A and B, the number 2 , where of edges in DG− ( A , B ) ∪ DG− ( B , A ) is at least n− 2 n = | A | + | B |. This bound is tight for n ≥ 2. Proof. For the lower bound, color points of A black and points of B white. Consider DT( A ∪ B ), the Delaunay triangulation of A ∪ B. Each triangle has at least one monochromatic edge, i.e., an edge incident to two vertices of the same color. A monochromatic edge pq belongs to
Fig. 1. The union of the witness Delaunay graphs DG− ( A , B ) and 2 DG− ( B , A ) in each of the figures has n− monochromatic edges and 2
n+2 1 components. Monochromatic edges of DT( A ∪ B ) are drawn solid and bichromatic ones dashed.
DG− ( A , B ) ∪ DG− ( B , A ) as there exists a disk empty of points of A ∪ B whose boundary circle passes through p and q. Therefore, the number of monochromatic edges in DT( A ∪ B ) is a lower bound on the number of edges in DG− ( A , B ) ∪ DG− ( B , A ). Any Delaunay triangulation has at least n − 2 triangles, each with at least one monochromatic edge. Since a monochromatic edge might appear in two faces, and if the number of faces is odd at least one such edge is not counted twice, the number of monochromatic edges is at 2 least n− . 2 To see that this bound is tight, we present a concrete construction; refer to Fig. 1. Place black points and m white points, with + m = n and = m ( = m − 1 in the odd case), spaced on the boundary of an ellipse in an alternating manner and such that, except for the leftmost point (for the even case only) and rightmost point, pairs of black points and pairs of white points are placed at the same x-coordinate. Consider DT( A ∪ B ). The vertical edges between pair of points at the same x-coordinate define the only monochromatic edges in DT( A ∪ B ). Indeed, for any pair of points a and b of the same color on different sides of a vertical monochromatic edge, the edge ab would cross a monochromatic edge de of DT( A ∪ B ). As de is in DT( A ∪ B ), there exists a disk with d and e on its boundary that does not contain a or b. Hence, any disk with a and b on its boundary would contain either d or e or both. Therefore, there is no non-vertical monochromatic edge ab in DG− ( A , B ). Thus the only edges in DG− ( A , B ) ∪ DG− ( B , A ) are the vertical ones connecting same-color vertices; and hence, the number of edges of DG− ( A , B ) ∪ DG− ( B , A ) is exactly n−2 2 . 2 Our next result, on a particular kind of geometric separability, proves that mutual witness Delaunay graphs contain useful information about interclass structure. Theorem 2. For disjoint sets of points A and B, if DG− ( A , B ) and DG− ( B , A ) are complete, then the sets A, B are circularly separable. The converse is not true. Proof. Consider the convex hulls CH( A ) and CH( B ). If they don’t intersect, A and B are linearly separable and therefore circularly separable. Suppose CH( A ) and CH( B ) intersect and an edge ab of DG− ( A , B ) crosses an edge cd of DG− ( B , A ). If the points a, c , b, d are in this clockwise order, consider CH({a, c , b, d}): either acb + bda > 180◦ , or cbd + dac > 180◦ ; hence, all disks D ab with a, b on their
B. Aronov et al. / Information Processing Letters 114 (2014) 519–523
boundary contain a point of B, or all disks D cd with c , d on their boundary contain a point of A, a contradiction. Therefore we may assume that CH( A ) and CH( B ) intersect but no edge of DG− ( A , B ) intersects an edge of DG− ( B , A ). This means that DG− ( B , A ) must be contained in one of the interior cells of the arrangement of the segments connecting all pairs of points of A (or conversely). Suppose DG− ( B , A ) is contained in one of the interior cells of the arrangement induced by DG− ( A , B ). Consider the Delaunay triangulation DT( A ) of A; DG− ( B , A ) is entirely contained in one of its triangles e f g, as DT( A ) is a subgraph of DG− ( A , B ). By definition of the Delaunay triangulation, there is an open disk D e f g empty of vertices of A that contains on its boundary e , f , g ∈ A. Since D e f g contains B in its interior, the boundary of a slightly shrunken version of D e f g separates A from B, as claimed. To prove that the converse is not true, place m points of A and m points of B equally spaced on the boundary C of a disk in an alternating manner. Shift them very slightly so they are still in convex position (and in general position) moving the points from A outside C , and the points from B inside C . Then the two point sets are circularly separable, but the graph DG− ( A , B ) is empty. 2 Observation 1. For disjoint sets of points A and B, even if DG− ( A , B ) and DG− ( B , A ) are complete, the sets A, B are not necessarily linearly separable. Indeed A could be entirely contained in an interior face of the arrangement of the segments connecting all pairs of points of B. In the following, we present a theorem about the number of connected components in DG− ( A , B ) ∪ DG− ( B , A ), derive a result on bicolored triangulations, and pose a problem on Delaunay triangulations. Let A and B be disjoint point sets with | A | + | B | = n. As in the proof of Theorem 1, we consider the points in A white and those in B black, and build a Delaunay triangulation DT( A ∪ B ) in which we again call the edges joining two points of different colors “bichromatic,” and the edges joining two points of the same color “monochromatic.” Let hm and hb be the numbers of monochromatic and bichromatic edges on the convex hull of A ∪ B, respectively. Theorem 3. The maximum number of connected components in DG− ( A , B ) ∪ DG− ( B , A ) is at most 2n3+1 , and there exist 1 sets A and B for which this value is n+ . 2
Proof. For the lower bound, depending on whether n is even or odd we place the points as in Fig. 1, left or right, respectively. Now we turn to the upper bound. As mentioned in the proof of Theorem 1, the monochromatic edges in G = DT( A ∪ B ) are contained in DG− ( A , B ) ∪ DG− ( B , A ), therefore the number of connected components of DG− ( A , B ) ∪ DG− ( B , A ) is at most the number of monochromatic connected components of G, which we proceed to bound. We construct an auxiliary planar straight-line embedded graph P . Its vertices are the bichromatic edges of G (drawn as midpoints of the edges) and we connect two such edges if they belong to the same triangle of G (see
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Fig. 2. Edges of P are drawn dotted. The regions are represented by different shades of gray.
