We present a simple dynamic programming based method for counting straight-edge triangulations of planar point sets. This method can be adapted to solve related problems such as  nding the best triangulation of a point set according to certain optimality criteria, or generating a triangulation of a point set uniformly at random. We have implemented our counting method. It appears to be substantially less slow than previous methods:
instances with 20 points, which used to take minutes, can now be handled in less than a second, and instances with 30 points, which used to be solvable only by employing several workstations in parallel over a substantial amount of time, an now be solved in about one minute on a single standard workstation.International Max Planck Research Schoo

Ray, Saurabh

Seidel, Raimund

English

idUS. Depósito de Investigación Universidad de Sevilla

ASimple and Less SlowMethodfor Counting Triangulations and for Related ProblemsExtended AbstratSaurabh Raya;1and Raimund SeidelbaMax-Plank-Institut fur Informatik Stuhlsatzenhausweg 85, 66123 Saarbruken, GermanybUniversitat des Saarlandes, Im Stadtwald, 66123 Saarbruken, GermanyAbstratWe present a simple dynami programming based method for ounting straight-edge triangulations of planarpoint sets. This method an be adapted to solve related problems suh as nding the best triangulation of a pointset aording to ertain optimality riteria, or generating a triangulation of a point set uniformly at random.We have implemented our ounting method. It appears to be substantially less slow than previous methods:instanes with 20 points, whih used to take minutes, an now be handled in less than a seond, and instaneswith 30 points, whih used to be solvable only by employing several workstations in parallel over a substantialamount of time, an now be solved in about one minute on a single standard workstation.1. IntrodutionIn reent years there has been some interest instudying the set T (S) of straight-edge triangula-tions assoiated with a nite planar point set S.Typial problems are ounting, i.e. given S deter-mine jT (S)j, random generation, i.e. given S ran-domly generate a triangulation in T (S) with uni-form probability, or optimization, i.e. given S ndthe triangulation in T (S) that satises some opti-mality riterion.The ounting problem is at this point not knownto be in P . It is known that jT (S)j is exponentialin n = jSj with the best urrently known lower [4℄and upper [14℄ bounds of roughly 2:33nand 59n.Some previous work on the ounting problem ad-dressed ases where S has some speial struture[13,10,11,6℄. An algorithm for the general ase wasgiven by Avis and Fukuda [5℄ whih was based ontheir reverse-searhparadigm and hene ounts viaenumeration. For another enumerative method seeRambau's TOPCOM page [12℄. The so far best al-Email addresses: saurabhmpi-sb.mpg.de (SaurabhRay), rseidels.uni-sb.de (Raimund Seidel).1The rst author is supported by the International-Max-Plank-Researh-Shool in Saarbruken.gorithm was given by Aihholzer [2℄. It is a divide-and-onquer type algorithm based on the notion ofso-alled paths of a triangulation. Erkinger [8℄ im-plemented a prototype of this algorithm and sub-sequently Gimpl [9℄ wrote a \prodution type" im-plementation that even inludes parallelization inthe sense of distributing work to several mahines.For usage see \The Triangulation Homepage" [3℄.The random generation and the optimizationproblem an learly be solved, though expensively,using the enumeration based method of Avis andFukuda [5℄. Aihholzer's method an as well beadapted to solve the random generation problemand also ertain versions of the optimization prob-lem, namely those that are \deomposable" in thefollowing sense: Suppose one wishes to nd the op-timum triangulation of a polygon P with interiorpoints and subjet to the onstraint that some setE of edges must be inluded in the triangulation.Suppose that E is suh that it indues a partitionof P into two polygons P1and P2. Then the op-timum triangulation of P subjet ot E must beomputable from the optimum triangulations of Pisubjet to E \ Pi, with i = 1; 2. Examples of opti-mization problems that satisfy this deomposabil-ity ondition are MinMax-Area [7℄[p. 142℄ or min-imum total edge-length (a.k.a. minimum weight).20th EWCG Seville, Spain (2004)20th European Workshop on Computational GeometryWe present a simple dynami-programmingbased algorithm for the ounting problem. Itan be viewed as a divide-and-onquer algorithmthat in addition stores omputed results of sub-problems for later reuse. The algorithm an benaturally adapted to solve the random generationproblem and to solve optimization problems thatare deomposable in the sense desribed above.We report on a prototype implementation of theounting algorithm.2. The AlgorithmWe onsider the slightly more general problemof omputing the number T (P ) of triangulationsof a simple polygon P that ontains the point setS and whose orners are all in S. In the originalproblem polygon P is just the onvex hull of S.In ourses