n If S is in convex position, then thee ae exactly k convex k-gons detemined by subsets of S. In geneal, howeve, S may detemine fa fewe convex k-gons.

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1 Counting Convex Polygons in Plana Point Sets Joseph S. B. Mitchell a;1, Günte Rote b, Gopalakishnan Sundaam c, and Gehad Woeginge b a Applied Mathematics and Statistics, SUNY Stony Book, NY jsbm@ams.sunysb.edu b Technische Univesität Gaz, Institut fü Mathematik, Steyegasse 30, A-8010 Gaz, Austia. ote@ftug.dnet.tu-gaz.ac.at, woeginge@fmatbds01.tu-gaz.ac.at c Envionmental Systems Reseach Institute, 380 New Yok St., Redlands, CA gsundaam@esi.com Abstact Given a set S of n points in the plane, we compute in time O(n 3 ) the total numbe of convex polygons whose vetices ae a subset of S. We give ano(m n 3 ) algoithm fo computing the numbe of convex k-gons with vetices in S, fo all values k = 3;:::;m; peviously known bounds wee exponential (O(n dk=2e )). We also compute the numbe of empty convex polygons (esp., k-gons, k» m) with vetices in S in time O(n 3 ) (esp., O(m n 3 )). Key wods: Computational geomety, convexity, combinatoics, dynamic pogamming 1 Intoduction Let S denote a set of n points in the plane. A subset T S is said to be in convex position if T is the vetex set of a convex polygon, and we then say that T detemines a convex k-gon, whee k = jt j. We say that a polygon is empty if it contains no point of Sin its inteio. 1 Patially suppoted by gants fom Boeing Compute Sevices, Hughes Reseach Laboatoies, Ai Foce Office of Scientific Reseach contact AFOSR , and by NSF Gants ECSE and CCR Pepint submitted to Elsevie Pepint 27 Apil 1999

2 n If S is in convex position, then thee ae exactly k convex k-gons detemined by subsets of S. In geneal, howeve, S may detemine fa fewe convex k-gons. Conside, fo example, a set S of n =3Kpoints, with K points along each of thee ays emanating fom the oigin, such that the thee ays positively span the plane. Then thee is no convex k-gon detemined by a subset of S fo any value of k 5. In this note we show that the total numbe of convex polygons detemined by S can be computed in time O(n 3 ). Futhe, we show that the numbe of convex k-gons detemined by S can be tabulated, fo all values of k = 3;:::;m, in total time O(m n 3 ). Within these same time bounds, we can compute the total numbe of empty convex polygons detemined by S (in time O(n 3 )) o tabulate, fo k = 3;:::;m, the numbe of empty convex k-gons detemined by S (in time O(m n 3 )). Finally, we can compute fo a given point u (not necessaily fom the set S) the numbe of convex k-gons detemined by S that contain u in total time O(m n 3 ), fo all k» m. In geneal, the numbes that we compute can be lage e.g., n points in convex position detemine oughly 2 n convex polygons. We assume a eal RAM model of computation in which aithmetic opeations on lage integes can be done in constant time. Relation to pevious wok. Khulle and Mitchell [6] showed how to compute the numbe of tiangles detemined by S containing each point p 2 S in total time O(n 2 ). Rote et al. [8] showed how to count the total numbe of convex k-gons detemined by S in time O(n k 2 ), impoving ove the tivial bound of O(n k ). This esult has ecently been impoved to O(n dk=2e )byrote and Woeginge [9]. Unfotunately, these bounds ae exponential in k. Ou bounds ae polynomial in n and k. Counting poblems ae closely elated to optimization poblems fo convex polygons, because both types of poblems show diffeent aspects of thei natual common genealization: the enumeation of convex polygons. Futhemoe, both types of poblems can be solved by dynamic pogamming. This is also the appoach that we will take hee. The details of ou ecusions ae simila to some ecusions applied by Akin, Khulle, and Mitchell [1] in the context of vaious optimization poblems associated with selecting subsets of S to enclose with a fence", o simila ecusions of Eppstein et al. [5] fo computing polygons of smallest aea. Chvátal and Klincsek [2] also apply simila ecusions to the poblem of seaching fo maximum weight convex point sets (see also the book of Kote, Lovász, and Schade [7], pp ). An oveview of vaious optimization poblems associated with polygons detemined by a given point set can be found in Eppstein [4]. 2