Fig. 2). Since each triangle of G has zero or two bichromatic edges, it gives rise to at most one edge in P , drawn as a line segment within the triangle; hence P is indeed properly embedded and crossing-free. Since every bichromatic edge of G is adjacent to two bichromatic triangles, unless it appears on the boundary of the convex hull H = CH( A ∪ B ), P is a disjoint union of p paths and c cycles, whose geometric realizations are pairwise disjoint simple polygonal curves, with cycles lying in the interior of H and paths connecting pairs of points on the boundary of H ; slightly abusing the terminology we will refer to such a path or cycle as a curve in P . The curves of P partition H into 1 + p + c connected regions, R 1 , . . . , R 1+ p +c and we will prove below that each region contains exactly one monochromatic connected component of G; once this is established, bounding the quantity 1 + p + c completes the proof. We assign a monochromatic triangle to white or black zone according to the color of its vertices. A bichromatic triangle is cut by an edge of P into a triangle and a trapezoid, one black and the other white. Let white zone be the union of the white sets thus formed and define black zone symmetrically. A region is a connected component of white zone or of black zone. As curves of P and the boundary of H form the only boundaries of regions, they coincide precisely with R 1 , . . . , R 1+ p +c . Let us prove that each of the regions R i separated by curves of P contains exactly one monochromatic component of G. The second definition of regions implies that each region R i contains at least one vertex of G and all such vertices have the same color. So it is sufficient to prove that the subgraph G [ V i ] induced by the set V i of vertices lying in R i is connected. Consider a path π connecting two vertices v , w ∈ V i . If the polygonal path corresponding to π stays in R i , we are / Vi, done. Otherwise, let v 1 w 1 ∈ E (G ), with v 1 ∈ V i , w 1 ∈ be the first edge of π leaving R i by crossing a boundary component γ of R i . As removal of γ splits H into two connected sets, π has to cross γ again for the first time, say, / V i . Recall via an edge w k v k ∈ γ , with v k ∈ V i and w k ∈ that γ is a curve in P , so since v 1 w 1 , v k w k ∈ γ , γ contains a path v 1 w 1 , v 2 w 2 , . . . , v k w k , where all v j ∈ V i and / V i . Note now that v 1 v 2 . . . v k is an, in general, nonwj ∈ simple, path in G [ V i ]: for every j < k, ether v j = v j +1 or v j v j +1 is a monochromatic edge of the triangle w j v j v j +1 and thus contained in R i and G [ V i ]. Hence by replacing the following portion of π : v 1 w 1 . . . w k v k by v 1 v 2 . . . v k , we eliminate one contiguous section of π where it lies outside of G [ V i ]. Repeating this for each such section, we obtain a new path π that connects v to w and stays within G [ V i ], completing the proof of connectivity of this induced subgraph.
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We now proceed to count paths and cycles of P in order to bound the value of 1 + p + c. By Euler’s formula, G has
3n − hm − hb − 3
(1)
edges and
2n − hm − hb − 2
(2)
triangles. As every triangle contains at least one monochromatic edge, which is counted in two triangles, unless it lies on the boundary of H , from (2) we deduce that the number of monochromatic edges in G is at least
2n − hm − hb − 2 2
+
hm 2
=n−1−
hb 2
(3)
.
Combining (1) and (3), we conclude that the number of bichromatic edges in G is at most
hb
(3n − hm − hb − 3) − n − 1 − 2 hb 3hb − 2 = hb + 2n − hm − −2 , = 2n − hm − 2
2
from which we deduce that the number of interior bichromatic edges in G is at most
2n − hm −
3hb 2
− 2.