on algorithm design the ase that Phas no \interior points" (i.e. the orners of P arepreisely S) is a ommon homework problem forpratiing the appliation of dynami program-ming: Choose an arbitrary edge e = [a; b℄ of P as\base edge." Let C(e) be the set of all \andidate"verties  of P so that the triangle spanned bya; b;  is ontained in P . Consider some  2 C(e).The diagonals [a; ℄ and [b; ℄ partition polygon Pinto three parts: the trianlge (a; b; ) and two sub-polygons Paand Pb(whih may be trivial, i.e. anedge). The number of triangulations of P that in-lude triangle (a; b; ) is then given by the produtT (Pa)T (Pb). Sine in any triangulation of P thebase edge [a; b℄ has to be part of some triangle wegetT (P ) =X2C(e)T (Pa)T (Pb) :Of ourse the subproblems like T (Pa) are solvedreursively using the same method and using [a; ℄as base edge. This hoie of base edge ensures thatall subproblems ever onsidered only involve sub-polygons of P formed by utting P along just onediagonal. Sine there are only O(n2) suh sub-polygons, only O(n2) many subproblems need tobe solved in total and omputed results an bestored and reused. This leads to an O(n3) \time"and O(n2) \spae" algorithm, where the quotationmarks are to remind the reader that the statedbounds only hold in a model where the ost of stor-ing of and operating on arbitrarily large integers isonstant. In our ase we will be dealing with inte-gers not larger than 59n, i.e. representable by O(n)many bits. This leads to anO(n4logn) time boundand anO(n3) spae bound in the more appropriateword model of omputation.Let us now onsider the general ase where thereare \interior point," i.e. S onsists of more thanjust the orners of P . We will proeed as in themethod outlined above: Choose some base edgee = [a; b℄ from the boundary of P . The andidateset C(e) now must ontain all points  2 S thattogether with e an span a triangle in some tri-angulation of P . Thus C(e) onsists of all  2 Sso that the triangle Dspanned by a; b;  is on-tained in P and no point of S lies in the interiorof triangle D. (In this abstrat we assume non-degeneray.) If a andidate point  is a orner of Pwe an proeed exatly as in the simple ase out-lined above and ompute the number of triangu-lations of P ontaining the triangle Dby solvingtwo reursive subproblems. If  lies in the interiorof P the number of triangulations of P ontainingDis given by the number of triangulations of thepolygon P= P nD, i.e. the polygon P with edge[a; b℄ replaed by the hain of two edges [a; ℄; [; b℄.Thus we only need to solve one reursive subprob-lem, namely nding the number of triangulationsof P. Note that this problem is easier than theoriginal problem in the sense that there are onefewer interior points (and if there are no interiorpoints we know how to solve the problem). Thusour algorithm proeeds as follows:(i) Choose some base edge e = [a; b℄.(ii) Determine the andidate set C(e).(iii) For eah  2 C(e) that is not a orner of Pompute T (P) reursively.(iv) For eah  2 C(e) that is a orner of Psolve two reursive subproblems and om-pute T (Pa)T (Pb).(v) Return the sum of the omputed numbers.In this proedure some subpolygon Q of P maybe onsidered many times. In the usual top-downdynami programming fashion we will omputeT (Q) only one and store the result for later reuse.Unfortunately the arising subpolygons do not havein general suh a nie form as they do in the sim-ple ase without interior points. Thus we annotuse an array for storage but need to resort to ahash table with a anoni sequene of vertex namesaround polygon Q serving as the key for Q.Marh 24{26, 2004 Seville (Spain)n h m avg. #triangulations avg. time(seonds) avg. time(seonds)Dynami Programming Path Method13 3 10 117017.2 0.009 0.25018 3 15 434650561.2 0.064 43.51223 3 20 2175541362109.0 0.982 28 3 25 17295702671778911.6 24.973 33 3 30 9064438879955990031.2 537.890 18 15 3 61156327.0 0.030 1.57023 15 8 148363536731.6 0.178 262.23228 15 13 596631344845165.6 2.569 33 15 18 2615696070967273559.2 55.623 23 20 3 49106130174.0 0.077 69.63428 20 8 124263097179506.8 0.480 33 20 13 1303793620633224385.4 10.793 Table 1: Results for n = h+m points, withm points randomly hosen in a onvex h-gon; average taken over 5runs. An asterisk indiates that the program did not omplete a single instane of that size within 90 minutes.3. HeuristisThe algorithm outlined above makes no restri-ions on the hoie of the base edge. We havefound the following two heuristis protable forthis hoie.If P has an edge e with jC(e)j  2, then hoosee as base edge.Otherwise x a line ` that has approximately onehalf of the points of S on eah side and as long asinterior points are onsidered as andidate pointshoose as base edge always a boundary edge of theurrent polygon that intersets `. For dividing linesof subproblems always hoose lines parallel to theinitial `.The reasonability of the rst heuristi is obvi-ous. The seond heuristi is reminisent of the pathmethod of Aihholzer [2℄. It aims to steer the al-gorithm towards a rapid breakup of the polygonsonsidered.4. Preliminary