We solve two open problems posed by Goodman and Pollack[GP84] about sets of signed circular permutations (clusters of stars) arising from generalized configurations of points: recognition and efficient reconstruction (drawing). As a biproduct we get an O(n2) space data structure constructible in O(n2) time, representing the order type of a (generalized) configuration of points and from which the orientation of each triple can be found in constant time, a problem posed in [EHN]

Streinu, Ileana

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Smith ScholarWorks Computer Science: Faculty Publications Computer Science 1-1-1997 Clusters of Stars Ileana Streinu Smith College, istreinu@smith.edu Follow this and additional works at: https://scholarworks.smith.edu/csc_facpubs  Part of the Computer Sciences Commons Recommended Citation Streinu, Ileana, "Clusters of Stars" (1997). Computer Science: Faculty Publications, Smith College, Northampton, MA. https://scholarworks.smith.edu/csc_facpubs/234 This Conference Proceeding has been accepted for inclusion in Computer Science: Faculty Publications by an authorized administrator of Smith ScholarWorks. For more information, please contact scholarworks@smith.edu Clusters of StarsIleana Streinu”AbstractWe solve two open problems posed by Goodman andPollack[GP84] about sets of signed circular permuta-tions (clusters of stars) arising from generalized con-figurations of points: recognition and efficient recon-struction (drawing). As a biproduct we get an (7(n2 )space data structure constructible in 0(n2 ) time, rep-resenting the order type of a (generalized) configura-tion of points and from which the orientation of eachtride can be found in constant time. a moblem Dosedin ‘~HNJ.1 Introduction and motivationImagine that robots equiped withing visibility information from anment. The goal is to reconstruct,.cameras are send-unknown environ-a global model ofwhat the en-tironment might be like: In a simpli-fied model, assume that the observers are placed atthe vertices of a polygon, with each observer send-ing information about the combinatorial structure ofits uisibikt~ poiygon: an ordered list of vertices andedges. The receiver will verify the consistency of thedata and dmw a model matching the given combi-natorial information. As a first approximation oneis interested in reconstructing a pseudo-polygon (see[0 S96]) without regard as to whether its sides can orcarmot be stretched into straight lines, since there isevidence that stretchability is considerably harder toachieve. In this paper we conclude the work startedin [0S96] by providing efficient algorithms for recog-nizing and drawing clusters of stars, from which thepseudo-polygon drawing problem is a simple conse-quence. This way we also answer two problems posed‘Dept. of Computer Science, Smith College, Northamp-ton, MA 01063, USA. streinu~cs.smith. edu.by Goodman and Pollack in [GP84]: finding condi-tions for a cluster of stars to fit into a generalized con-figuration of points, and finding a simple procedure fordrawing the generalized configuration of points with-out going through a doubl~dualizing process.2 Definitions, Problems, RezultaWe refer to [0 S96], [GP84], [Bj93] for definitions ofpseudAine (p-line) arrangements, generalized config-urations of points, clusters of stars, local (or i-) se-quences, allowable sequences and wiring diagrams. Weassume general position throughout.Given a finite set of planar points @l,... ,pn},and a direction for each line lij through the pointspi and pj, we associate a directed cluster of stars byrecording Vi the circular sequence of all the directedlines iij around vertex pi. This doubly irfuxite peri-odic sequence is fully characterized by a half-period.A signed permutations (called a star permutation) ofindices of points different from i su%ices to representit: indices occur positively or negatively, dependingon whether the corresponding line is encountered ina direction coinciding or not to its orientation. Wewill also work with undirected clusters of stars as in[0S96], where no predefine orientation of the con-necting lines lij is assumed. The concepts can be in-troduced for generalized configurations of points aswell. For the example in Fig.1 the directed star per-mutations can be taken as: 1 :423, 2: 1~, 3: 4fi and4: 13~. The undirected stars are similar but sigrdess.Goodman and Pollack’s first problem is: Underwhat conditions can one complete a picture such asFig. 