As each cycle in P contains at least three interior bichromatic edges of G, there are c ≤
2n−hm − 3
3hb 2
−2
such
cycles; and as the endpoints of each path in P are bichroh matic edges of H , there are p = 2b such paths. Therefore, the number of monochromatic components in G is 2n−hm − 3
3hb 2
−2
+ 1 = 2n−h3m +1 . Being an integer, this number is equal to 2n−h3m +1 , and hence, at most 2n3+1 . 2 p+c+1≤
+
hb 2
Corollary 4. Let S be any set of n points in general position, and let T be any triangulation of S. For any bicoloring of the points of S that produces hm monochromatic convex hull edges, the number of monochromatic connected components in the graph T is at most 2n−h3m +1 . There is an infinite set of pairs of values n and hm , for which there exist sets of n points with convex hull of size hm , admitting a bicoloring that has exactly
2n−h3m +1 monochromatic connected components. Proof. As the proof for the upper bound in the previous theorem starts with the triangulation G = DT( A ∪ B ) but makes no use of any Delaunay property, it applies to any bicolored triangulation on S = A ∪ B. For the second claim, consider a set A of hm white points in convex position, and place m − hm additional white points in the interior of their convex hull. Any triangulation of A will consist of 2m − hm − 2 triangles. Now, place one black point inside each of those triangles, obtaining a set B of black points. Connect each black point to the three white vertices of the triangle containing it. We obtain a bicolored triangulation T of S = A ∪ B, with
Fig. 3. Illustration of the proof of linear separability for mutual witness Gabriel graphs.
n = | S | = m + (2m − hm − 2) = 3m − hm − 2, and the T is number of monochromatic connected components in 1 + (2m − hm − 2) = 2m − hm − 1 = 2n−h3m +1 . 2 The upper bound in the preceding corollary applies to any triangulation, and is bounded above by 2n3+1 , a value that can certainly be reached, up to an additive constant, for sets whose convex hull has constant size. However, we believe that for Delaunay triangulations the number of components cannot be that large, which we express more precisely as an open question: Question 1. Let S be any set of n points in general position, and let DT( S ) be the Delaunay triangulation of S. Is it true that for any bicoloring of the points of S, the number of monochromatic connected components in DT( S ) is 1 ? at most n+ 2
3. Mutual witness Gabriel graphs In this section, we prove a theorem showing that mutual witness Gabriel graphs also contain information about interclass structure. Theorem 5. For two disjoint sets of points A and B, if GG− ( A , B ) and GG− ( B , A ) are complete, the sets A , B are linearly separable. The converse is not true. Proof. Suppose GG− ( A , B ) and GG− ( B , A ) are complete but the sets A , B are not linearly separable. Therefore CH( A ) ∩ CH( B ) = ∅, as two disjoint convex sets are linearly separable. Either CH( A ) ⊂ CH( B ), or CH( B ) ⊂ CH( A ), or an edge ab of GG− ( A , B ) intersects an edge cd of GG− ( B , A ). However, GG− ( A , B )⊂ DG− ( A , B ), GG− ( B , A )⊂ DG− ( B , A ), and the edges of DG− ( A , B ) and DG− ( B , A ) do not cross, as observed in the proof of Theorem 2, so the last case cannot arise. Thus, suppose CH( A ) ⊂ CH( B ), without loss of generality. Pick a point a ∈ A and a point b ∈ B. Draw a line through a perpendicular to ab. Suppose that b lies to the right of . As A is enclosed within CH( B ), there must be another point c from B to the left of (see Fig. 3). Therefore bac > 90◦ and hence a ∈ D bc , contradicting our assumption that GG− ( B , A ) is complete. To prove that the converse is not true, place points of A and of B alternating on a vertical line. Move the points
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10-1-0210. Ferran Hurtado is partially supported by projects MEC MTM2006-01267, MICINN MTM2009-07242, MINECO MTM2012-30951, Generalitat de Catalunya DGR 2009SGR1040, and ESF EUROCORES programme EuroGIGA, CRP ComPoSe: MICINN Project EUI-EURC-2011-4306, for Spain. References
Fig. 4. RG− ( A , B ) and RG− ( B , A ) are complete ( A is the set of white points and B the set of black points), but the pair ( A , B ) is not linearly separable.
