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Generalized Ham-Sandwich Cuts for Well Separated Point Sets

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Generalized Ham-Sandwich Cuts for Well Separated Point Sets CCCG 2008, Montr´eal, Qu´ebec, August 13{15, 2008 Generalized Ham-Sandwich Cuts for Well Separated Point Sets William Steiger∗ Jihui Zhao‡ Abstract The present paper is in the same spirit. The starting point is a recent, interesting result about generalized B´ar´any, Hubard, and Jer´onimo recently showed that for ham-sandwich cuts. d given well separated convex bodies S1;::: ;Sd in R and Definition 1: (see [8]) Given k d +1,afamily constants βi [0; 1], there exists a unique hyperplane h ∈ Rd ≤ with the property that Vol(h+ S )=β Vol(S ); h+ is S1;::: ;Sk of connected sets in is well-separated if, ∩ i i· i the closed positive transversal halfspace of h,andh is for every choice of xi Si, the affine hull of x1;::: ;xk is a (k 1)-dimensional∈ flat in Rd. a “generalized ham-sandwich cut”. We give a discrete − Rd analogue for a set S of n points in which is parti- B´ar´any et.al. [1] proved tioned into a family S = P P of well separated 1 ∪···∪ d sets and are in weak general position. The combinato- Proposition 1 Let K1; :::; Kd be well separated convex d 3 d rial proof inspires an O(n(log n) − ) algorithm which, bodies in R and β1;::: ,βd given constants with 0 ≤d given positive integers ai Pi , finds the unique hy- βi 1. Then there is a unique hyperplane h R ≤| | ≤ + ⊂ perplane h incident with a point in each Pi and having with the property that Vol(Ki h )=βi Vol(Ki), i = + ∩ · h Pi = ai. Finally we show that the conditions as- 1;::: ;d. |suring∩ existence| and uniqueness of generalized cuts are + also necessary. Here h denotes the closed, positive transversal halfs- pace defined by h: that is the halfspace where, if Q is an interior point of h+ and z K h,thed-simplex i ∈ i ∩ ∆(z1; :::; zd;Q)isnegatively oriented [1]. Specifying this 1 Introduction. choice of halfspaces is what forces h to be uniquely de- termined. B´ar´any et. al. give analogous results for Given d sets S ;S ; :::; S Rd, a ham-sandwich cut is a 1 2 d ∈ such generalized ham-sandwich cuts for other kinds of hyperplane h that simultaneously bisects each Si.“Bi- + + well separated sets that support suitable measures. sect” means that µ(Si h )=µ(Si h−) < , h ;h− We are interested in a version of Proposition 1 for the closed halfspaces∩ defined by ∩and a∞ suitable, h µ n points partitioned into d sets in Rd; i.e., points in “nice” measure on Borel sets in Rd, e.g., the volume. S = P1 Pd, Pi Pj = φ, i = j, S = n. For this The well known ham-sandwich theorem guarantees the context∪···∪ we use ∩ 6 | | existence of such a cut. As with other consequences of the Borsuk-Ulam theorem [10] there is a discrete ver- Definition 2: Point sets P1; :::; Pd are well separated sion that applies to sets P1;::: ;Pd of points in general if their convex hulls, Conv(P1);::: ;Conv(Pn), are well position in Rd. For example Lo et. al [9] gave a direct separated. proof of a discrete version of the ham-sandwich theorem We need some kind of general position, and will assume which inspired an efficient algorithm to compute a cut. the following weaker form. More recently Bereg [4] studied a discrete version of a result of B´ar´any and Matouˇsek [2] that showed the exis- Definition 3: Points in S = P1 ::: Pd have weak general position if, for each (x ;:::∪ ;x∪), x P , tence of wedges that simultaneously equipartition three 1 d i ∈ i R2 aff(x1;::: ;xd) is a (d 1)-flat that contains no other measures on (they are called equitable two-fans). By − seeking a direct, combinatorial proof of a discrete ver- point of S. sion (for counting measure on points sets in R2)hewas This does not prohibit more than d data points from able to strengthen the original result and also obtained a being in a hyperplane, e.g. if they are all in the same beautiful, nearly optimal algorithm to construct an eq- Pi. For the discrete analogue of a generalized cut we uitable two-fan. Finally, Roy and Steiger [12] followed use a similar path to obtain complexity results for several Definition 4: Given positive integers a P ,an other combinatorial consequences of the Borsuk-Ulam i i (a ; :::; a )-cut is a hyperplane h for which h≤|P | = φ theorem. 