3 2 Counting all convex polygons Conside fist the poblem of computing the total numbe of convex polygons detemined by S. Fo simplicity of pesentation, we assume that no thee points of S ae colinea and that no two points of S have the same y-coodinate; ou methods easily extend to the degeneate cases. Fix attention on one point s 2 S; we will compute in time O(n 2 ) the numbe of convex polygons detemined by S such that s is the lowest vetex of the polygon. Let U s denote the open halfspace of all points whose y-coodinate is geate than that of s. Following the sweep-line" algoithm of [1] fo computing maximum value enclosues, we devise a set of ecusions based on imagining a sweep-ay" otating clockwise about s, stating with a leftwad ay and ending with a ightwad ay out of s. Let H p;q denote the open halfspace to the left of the oiented line pq. Let p and q be points of S U s, such that q 2 H s;p (i.e., q is counteclockwise fom p with espect to s, and p 6= q 6= s). Then, we define f(p; q; s) tobe the numbe of convex polygons which (a) have vetices among the points S; (b) lie in the closue of the cone U s H s;p ; and (c) have qp and ps as two edges. Refe to Figue 1. Then, the numbe f(p; q; s) isobtained as follows: We test p q s Fig. 1. Notation used in fomulating the ecusions. each point 2 U s ; if 2 H p;q H s;q, then we add up the numbes f(q; ; s), obtaining f(p; q; s) =1+ 2Us Hp;q Hs;q f(q; ; s): (1) 3

4 This ecusion can be evaluated systematically, if we tabulate the values of f(p; q; s), fo all q 2 S U s H s;p, fo p in clockwise angula ode about s. The justification of the expession fo f(p; q; s) is simple: A convex polygon satisfying (a) (c) is eithe a tiangle (with thid edge qs) o is obtained by attaching the tiangle pqs to a convex polygon counted in f(q; ; s), fo some 2 S in the cone U s H s;q that lies left of the oiented line pq. As witten, these ecusions can be evaluated in O(n) time fo each choice of p; q; s, and oveall O(n 4 ) time. But this can be impoved by noting that, fo fixed values of q and s, we can evaluate f(p; q; s) incementally fo points p 2 U s H q;s in clockwise ode about q. Specifically, if the points of S in H q;s ae labelled p 1 ;p 2 ;:::, in clockwise ode about q, then we can compute f(p i ;q;s) fom f(p i 1 ;q;s) accoding to f(p i ;q;s)=f(p i 1 ;q;s)+ 2H pi ;q H q;pi 1 f(q; ; s): (2) Refe to Figue 2. Fist, f(p 1 ;q;s) is evaluated diectly, using (1), and then all othe values ae obtained using (2). We can chage off the wok involved in the summation to the points, each of which is consideed at most once duing the angula sweep about q. The esult is that, fo the fixed choice of s, the values f(p; q; s) can be tabulated in total time O(n 2 ). (The O(n) angula sots of points about each choice of q can be done, using the standad method of computing the aangement of dual lines, in total time O(n 2 ); see [3].) p p 2 1 p i 1 q p i s Fig. 2. Notation used in ecusion (2). The total numbe of convex polygons detemined by S is obtained simply by 4

5 summing: s2s p2u s q2u s H s;p f(p; q; s): 3 Counting convex k-gons Now conside the poblem in which we count only convex k-gons, fo a given intege k. (In the pocess of counting k-gons, we will in fact also count convex j-gons, fo j» k.) Let p and q be points of U s, such that q 2 H s;p. We define g(p; q; j; s) to be the numbe of convex polygons which fulfill the conditions (a) (c), as in the pevious section, and have exactly j edges. Then the numbe g(p; q; j; s) is obtained as follows: g(p; q; j; s) = 8>< >: 1; if j =3; 2Us Hp;q Hs;q g(q; ;j 1; s); othewise; (3) and the total numbe of convex k-gons is given by summing: s2s p2us q2us Hs;p g(p; q; k 1; s): As in (2), we can impove the efficiency of evaluating the ecusion in (3) incementally, obtaining g(p i ;q;j;s) fom g(p i 1 ;q;j;s), plus a sum (ove ) of g(q; ;j 1; s), fo points p 1 ;p 2 ;::: soted in clockwise ode about q. 4 Counting empty convex polygons If we estict attention to empty convex polygons we only have to ewite the ecusions (1) and (3), adding the estiction that tiangle pqs have no points of S in its inteio. Fo example, the numbe of j-edge convex polygons (with lowest point s) whose convex hull encloses no points of S is given by the following ecusion: G(p; q; j; s) = 8>< >: 2Us Hp;q Hs;q 0; if pqs S 6= ;; 1; if pqs S = ; and j =3; G(q; ;j 1; s); othewise. 5