of A slightly to the left, and the points of B slightly to the right, so they are in general position. The two sets of points are linearly separable but the union of the two mutual Gabriel graphs is the empty graph. 2 4. Mutual witness rectangle graphs In [9], Ichino and Sklansky introduced the mutual neighborhood graph defined by a pair of point sets ( A , B ) in the plane. This concept coincides with that of the negative witness rectangle graph RG− ( P , W ) in which there is an edge between two points p, q in P if, and only if, the rectangle B ( p , q) does not contain any witness w ∈ W . For a fixed pair of sets A, B, Ichino and Slansky focus on the relation between the two negative witness rectangle graphs RG− ( A , B ) and RG− ( B , A ). In particular, they claim that if RG− ( A , B ) and RG− ( B , A ) are complete graphs, then the pair ( A , B ) is linearly separable. Unfortunately this is not true, as demonstrated by the counterexample in Fig. 4. Acknowledgements Boris Aronov and Muriel Dulieu were partially supported by grant No. 2006/194 from the United States – Israel Binational Science Foundation and by NSF Grants CCF-08-30691, CCF-11-17336, and CCF-12-18791. Boris Aronov was also supported by NSA MSP Grant H98230-
[1] A. Getis, B. Boots, Models of Spatial Processes: An Approach to the Study of Point, Line and Area Patterns, Cambridge University Press, 1977. [2] A. Okabe, B. Boots, K. Sugihara, S.N. Chiu, Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 2nd ed., Wiley, 2000. [3] J.W. Jaromczyk, G.T. Toussaint, Relative neighborhood graphs and their relatives, Proc. IEEE 80 (9) (1992) 1502–1517. [4] G.T. Toussaint, Proximity graphs for nearest neighbor decision rules: recent progress, in: Interface-2002, 34th Symposium on Computing and Statistics, Montréal, Canada, 2002, pp. 17–20. [5] J. Han, M. Kamber, J. Pei, Data Mining: Concepts and Techniques, 3rd ed., Morgan Kaufmann Publishers, 2012. [6] B. Bhattacharya, K. Mukherjee, G.T. Toussaint, Geometric decision rules for instance-based learning algorithms, in: Proc. Pattern Recognition and Machine Intelligence 2005, in: Lect. Notes Comput. Sci., vol. 3776, 2005, pp. 60–69. [7] G. Di Battista, W. Lenhart, G. Liotta, Proximity drawability: a survey, in: Proc. Graph Drawing’94, in: Lect. Notes Comput. Sci., vol. 894, 1994, pp. 328–339. [8] G. Liotta, Proximity drawings, Chapter 4 in: R. Tamassia (Ed.), Handbook of Graph Drawing and Visualization, Chapman & Hall/CRC Press, 2013, pp. 115–154. [9] M. Ichino, J. Sklansky, The relative neighborhood graph for mixed feature variables, Pattern Recognit. 18 (2) (1985) 161–167. [10] B. Aronov, M. Dulieu, F. Hurtado, Witness (Delaunay) graphs, Comput. Geom. 44 (6–7) (2011) 329–344. [11] B. Aronov, M. Dulieu, F. Hurtado, Witness Gabriel graphs, Comput. Geom. 46 (7) (2013) 894–908. [12] B. Aronov, M. Dulieu, F. Hurtado, Witness rectangle graphs, Graphs Comb., in press. Appeared online at: http://doi.org/10.1007/s00373013-1316-x, May 2013. [13] M. Dulieu, Witness proximity graphs and other geometric problems, PhD thesis, Polytechnic Institute of NYU, Brooklyn, New York, 2012. [14] O. Aichholzer, R. Fabila-Monroy, T. Hackl, M. van Kreveld, A. Pilz, P. Ramos, B. Vogtenhuber, Blocking Delaunay triangulations, Comput. Geom. 46 (2) (2013) 154–159. [15] S. Ishtiaque Ahmed, M. Hasan, A. Sopan, Vindictive Voronoi games and stabbing Delaunay circles, in: Proc. International Symposium on Voronoi Diagrams in Science and Engineering, 2010, pp. 124–131.
Contents lists available at ScienceDirect
Information Processing Letters www.elsevier.com/locate/ipl
Mutual witness proximity graphs Boris Aronov a,∗ , Muriel Dulieu a , Ferran Hurtado b a b
Dept. of Computer Science and Engineering, Polytechnic School of Engineering, New York University, Brooklyn, NY, USA Dept. de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Barcelona, Spain
a r t i c l e
i n f o
Article history: Received 18 August 2013 Received in revised form 8 March 2014 Accepted 2 April 2014 Available online 16 April 2014 Communicated by R. Uehara Keywords: Computational geometry Proximity graphs Witness graphs Delaunay graphs Gabriel graphs Rectangle of influence graphs
a b s t r a c t This paper describes one variation on witness proximity graphs called mutual witness proximity graphs. Two witness proximity graphs are said to be mutual when, given two sets of points A and B, A is the vertex set of the first graph and the witness set of the second one, while B is the witness set of the first graph and the vertex set of the second one. We show that in the union of two mutual witness Delaunay graphs, there are 2 edges, where n = | A | + | B |, which is tight in the worst case. We also always at least n− 2 show that if two mutual witness Delaunay graphs are complete, then the sets A and B are circularly separable; if two mutual witness Gabriel graphs are complete, then the sets A and B are linearly separable; but two mutual witness rectangle graphs might be complete, with A and B not linearly separable. © 2014 Elsevier B.V. All rights reserved.
