Many Order Types on Integer Grids of Polynomial Size
MMany Order Types on Integer Gridsof Polynomial Size
Manfred ScheucherJuly 31, 2020
Fakult¨at f¨ur Mathematik und Informatik,FernUniversit¨at in Hagen, Germany, { manfred.scheucher } @fernuni-hagen.de Two point configurations { p , . . . , p n } and { q , . . . , q n } are of the same or-der type if, for every i, j, k , the triples ( p i , p j , p k ) and ( q i , q j , q k ) have thesame orientation. In the 1980’s, Goodman, Pollack and Sturmfels showedthat (i) the number of order types on n points is of order 4 n + o ( n ) , (ii) allorder types can be realized with double-exponential integer coordinates, andthat (iii) certain order types indeed require double-exponential integer coor-dinates. In 2018, Caraballo, D´ıaz-B´a˜nez, Fabila-Monroy, Hidalgo-Toscano,Lea˜nos, Montejano showed that n n − o ( n ) order types can be realized on aninteger grid of polynomial size. In this article, we show that n n − o ( n ) ordertypes can be realized on an integer grid of polynomial size, which is essentiallybest-possible. A set of n labeled points { p , . . . , p n } in the plane with p i = ( x i , y i ) induces a chirotope ,that is, a mapping χ : [ n ] → { + , , −} which assigns an orientation χ ( a, b, c ) to eachtriple of points ( p a , p b , p c ) with χ ( a, b, c ) = sgn det x a x b x c y a y b y c . Geometrically this means χ ( a, b, c ) is positive (negative) if the point p c lies to the left(right) of the directed line −−→ p a p b through p a directed towards p b . Figure 1 gives anillustration. We say that two point sets are equivalent if they induce the same chirotopeand denote the equivalence classes as order types . An order type in which three or morepoints lie on a common line is called degenerate .1 a r X i v : . [ c s . C G ] J u l – p a p b p c p d Figure 1:
A chirotope with χ ( a, b, c ) = + and χ ( a, b, d ) = − . Goodman and Pollack [11] (cf. [16, Section 6.2]) showed the number of order types on n points is of order exp(4 n log n + O ( n )) = n n + o ( n ) . While the lower bound follows from asimple recursive construction of non-degenerate order types, the proof of the upper bounduses the Milnor–Thom theorem [17, 21] (cf. [19, 22]) - a powerful tool from real algebraicgeometry. The precise number of non-degenerate order types has been determined forup to 11 points by Aichholzer, Aurenhammer, and Krasser [1, 2] (cf. [15]). For theirinvestigations, they used computer-assistance to enumerate all “abstract” order typesand heuristics to either find a point set representation or to decide non-realizability.Similar approaches have been taken by Fukuda, Miyata, and Moriyama [10] (cf. [9]) toinvestigate order types with degeneracies for up to 8 points. It is interesting to notethat deciding realizability is an ETR-hard problem [18] and, since there are exp(Θ( n ))abstract order types (cf. [6] and [8]), most of them are non-realizable. For more details,we refer the interested reader to the handbook article by Felsner and Goodman [7].Gr¨unbaum and Perles [13, pp. 93–94] (cf. [4, pp. 355]) showed that there exist de-generate order types that are only realizable with irrational coordinates. Since we aremainly interested in order type representations with integer coordinates in this article,we will restrict our attention in the following to the non-degenerate setting.Goodman, Pollack, and Sturmfels [12] showed that all non-degenerate order types canbe realized with double-exponential integer coordinates and that certain order types in-deed require double-exponential integer coordinates. Moreover, from their constructionone can also conclude that n n − o ( n ) order types on n points require integer coordinatesof almost doubly-exponential size as outlined: For a slowly growing function f : N → R with f ( n ) → ∞ as n → ∞ and m = (cid:98) n/f ( n ) (cid:99) (cid:28) n , we can combine each of the( n − m ) n − m ) − o ( n − m ) = n n − o ( n ) order types of n − m points with the m -point con-struction from [12], which requires integer coordinates of size exp(exp(Ω( m ))). Anotherinfinite family that requires integer coordinates of super-polynomial size are the so-called Horton sets [3] (cf. [14]), which play a central role in the study of Erd˝os–Szekeres–typeproblems.In 2018, Caraballo et al. [5] showed that at least n n − o ( n ) non-degenerate order typescan be realized on an integer grid of size Θ( n . ) × Θ( n . ). In this article, we im-prove their result by showing that n n − o ( n ) order types can be realized on a grid of sizeΘ( n ) × Θ( n ), which is essentially best-possible up to a lower order error term. Theorem 1