1 d i and h+ P = a , 1 i d. ∩ 6 | ∩ i| i ≤ ≤ ∗Department of Computer Science, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8004; As in Proposition 1, a cut is a transversal hyperplane ( steiger,zhaojih @cs.rutgers.edu.) (here incident with at least one data point in each P ) f g i 20th Canadian Conference on Computational Geometry, 2008 and h+ its positive halfspace. The discrete version of We can write h as p Rd : p; v = t ,andh+,the Proposition 1 is positive transversal{ halfspace∈ ,ash i } h+ = p Rd : p; v t : Theorem 2 If P1;::: ;Pd are well separated point sets { ∈ h i≤ } in Rd, then (i) if an (a ; :::; a )-cut exists, it is unique. 1 d The relation p h+ is invariant under translation and Also (ii) if the points have weak general position, cuts rotation. ∈ exist for every (a1; :::; ad), 1 ai Pi . ≤ ≤| | Proof of Theorem 2: The proof is by induction. The This might be proved using some results of [1] via a base case d = 2 is probably folklore (but see [11]). Well standard argument that takes the average of n proba- separated implies that points in P1 may be dualized to bility measures, one centered at each data point. The (red) lines having positive slopes and those in P2,to variance of the measures is decreased to zero, and one (blue) lines having negative slope. If a red/blue inter- argues about the limit (see [7]). Instead we give a direct section q has a1 red lines and a2 blue lines above it, combinatorial proof in Section 2. In addition, and of in- vertex q is the dual of an (a1;a2)-cut. It must be the dependent interest, we show that both well-separated unique one because the red levels have positive slope and weak general position are also necessary for every and blue ones have negative slope, proving (i). possible (a1; :::; ad)-cut to exist and be unique. An anal- If P1 and P2 also have weak general position, every ogous converse is likely to hold for Proposition 1. red/blue intersection in the dual is a distinct vertex, There is also interest in the algorithmic problem P P of them in all, and each is incident with just | 1|·| 2| where, given n points distributed among d well sepa- those two lines. This implies that each level in the first rated sets in Rd, and in weak general position, the ob- arrangement has a unique intersection with every level ject is to find the cut for given a1;::: ;ad. Our proof of the second, proving (ii). In fact the unique inter- of Theorem 2 leads to the formulation of an efficient, section can be found in linear time by adapting the d 3 O(n(log n) − ) algorithm for generalized cuts. This ap- prune-and-search algorithm given in [11] for intersection pears in Section 3. Throughout, because of space limi- of median levels. tations, some details are omitted. Next, suppose the claim holds for dimension j<d. d Let π be a hyperplane that separates P1 from Si=2 Pi. Fix a point x Conv(P1), project each data point z 2 Proof of the Discrete Version. d ∈ ∈ Si=2 Pi onto π, and write Pi0 for the set of images in π of the points z P . There are several equivalent forms of the well separated ∈ i property for connected sets [3], in particular the fact Fact 1: P20;::: ;Pd0 are d 1 well-separated sets in π. that such a family is well separated if and only if the − If not there is a d 3flatρ π that meets all convex hulls are well separated. Others include − ⊂ Conv(Pi0);i 2. But the span of x and ρ is a d 2 flat that meets≥ all , a contradiction. − 1. Sets S1;::: ;Sk;k d + 1 are well separated if P1;::: ;Pd and only if, when I≤and J are disjoint subsets of Fact 2: If P ;::: ;P have weak general position and 1;::: ;d+ 1, there is a hyperplane separating the 1 d if x P1 then P20;::: ;Pd0 have weak general position in sets Si;i I from the sets Sj;j J. ∈ ∈ ∈ π. Rd 2. S1;::: ;Sd are well separated in if and only if A transversal flat ρx π has dimension d 2byFact1. there is no (d 2)-dimensional flat that meets all If it contains one point⊂ x from each P ;i− > 1 and any − i i0 Conv(Si);i=1;::: ;d. d other z Si=2 Pi0,thenx and ρx span a hyperplane that violates∈ weak general position for P ;::: ;P . In view of Definition 2, they hold for the discrete context 1 d as well. Therefore the induction hypotheses apply to P20;::: ;Pd0 . Given points pi Conv(Pi);i =1;::: ;d (not Given a point x Conv(P )and(a ;::: ;a ), a hy- ∈ 1 2 d necessarily data points in S), the hyperplane h perplane h containing∈ x is an (a ;::: ;a ) semi-cut (or ≡ x 2 d aff p1;::: ;pd is a transversal hyperplane of dimension just a semi-cut) if, for each i>1, it is incident with a { } d 1.
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