1. Introduction A proximity graph is a graph whose vertices correspond to some geometric objects, and an edge connects two of them when the objects are considered to be neighbors according to some proximity criterion. These graphs have been widely used for spatial data analysis [1,2], pattern recognition (see [3,4] and references therein), data mining [5], and in many disciplines where classification or interpolation is required [2,4,6]. On the other hand, as connecting neighbors naturally yields an aesthetically appealing layout of a graph, proximity graphs have also been a topic thoroughly studied in the area of graph drawing [7,8]. We consider proximity graphs on point sets on the plane. Given a point set S, the Delaunay graph DG( S ), the Gabriel graph GG( S ), and the rectangle of influence graph RIG( S ) of S all have vertex set S. Given p , q ∈ S, there is an edge connecting p and q in DG( S ) if there is some disk with p and q on its boundary, whose interior is empty of
*
Corresponding author. E-mail addresses: [email protected] (B. Aronov), [email protected] (M. Dulieu), [email protected] (F. Hurtado). http://dx.doi.org/10.1016/j.ipl.2014.04.001 0020-0190/© 2014 Elsevier B.V. All rights reserved.
points from S. The vertices p and q are adjacent in GG( S ) if the closed disk with diameter pq covers no other points from S. Finally, there is an edge connecting p and q in RIG( S ) if the closed axis-aligned rectangle B ( p , q) with p and q as opposite corners contains no other points from S. These graphs belong to the Delaunay family of graphs, and define neighbor relationships that describe how the points interact; see [3] for a survey on the topic and its applications. The rectangle of influence graph was introduced by Ichino and Sklansky in [9]. In the same work they also defined the mutual neighborhood graph of a point set A against a second point set B, denoted MNG( A | B ): its vertex set is A, and there is an edge connecting p , q ∈ A when the rectangle B ( p , q) contains no points from B (with the exception of p and q themselves, should they belong to B). The motivation behind their definition was to consider the union of the graphs MNG( A | B ) and MNG( B | A ) as a descriptor of the interaction between the sets A and B. A systematic study of witness proximity graphs, which are proximity graphs on a point set with respect to a second point set, was undertaken in [10–13] and also studied later in other works such as [14] and [15]. For the sake of
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conciseness we don’t give here the general definition but concentrate on the two graphs we study in the following sections. We define the witness Delaunay graph of a point set A of vertices in the plane, with respect to a point set B of witnesses, denoted DG− ( A , B ), as the graph with vertex set A in which two points x, y ∈ A are adjacent when there is an open disk whose boundary passes through x and y and that does not contain any witness w ∈ B. Notice that we don’t care about the presence of elements from A \ B inside the disk. DG− ( A , B ) is a negative-witness graph because the witnesses from B “prevent” adjacencies. DG− ( A , ∅) is simply the complete graph K | A | . DG− ( A , A ) is precisely the Delaunay graph DG( A ), which under standard non-degeneracy assumptions is a triangulation, denoted by DT( A ). For a pair of points p, q in the plane, we denote by D pq the closed disk with diameter pq. The witness Gabriel graph GG− ( A , B ) is defined by two sets of points A and B; A is the set of vertices of the graph and B is the set of witnesses. There is an edge xy in GG− ( A , B ) if, and only if, there is no point of B in D xy \ {x, y }. It is a negative witness graph. GG− ( A , A ) is precisely the original Gabriel graph GG( A ). Following the rationale of Ichino and Sklansky [9], it is natural to wonder what would be the characteristics of the two interdependent witness graphs that we call mutual witness graphs, corresponding to the witness Delaunay graphs and to the witness Gabriel graphs. In both cases the two graphs would be mutual in the following sense: Given two sets of points A and B, A is the vertex set of the first graph and the witness set of the second one, while B is the witness set of the first graph and the vertex set of the second one. In Sections 2 and 3 we consider the properties of the mutual witness Delaunay graphs and the mutual witness Gabriel graphs, respectively, the first one being the main subject of this work. We conclude in Section 4 with an observation on the mutual neighborhood graphs originally introduced by Ichino and Sklansky. We assume hereafter strong general position of the point sets A and B: no three collinear points and no four concyclic points. Removing this assumption involves complicating some arguments and invalidating other theorems below; for several of the objects there are no agreed upon standard definitions in presence of degeneracies. 2. Mutual witness Delaunay graphs We first consider the joint size of the mutual witness Delaunay graphs DG− ( A , B ) and DG− ( B , A ): Theorem 1. Given disjoint sets of points A and B, the number 2 , where of edges in DG− ( A , B ) ∪ DG− ( B , A ) is at least n− 2 n = | A | + | B |. This bound is tight for n ≥ 2. Proof. For the lower bound, color points of A black and points of B white. Consider DT( A ∪ B ), the Delaunay triangulation of A ∪ B. Each triangle has at least one monochromatic edge, i.e., an edge incident to two vertices of the same color. A monochromatic edge pq belongs to
Fig. 1. The union of the witness Delaunay graphs DG− ( A , B ) and 2 DG− ( B , A ) in each of the figures has n− monochromatic edges and 2
n+2 1 components. Monochromatic edges of DT( A ∪ B ) are drawn solid and bichromatic ones dashed.