The number of non-degenerate order types which can be realized on aninteger grid of size (3 n ) × (3 n ) is of order exp(4 n log n − O ( n log log n )) . Proof of Theorem 1
Let n be a sufficiently large positive integer. According to Bertrand’s postulate, we canfind a prime number p satisfying n (cid:98) log n (cid:99) < p < n (cid:98) log n (cid:99) . As an auxiliary point set, we let Q p = { ( x, y ) ∈ { , . . . , p } : y = x mod p } . The point set Q p contains p points from the p × p integer grid, and each two points havedistinct x -coordinates. Moreover, Q p is non-degenerate because, by the Vandermondedeterminant, we havedet a b ca b c = ( b − a )( c − a )( c − b ) (cid:54) = 0 mod p, and hence χ ( a, b, c ) (cid:54) = 0 for any pairwise distinct a, b, c ∈ { , . . . , p } (cf. [20]). In thefollowing, we denote by R ( Q p ) = { ( y, x ) : ( x, y ) ∈ Q p } the reflection of Q p with respectto the line x = y .Let α = 2 n and m = α · (2 n + n ). For sufficiently large n , we have 2 n + n ≤ n and m + 2 p ≤ n . Our goal is to construct 4 n − o ( n ) different n -point order types on theinteger grid G = {− p, . . . , m + p } × {− p, . . . , m + p } . We start with placing four scaled and translated copies of Q p , which we denote by D, U, L, R , as follows: • To obtain D , we scale Q p in x -direction by a factor αn (cid:98) log n (cid:99) and translate by( αn , − p ). All points from D have x -coordinates between αn and 2 αn and y -coordinates between − p and 0; • To obtain U , we scale Q p in x -direction by a factor αn (cid:98) log n (cid:99) and translate by( αn , m ). All points from U have x -coordinates between αn and 2 αn and y -coordinates between m and m + p ; • To obtain L , we scale R ( Q p ) in y -direction by a factor αn (cid:98) log n (cid:99) and translateby ( − p, αn ). All points from L have y -coordinates between αn and 2 αn and x -coordinates between − p and 0; • To obtain R , we scale R ( Q p ) in y -direction by a factor αn (cid:98) log n (cid:99) and translateby ( m, αn ). All points from R have y -coordinates between αn and 2 αn and x -coordinates between m and m + p .Each pair of points ( l, r ) ∈ L × R spans an almost-horizontal line-segment with absoluteslope less than n . Similarly, each pair of points from D × U spans an almost-vertical line-segment with absolute reciprocal slope less than n . As depicted in Figure 2, these line-segments bound ( p − p ) almost-square regions. Later, we will distribute the remaining3 n αn αn αn (cid:98) log n (cid:99) DpL RU p α n α n α n (cid:98) l og n (cid:99) α n Figure 2:
An illustration of the construction. The four copies
D, U, L, R of Q p are highlightedgray and the ( p − p ) almost-square regions are highlighted green. n − p points among these almost-square regions in all possible way to obtain manydifferent order types.For every pair of distinct points d , d from D , the x -distance between them is at least αn (cid:98) log n (cid:99) and their y -distance is less than p . Since both points d , d have x -coordinatesfrom { αn , . . . , αn } and non-positive y -coordinates, the line d d can only pass throughpoints of G with y -coordinate less than p · mαn (cid:98) log n (cid:99) < n ≤ αn . (Recall that m = α · (2 n + n ) ≤ αn and p ≤ n (cid:98) log n (cid:99) .) We conclude that every point from U ∪ L ∪ R or from the almost-square regions lies strictly above the line d d . Similar argumentsapply to lines spanned by pairs of points from U , L , and R , respectively. Note that, inparticular, our construction has the property that for any point q from an almost-squareregion, the point set D ∪ U ∪ L ∪ R ∪ { q } is non-degenerate.4 b cdαn αn αn α n (cid:98) l og n (cid:99) l r r (cid:96) (cid:96) δ Figure 3:
An almost-square region.