1a to a genemlized conjigumtion, with each j-linemdiatingfrom point i connecting to the i-line mdiatingfrom point j in such a way that the directions match?The goal is to find conditions verifyable in low poly-nomial time. If for every pair of points (i, j) we knewthat the direction of the underlying pseuddne was,say, from i to j, then the information in the cluster ofstars would be equivalent to the information in a chi-rotope, which can be verified by looking at all 5-tuplesand checlcing in constant time that the orientation ofall its triples is compatible to the sign patterns re-439(a) (b)Figure 1: A cluster of stars and a generalized config-uration of points from it.suking from the Grassman-Plucker relationships (see[Bj93]). This can be done naively in O(n’) time. Try-ing ail possible ways of orienting the paira, and check-ing the chirotope conditions for each would be expo-nential. We will give a simple O(n2 ) time algorithmto carry on this verification.One way of drawing the underlying p-line arrange-ment of a generalized configuration of points as a wiringdiagram is described in [GP84]. It works by using du-ality twice. Firat they construct the dual m-rangementas a wiring diagram, then apply Levy’s EnlargementLemma to connect the vertices of the arrangement byp-lines, then record the order of all the plines (fromthe wiring diagram and from Levy’s lemma) at z.~and use that as the allowable sequence for a dualarrangement. The process is quite complicated andbecause it uses Levy’s Lemma, it is not algorithmic.Hence the second open problem from [GP84]: Givena cluster of stars, realize it as a genemlized configura-tion of points without going through the double dualiz-ing procedure. Here weprovide an 0(n2 ) algorithm fordrawing the connecting plines of a generalized config-uration of points. Along the way we answer a problemmentioned in [EHN]: Find an 0(n2 ) space data struc-ture, constructible in (7(n2 ) time, for storing the ordertype of a configumtion o~points, from which the orien-tation of each triplet of points can be found in constanttime. We have realized recently that this data struc-ture has also been defied in Knuth[Kn92], page 16,but its usage seems to be novel.3 Realizability of Clusters of Stars as GeneralizedConfigurations of PointsWe identify two variants of the problem, correspond-ing to the undirected and directed cluster of staraproblem. We give an efficient algorithm for recog-nizing (undirected) clusters of stars and show thatthe original problem of Goodman and Pollack (thedirected stars) can be reduced to it at the expense ofan additional O(n2 ) factor.Problem 1. (lJndirected) cluster of stars. Givenn + 1 star permutations, the first one nO being theidentity of {1, . . . . n} and each ~i starting with O, findnecessary and sufficient conditions for them to be theA40 3 21Clusters of stars for Problem 1.Figure 2:cluster of star ~ermutations of a Generalized planarconfiguration of points with undirected underl~ing p-Iines, so that the two points O and 1 form an edge ofthe convex hull.For example, the set of star permutations: O :1234,1:0234, 2:0143, 3:0142, 4:0132 can be realized asin Fig.2.Problem 2. Directed cluster of stars. Given aset of signed star permutations, tind an efficient alg~rithm to decide when they can fit into a generalizedconfiguration of points with matching directions.We solve Problem 2 using Problem 1 as follows.For all the pairs of points, associate an instance ofProblem 1 where the two points are the distinguishededge of the convex hull. Ignore the signs for the timebeing. If the instance doesn’t have a solution, try nextpair. If it does, then check the signa (directions) forcompatibility. If they do not match for any of thepairs for which the problem 1 has a solution, thenproblem 2 does not have a solution. As an aaide weshould mention that solutions for Problems 1 and 2,when they exists, are unique with respect to the ordertype.The characterization needed to solve Problem 1 isbased on a relationship + on the set of pairs ij (whereij = ji): if in the permutation k, i occwa somewherebefore j (and after O), then detine ik + jk.Theorem 3.1 Given a cluster of stars, it is realizableas a generalized conjigumtion of points ifl the followingproperty (the A-property) holds:For an~ triple O < i < j < k ~ n – 1, exactly oneof the following is true:1. jk + ik + ij, or2. ij + ik + jk.Proof. (Sketch) The necessity is straightforward.For the sufficiency we will prove by a double inductiona dual statement: Given a cluster of stars satisfyingthe A-property, there ezists a p-line arrangement withthe p-lines