DG− ( A , B ) ∪ DG− ( B , A ) as there exists a disk empty of points of A ∪ B whose boundary circle passes through p and q. Therefore, the number of monochromatic edges in DT( A ∪ B ) is a lower bound on the number of edges in DG− ( A , B ) ∪ DG− ( B , A ). Any Delaunay triangulation has at least n − 2 triangles, each with at least one monochromatic edge. Since a monochromatic edge might appear in two faces, and if the number of faces is odd at least one such edge is not counted twice, the number of monochromatic edges is at 2 least n− . 2 To see that this bound is tight, we present a concrete construction; refer to Fig. 1. Place black points and m white points, with + m = n and = m ( = m − 1 in the odd case), spaced on the boundary of an ellipse in an alternating manner and such that, except for the leftmost point (for the even case only) and rightmost point, pairs of black points and pairs of white points are placed at the same x-coordinate. Consider DT( A ∪ B ). The vertical edges between pair of points at the same x-coordinate define the only monochromatic edges in DT( A ∪ B ). Indeed, for any pair of points a and b of the same color on different sides of a vertical monochromatic edge, the edge ab would cross a monochromatic edge de of DT( A ∪ B ). As de is in DT( A ∪ B ), there exists a disk with d and e on its boundary that does not contain a or b. Hence, any disk with a and b on its boundary would contain either d or e or both. Therefore, there is no non-vertical monochromatic edge ab in DG− ( A , B ). Thus the only edges in DG− ( A , B ) ∪ DG− ( B , A ) are the vertical ones connecting same-color vertices; and hence, the number of edges of DG− ( A , B ) ∪ DG− ( B , A ) is exactly n−2 2 . 2 Our next result, on a particular kind of geometric separability, proves that mutual witness Delaunay graphs contain useful information about interclass structure. Theorem 2. For disjoint sets of points A and B, if DG− ( A , B ) and DG− ( B , A ) are complete, then the sets A, B are circularly separable. The converse is not true. Proof. Consider the convex hulls CH( A ) and CH( B ). If they don’t intersect, A and B are linearly separable and therefore circularly separable. Suppose CH( A ) and CH( B ) intersect and an edge ab of DG− ( A , B ) crosses an edge cd of DG− ( B , A ). If the points a, c , b, d are in this clockwise order, consider CH({a, c , b, d}): either acb + bda > 180◦ , or cbd + dac > 180◦ ; hence, all disks D ab with a, b on their
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boundary contain a point of B, or all disks D cd with c , d on their boundary contain a point of A, a contradiction. Therefore we may assume that CH( A ) and CH( B ) intersect but no edge of DG− ( A , B ) intersects an edge of DG− ( B , A ). This means that DG− ( B , A ) must be contained in one of the interior cells of the arrangement of the segments connecting all pairs of points of A (or conversely). Suppose DG− ( B , A ) is contained in one of the interior cells of the arrangement induced by DG− ( A , B ). Consider the Delaunay triangulation DT( A ) of A; DG− ( B , A ) is entirely contained in one of its triangles e f g, as DT( A ) is a subgraph of DG− ( A , B ). By definition of the Delaunay triangulation, there is an open disk D e f g empty of vertices of A that contains on its boundary e , f , g ∈ A. Since D e f g contains B in its interior, the boundary of a slightly shrunken version of D e f g separates A from B, as claimed. To prove that the converse is not true, place m points of A and m points of B equally spaced on the boundary C of a disk in an alternating manner. Shift them very slightly so they are still in convex position (and in general position) moving the points from A outside C , and the points from B inside C . Then the two point sets are circularly separable, but the graph DG− ( A , B ) is empty. 2 Observation 1. For disjoint sets of points A and B, even if DG− ( A , B ) and DG− ( B , A ) are complete, the sets A, B are not necessarily linearly separable. Indeed A could be entirely contained in an interior face of the arrangement of the segments connecting all pairs of points of B. In the following, we present a theorem about the number of connected components in DG− ( A , B ) ∪ DG− ( B , A ), derive a result on bicolored triangulations, and pose a problem on Delaunay triangulations. Let A and B be disjoint point sets with | A | + | B | = n. As in the proof of Theorem 1, we consider the points in A white and those in B black, and build a Delaunay triangulation DT( A ∪ B ) in which we again call the edges joining two points of different colors “bichromatic,” and the edges joining two points of the same color “monochromatic.” Let hm and hb be the numbers of monochromatic and bichromatic edges on the convex hull of A ∪ B, respectively. Theorem 3. The maximum number of connected components in DG− ( A , B ) ∪ DG− ( B , A ) is at most 2n3+1 , and there exist 1 sets A and B for which this value is n+ . 2
Proof. For the lower bound, depending on whether n is even or odd we place the points as in Fig. 1, left or right, respectively. Now we turn to the upper bound. As mentioned in the proof of Theorem 1, the monochromatic edges in G = DT( A ∪ B ) are contained in DG− ( A , B ) ∪ DG− ( B , A ), therefore the number of connected components of DG− ( A , B ) ∪ DG− ( B , A ) is at most the number of monochromatic connected components of G, which we proceed to bound. We construct an auxiliary planar straight-line embedded graph P . Its vertices are the bichromatic edges of G (drawn as midpoints of the edges) and we connect two such edges if they belong to the same triangle of G (see
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Fig. 2. Edges of P are drawn dotted. The regions are represented by different shades of gray.