Almost-square regions
Consider an almost-square region A with top-left vertex a ,bottom-left vertex b , top-right vertex c , and bottom-right vertex d , as depicted in Fig-ure 3. The two almost-horizontal line-segments (cid:96) , (cid:96) bounding A meet in a commonend-point l ∈ L . Let r , r ∈ R denote the other end-points of (cid:96) and (cid:96) , respectively,which have y -distance αn (cid:98) log n (cid:99) .Since we assumed n to be sufficiently large, the points l and r i (for i = 1 ,
2) have x -distance between αn and 2 αn , and a and b have x -distance between αn and 2 αn .Hence, we can bound the y -distance δ between a and b by12 α (cid:98) log n (cid:99) = αn (cid:98) log n (cid:99) · αn α n ≤ δ ≤ αn (cid:98) log n (cid:99) · α n αn = 2 α (cid:98) log n (cid:99) . Moreover, since a and b lie on an almost-vertical line (i.e., absolute reciprocal slope lessthan n ), the x -distance between a and b is less than α (cid:98) log n (cid:99) n . An analogous argumentapplies to the pairs ( a, c ), ( a, c ), and ( a, c ), and hence we can conclude that the almost-square region A contains at least (cid:18) α (cid:98) log n (cid:99) − α (cid:98) log n (cid:99) n (cid:19) ≥
15 ( α (cid:98) log n (cid:99) ) points from the integer grid G , provided that n is sufficiently large. Placing the remaining points
We have already placed p points in each of the four sets D, U, L, R . For each of the remaining n − p points, we can iteratively choose one of the5 p − p ) almost-square regions and place it, unless our point set becomes degenerate.To deal with these degeneracy-issue, we denote an almost-square region A alive if thereis at least one point from A which we can add to our current point configuration whilepreserving non-degeneracy. Otherwise we call A dead .Having k points placed (4 p ≤ k ≤ n − k points determine (cid:0) k (cid:1) lines which might kill points from our integer grid and some almost-square regions become dead . That is,if we add another point that lies on one of these (cid:0) k (cid:1) lines to our point configuration, weclearly have a degenerate order type.To obtain a lower bound the number of alive almost-square regions, note that allalmost-square regions lie in an ( αn ) × ( αn ) square and that each of the (cid:0) k (cid:1) lines killsat most αn grid points from almost-square regions. Moreover, since each almost-squareregion contains at least ( α (cid:98) log n (cid:99) ) grid points, we conclude that the number of alivealmost-square regions is at least( p − p ) − (cid:18) n (cid:19) · αn ( α (cid:98) log n (cid:99) ) ≥ (cid:18) n (cid:98) log n (cid:99) (cid:19) for sufficiently large n , since n (cid:98) log n (cid:99) ≤ p ≤ n (cid:98) log n (cid:99) and α = 2 n .Altogether, we have at least (cid:32) (cid:18) n (cid:98) log n (cid:99) (cid:19) (cid:33) n − p = n n − O (cid:16) n log log n log n (cid:17) possibilities to place the remaining n − p points. Each of these possibilities clearlygives us a different order type because, when we move a point q from one almost-squareregion into another, this point q moves over a line spanned by a pair ( l, r ) ∈ L × R or( d, u ) ∈ D × U , and this affects χ ( l, r, q ) or χ ( d, u, q ), respectively. This completes theproof of Theorem 1. References [1] O. Aichholzer, F. Aurenhammer, and H. Krasser. Enumerating order types for small pointsets with applications.
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