numbered O, . . . , n at z-~ and whose localsequences are the giwen permutations of the stars. Theproof can be turned into a verification and reconstruc-tion algorithm, which will attempt to construct the ar-rangement just like the classical ~OS] algorithm. Acareful implement ation turns this into an O (n2 ) timealgorithm. We omit the details.4404 Efficiently drawing a generalized configura-tion of pointsWe are given a cluster of stars as in Problem 1 andassume it satisfies the A-condition. Associate a di-rected graph G with vertices labeled by pairs ij withi,j # O and an arc ij + A whenever ij < ik. TheA condition implies, among others, that the digraphis acyclic. Perform a topological sort on it. This willbe the order of the “slopes” of the connecting p-linesin the final drawing, where the vertex O is at infinityand the p-lines through O are verticals.~%n even n oddFigure 3: A switch at a vertical reference line.The drawing is done using (~) horizontal referencelines and n vertical reference lines. The horizontallines are labeled top-down by the pairs ij in the orderprovided by the topological sort. The vertical lines arelabeled left to right by the point indices 1,2,. ~., n.All pseudo-lines labeled with pairs containing k willswitch at the kth vertical (on one or between two hor-izontals, see Fig. 3) and will not cross before or afterthe kth vertical reference line. This implies that theirrelative ordering stays the same before and after theswitch, even though they may move up and down onthe horizontal reference lines, generating other cross-ings.n4s1>IJ1>14141]13Y,F@re 4: A generalized configuration of points drawnby the algorithm. Type A and type B regions arerepresented.The main ingredient of the algorithm is a functionto compute how many p-lines are above a switch point.We can do this by counting the number of inversionsin its star permutation, which yields U(n log n) timeby sorting. To improve this to linear time we usean efficient data structure. It is based on having aninverse permutation for each star permutation. Theith inverse permutation stores in its j’th position therank of the p-line Jij around vertex i, counted from thepline lio. It is straightforward that this additionaldata occupies 0(n2 ) space and can be computed in0(n2 ) time. Fkom it we can answer in constant timequestions about the orientation of triples of points, aswell as compute the number of p-lines above/belowpoints in linear time.To complete the drawing we divide it into twotypes of vertical slabs: type A and type B. The typeA slabs are centered at the vertical lines correspond-ing to the n points. The B slabs are between typeA slabs. In the A slabs we place the “spiders” fromFig.3. In the type B slabs we draw a matching. Thewhole computation takes 0(n2 ) time. As for the draw-ing per se, although it looks like being on an 0(n3 )grid, it can be done in O(n2 ), too. Figure 4 is a com-plete example of the output of our algorithm on the al-lowable sequence 23,45,13,25,15,24, 14,12,35,34 de-rived from the cluster 1 :03542, 2 :03541, 3:02154,4 : 05213 and 5 : 04213. The inverse permutationsare: 1:0*4132,2:04*132,3:021 *43,4:0324*1and 5 : 03241x.The drawing algorithm has been implemented us-ing Leda and is accessible via ftp from:cs.smith.edu/pub/streinu/code/cluster.cc.Acknowledgements. We are grateful to Joe O’Rourkefor many stimulating discussions, and to Jorjeta Jetchevafor implementing the drawing algorithm.References[Bj93][EOS][EHNl[GP84][GP93]~n92][0S96]A. Bjorner, M. Las Vergnaa, B. Sturmfels,N. White and G. Ziegler. Oriented Ma-troids, Cambridge University Press} 1993H. Edelsbnmner, J. O’Rourke and R. Sei-del. Constructing arrangements of linesand hyperplanes with applications SIAMJ. Comput. 15, (1986), 341-363.H. Everett, F. Hurtado and M. Noy.Stabbing Information of a Simple Poly-gon 8th Canadian Conference of Compu-tational Geometry, Ottawa, to appear, Au-gust 1996.J.E. Goodman and R. Pollack. Semis-pace9 of configurations, cell complexes ofarrangements, J. of Combinatorial The-ory, Series A, 37:257–293, 1984.J.E. Goodman and R. Pollack. Allowablesequences and Order Types in Discrete andComputational Geometry, in New Trendsin Discrete and Computational Geometry,J. Path (cd.), Springer Verlag, 1993D.E. Knuth. Axioms and Hulls, SpringerVerlag, New York, 1992.J. O’Rourke and I. Streinu. Verter-EdgePseudo- Visibility Graphs: Characteriza-tion and Recognition, Smith College Techreport, 1996. Also these proceedings.441

Clusters of Stars

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