Fig. 2). Since each triangle of G has zero or two bichromatic edges, it gives rise to at most one edge in P , drawn as a line segment within the triangle; hence P is indeed properly embedded and crossing-free. Since every bichromatic edge of G is adjacent to two bichromatic triangles, unless it appears on the boundary of the convex hull H = CH( A ∪ B ), P is a disjoint union of p paths and c cycles, whose geometric realizations are pairwise disjoint simple polygonal curves, with cycles lying in the interior of H and paths connecting pairs of points on the boundary of H ; slightly abusing the terminology we will refer to such a path or cycle as a curve in P . The curves of P partition H into 1 + p + c connected regions, R 1 , . . . , R 1+ p +c and we will prove below that each region contains exactly one monochromatic connected component of G; once this is established, bounding the quantity 1 + p + c completes the proof. We assign a monochromatic triangle to white or black zone according to the color of its vertices. A bichromatic triangle is cut by an edge of P into a triangle and a trapezoid, one black and the other white. Let white zone be the union of the white sets thus formed and define black zone symmetrically. A region is a connected component of white zone or of black zone. As curves of P and the boundary of H form the only boundaries of regions, they coincide precisely with R 1 , . . . , R 1+ p +c . Let us prove that each of the regions R i separated by curves of P contains exactly one monochromatic component of G. The second definition of regions implies that each region R i contains at least one vertex of G and all such vertices have the same color. So it is sufficient to prove that the subgraph G [ V i ] induced by the set V i of vertices lying in R i is connected. Consider a path π connecting two vertices v , w ∈ V i . If the polygonal path corresponding to π stays in R i , we are / Vi, done. Otherwise, let v 1 w 1 ∈ E (G ), with v 1 ∈ V i , w 1 ∈ be the first edge of π leaving R i by crossing a boundary component γ of R i . As removal of γ splits H into two connected sets, π has to cross γ again for the first time, say, / V i . Recall via an edge w k v k ∈ γ , with v k ∈ V i and w k ∈ that γ is a curve in P , so since v 1 w 1 , v k w k ∈ γ , γ contains a path v 1 w 1 , v 2 w 2 , . . . , v k w k , where all v j ∈ V i and / V i . Note now that v 1 v 2 . . . v k is an, in general, nonwj ∈ simple, path in G [ V i ]: for every j < k, ether v j = v j +1 or v j v j +1 is a monochromatic edge of the triangle w j v j v j +1 and thus contained in R i and G [ V i ]. Hence by replacing the following portion of π : v 1 w 1 . . . w k v k by v 1 v 2 . . . v k , we eliminate one contiguous section of π where it lies outside of G [ V i ]. Repeating this for each such section, we obtain a new path π that connects v to w and stays within G [ V i ], completing the proof of connectivity of this induced subgraph.
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We now proceed to count paths and cycles of P in order to bound the value of 1 + p + c. By Euler’s formula, G has
3n − hm − hb − 3
(1)
edges and
2n − hm − hb − 2
(2)
triangles. As every triangle contains at least one monochromatic edge, which is counted in two triangles, unless it lies on the boundary of H , from (2) we deduce that the number of monochromatic edges in G is at least
2n − hm − hb − 2 2
+
hm 2
=n−1−
hb 2
(3)
.
Combining (1) and (3), we conclude that the number of bichromatic edges in G is at most
hb
(3n − hm − hb − 3) − n − 1 − 2 hb 3hb − 2 = hb + 2n − hm − −2 , = 2n − hm − 2
2
from which we deduce that the number of interior bichromatic edges in G is at most
2n − hm −
3hb 2
− 2.