6 Note that the test to see if pqs is empty can be done in constant time: As shown in Theoem 2.1 of [5] o in Theoem 1 of [1], we can pepocess the set S in time O(n 2 )into a data stuctue of size O(n 2 ), so that fo any quey consisting of thee points of S, we can check in constant time if the tiangle detemined by the thee points is empty o not. (Actually, the theoem allows us to sum the weights" of the contained points; but we do not need this geneality.) 5 Counting convex polygons containing a given point Anothe vaiation is the poblem of computing the numbe of convex k-gons containing a given point u (not necessaily fom the set S) in the inteio. The case k =3was consideed in [6], whee an O(n 2 ) algoithm was given fo computing, fo each u 2 S, the numbe of tiangles containing u. Fo each of the fou vesions of ou poblem given above (all convex polygons vs. convex k-gons; abitay convex polygons vs. empty convex polygons), we can solve the vaiant in which we equie point u to be inside the polygons that ae counted. We again fix attention on counting those polygons that have point s 2 S as thei lowest vetex. If u =2 U s the answe is 0. Othewise, we can compute, fo example, the numbe g 0 (p; q; j; s) ofj-edge convex polygons which fulfill the conditions (a) (c), as befoe, and contain point u, by solving the following ecusion: g 0 (p; q; j; s) = 8>< >: 2Us Hp;q Hs;q 0; if u 62 H s;p U s ; 1; if u 2 pqs and j =3; 2Us Hp;q Hs;q g 0 (q; ;j 1; s); g(q; ;j 1; s); if u 62 pqs; if u 2 pqs; whee g(p; q; j; s) has been aleady computed by equations (3). 6 Summay Fo the space complexity, note that we neve have need to stoe moe than O(n 2 )oo(kn 2 )numbes (fo the cases of counting all polygons, o all k-gons, espectively). The time complexity is obtained simply by multiplying O(n) 6

7 (the numbe of choices fo s) by the numbe of enties (O(n 2 )oo(kn 2 )) that we need to stoe fo each choice of s. Theoem 1 Given a set S of n points in the plane, in time O(n 3 ) (esp., O(mn 3 )) and space O(n 2 ) (esp., O(mn 2 )), the total numbe of convex polygons (esp., convex k-gons, fo all k» m) whose vetices ae a subset of S can be computed. Also, in the same time and space complexities, the total numbe of empty convex polygons (empty convex k-gons) can be computed, o the total numbe of convex polygons (convex k-gons) containing a given point can be computed. Note: We can extend the above ecusions to compute the numbe of t- subsets T S whose convex hull is a k-gon, in time O(k(t k +1) n 4 ). (This numbe is denoted by #(t; k) in [8,9].) Fo t = k this educes to the pat of the above theoem. It is staightfowad to decease the stoage equiement to O(n), at the expense of a facto of n in the unning time. Acknowledgement We thank the efeees fo obseving the impovement by a facto of n in the unning time fo the evaluation of the ecusions. Refeences [1] E.M. Akin, S. Khulle, and J.S.B. Mitchell, Geometic knapsack poblems, Algoithmica 10 (1993), [2] V. Chvátal and G. Klincsek, Finding lagest convex subsets, in: Poc. 11th Southeasten Conf. Combinatoics, Gaph Theoy, and Computing, (Boca Raton, Floida, 1980), Vol. II; Congessus Numeantium 29 (1980), [3] H. Edelsbunne and L. Guibas, Topologically sweeping an aangement, Jounal of Compute and System Sciences 38 (1989), [4] D. Eppstein, New algoithms fo minimum aea k-gons, in: Poc. 3d ACM-SIAM Sympos. Discete Algoithms (SODA), 1992, pp [5] D. Eppstein, M. Ovemas, G. Rote, and G. Woeginge, Finding minimum aea k-gons, Discete & Computational Geomety 7 (1992), [6] S. Khulle and J.S.B. Mitchell, On a tiangle counting poblem, Infomation Pocessing Lettes 33 (1990),

8 [7] B. Kote, L. Lovász, and R. Schade, Geedoids, in the seies Algoithms and Combinatoics, Vol. 4, Spinge-Velag, [8] G. Rote, Z. Wang, G. Woeginge, and B. Zhu, Counting k-subsets and convex k-gons in the plane, Infomation Pocessing Lettes 38 (1991), [9] G. Rote and G. Woeginge, Counting convex k-gons in plana point sets, Infomation Pocessing Lettes 41 (1992),