Experimental ResultsWe have implemented our dynami program-ming algorithm in C++ using the g++ Compilerand STL libraries. Oswin Aihholzer [1℄ kindly pro-vided us with Gimpl's C-implentation [9℄ of hispath-based method. Thus diret runtime ompar-isons were possible.Our experiments were run on a single mahinewith a 2.40 GHz Intel Pentium 4 Proessor, with512MBmemory and 512KBCahe, running underLinux 2.4.21.4.p4. We did not attempt to use theparallel feature of Gimpl's program that allows tospread work over several mahines.We ompared the two implemtations in a setof experiments where we onsidered m points dis-tributed uniformly at random in a onvex h-gon.We report here the results for pairs (h;m) withh 2 f3; 15; 20g and h+m 2 f13; 18; 23; 28; 33g. Foreah pair of parameters Table 1 reports the aver-age over 5 runs of the omputed number of trian-gulations and the average time (in seonds) takenfor the omputation by eah program. An asteriskindiates that the program did not solve a singleinstane of that size within 90 minutes.20th European Workshop on Computational GeometryWe also tried our method on the largest exam-ples that Gimpl reported on, namely a set of 32points representing European apitals and a set of30 random points [3℄. Gimpl ran these exampleson lusters of mahines with a 1GHz Athlon Thun-derbird Proessor and with 256 MB memory run-ning under Linux. He did not report any expliitrunning times for these example but stated thatthese was the largest he ould solve in \reasonabletime" employing parallelism and several mahines.We have been told that \reasonable time" in thisontext is to mean \several weeks."Our implementation solved the 32 point problemin 70.6 seonds and the 30 point problem in 42.9seonds.The largest example we tried was a set of 35points, with 32 points distributed randomly in atriangle. On our standard mahine this exampleled to exessive paging due to the large size of therequired hash table. However on a SPARC om-pute server we ould omplete this example in 15minutes 32 seonds using one proessor and 4 GBof main memory. Adjusting hash table size to avail-able main memory may lead to a more graefuldegradation of performane of our algorithm whenthe input size inreases.5. DisussionWe will not disuss in this abstrat how ourmethod an be applied to the random generationproblem and to the optimization problem. Theadaptions that need to bemade are fairly standard.Our method does not parallelize as easily asAihholzer's method sine reusing already om-puted results may require non-trivial ommunia-tion between proessors.We are in the proess of analyzing the runningtime of our method. Empirially we notied thatthe number of reursive alls when omputing T (S)is always aboutpT (S). So far we have been unableto prove this or any other non-trivial statementabout the running time.Of ourse determining the true omputationalomplexity of plane triangulation ounting still re-mains an open problem.6. AknowledgmentsWe would like to thank Oswin Aihholzer forproviding valuable information about the existingimplementations and also for kindly providing theimplementations themselves.Referenes[1℄ O. Aiholzer, Private Communiation.[2℄ O. Aiholzer, The path of a triangulation, in Pro.15th Symp. on Comp. Geometry 1999, pp. 14{23.[3℄ http://www.is.tugraz.at/igi/oaih/tri-angulations/ounting/ounting.html[4℄ O. Aihholzer, F. Hurtado and M. Noy, On theNumber of Triangulations Every Planar Point SetMust Have, in Pro. 13th Annual Canadian Confereneon Computational Geometry CCCG 2001, Waterloo,Canada, 2001, pp. 13{16. An improved version is toappear in CGTA.[5℄ D. Avis and K. Fukuda, Reverse Searh forEnumeration. Disrete Appl. Math. 65, 1996, pp. 21{46.[6℄ R. Baher, Counting triangulations of ongurations.Manunsript,http://arxiv.org/abs/math.CO/0310206.[7℄ H. Edelsbrunner, Geometry and Topology forMesh Generation. Cambridge University Press,2001.[8℄ B. Erkinger, Struktureigenshaften vonTriangulierungen. Master's Thesis, TU-Graz, 1998.[9℄ J. Gimpl, Enumeration von Triangulierungen.Master's Thesis, TU-Graz, 2002.[10℄ F. Hurtado and M. Noy, Counting triangulations ofalmost onvex polygons. Ars Combinatoria 45, 1997,pp. 169{179.[11℄ V. Kaibel and G.M. Ziegler, Counting LattieTriangulations. To appear in: \British CombinatorialSurveys" (C. D. Wensley, ed.), Cambridge UniversityPress.[12℄ http://www.zib.de/rambau/TOPCOM/[13℄ D. Randall, G. Rote, F. Santos, and J. Snoeyink,Counting triangulations and pseudo-triangulations ofwheels, in Pro. 13th Annual Canadian Confereneon Computational Geometry CCCG 2001, Waterloo,Canada, 2001, pp. 149{152.[14℄ F. Santos and R. Seidel, A better upper bound onthe number of triangulations of a planar point set.Journal of Combinatorial Theory, Series A 102(1),2003, pp. 186{193.

A simple and less slow method for counting triangulations and for related problems

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A simple and less slow method for counting triangulations and for related problems

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