As each cycle in P contains at least three interior bichromatic edges of G, there are c ≤
2n−hm − 3
3hb 2
−2
such
cycles; and as the endpoints of each path in P are bichroh matic edges of H , there are p = 2b such paths. Therefore, the number of monochromatic components in G is 2n−hm − 3
3hb 2
−2
+ 1 = 2n−h3m +1 . Being an integer, this number is equal to 2n−h3m +1 , and hence, at most 2n3+1 . 2 p+c+1≤
+
hb 2
Corollary 4. Let S be any set of n points in general position, and let T be any triangulation of S. For any bicoloring of the points of S that produces hm monochromatic convex hull edges, the number of monochromatic connected components in the graph T is at most 2n−h3m +1 . There is an infinite set of pairs of values n and hm , for which there exist sets of n points with convex hull of size hm , admitting a bicoloring that has exactly
2n−h3m +1 monochromatic connected components. Proof. As the proof for the upper bound in the previous theorem starts with the triangulation G = DT( A ∪ B ) but makes no use of any Delaunay property, it applies to any bicolored triangulation on S = A ∪ B. For the second claim, consider a set A of hm white points in convex position, and place m − hm additional white points in the interior of their convex hull. Any triangulation of A will consist of 2m − hm − 2 triangles. Now, place one black point inside each of those triangles, obtaining a set B of black points. Connect each black point to the three white vertices of the triangle containing it. We obtain a bicolored triangulation T of S = A ∪ B, with
Fig. 3. Illustration of the proof of linear separability for mutual witness Gabriel graphs.
n = | S | = m + (2m − hm − 2) = 3m − hm − 2, and the T is number of monochromatic connected components in 1 + (2m − hm − 2) = 2m − hm − 1 = 2n−h3m +1 . 2 The upper bound in the preceding corollary applies to any triangulation, and is bounded above by 2n3+1 , a value that can certainly be reached, up to an additive constant, for sets whose convex hull has constant size. However, we believe that for Delaunay triangulations the number of components cannot be that large, which we express more precisely as an open question: Question 1. Let S be any set of n points in general position, and let DT( S ) be the Delaunay triangulation of S. Is it true that for any bicoloring of the points of S, the number of monochromatic connected components in DT( S ) is 1 ? at most n+ 2
3. Mutual witness Gabriel graphs In this section, we prove a theorem showing that mutual witness Gabriel graphs also contain information about interclass structure. Theorem 5. For two disjoint sets of points A and B, if GG− ( A , B ) and GG− ( B , A ) are complete, the sets A , B are linearly separable. The converse is not true. Proof. Suppose GG− ( A , B ) and GG− ( B , A ) are complete but the sets A , B are not linearly separable. Therefore CH( A ) ∩ CH( B ) = ∅, as two disjoint convex sets are linearly separable. Either CH( A ) ⊂ CH( B ), or CH( B ) ⊂ CH( A ), or an edge ab of GG− ( A , B ) intersects an edge cd of GG− ( B , A ). However, GG− ( A , B )⊂ DG− ( A , B ), GG− ( B , A )⊂ DG− ( B , A ), and the edges of DG− ( A , B ) and DG− ( B , A ) do not cross, as observed in the proof of Theorem 2, so the last case cannot arise. Thus, suppose CH( A ) ⊂ CH( B ), without loss of generality. Pick a point a ∈ A and a point b ∈ B. Draw a line through a perpendicular to ab. Suppose that b lies to the right of . As A is enclosed within CH( B ), there must be another point c from B to the left of (see Fig. 3). Therefore bac > 90◦ and hence a ∈ D bc , contradicting our assumption that GG− ( B , A ) is complete. To prove that the converse is not true, place points of A and of B alternating on a vertical line. Move the points
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10-1-0210. Ferran Hurtado is partially supported by projects MEC MTM2006-01267, MICINN MTM2009-07242, MINECO MTM2012-30951, Generalitat de Catalunya DGR 2009SGR1040, and ESF EUROCORES programme EuroGIGA, CRP ComPoSe: MICINN Project EUI-EURC-2011-4306, for Spain. References
Fig. 4. RG− ( A , B ) and RG− ( B , A ) are complete ( A is the set of white points and B the set of black points), but the pair ( A , B ) is not linearly separable.
of A slightly to the left, and the points of B slightly to the right, so they are in general position. The two sets of points are linearly separable but the union of the two mutual Gabriel graphs is the empty graph. 2 4. Mutual witness rectangle graphs In [9], Ichino and Sklansky introduced the mutual neighborhood graph defined by a pair of point sets ( A , B ) in the plane. This concept coincides with that of the negative witness rectangle graph RG− ( P , W ) in which there is an edge between two points p, q in P if, and only if, the rectangle B ( p , q) does not contain any witness w ∈ W . For a fixed pair of sets A, B, Ichino and Slansky focus on the relation between the two negative witness rectangle graphs RG− ( A , B ) and RG− ( B , A ). In particular, they claim that if RG− ( A , B ) and RG− ( B , A ) are complete graphs, then the pair ( A , B ) is linearly separable. Unfortunately this is not true, as demonstrated by the counterexample in Fig. 4. Acknowledgements Boris Aronov and Muriel Dulieu were partially supported by grant No. 2006/194 from the United States – Israel Binational Science Foundation and by NSF Grants CCF-08-30691, CCF-11-17336, and CCF-12-18791. Boris Aronov was also supported by NSA MSP Grant H98230-
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