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Binary spa e partition (BSP) tree is one of the most popular data stru tures in omputational geometry. We proved that the lower bound on the exa t size of BSP trees for a set of n isotheti re tangles in the plane is 2n o(n) when the re tangles tile the underlying spa e, this losed the gap between the upper and lower bounds for this ase. Also, in general, for a set of n isotheti re tangles in the plane, we improved the lower bound from 94n o(n) to 73n o(n). A B C l 1 l 2 l 3 l1 l2 l3 D A C B D E Figure 1: Example of a BSP and its BSP tree 1 Introdu tion Given a set of obje ts S in an Eu lidean spa e, a hyperplane h is alled a guillotine ut if h separates S into two disjoint sets. Ea h fragment of the obje t belongs solely to one of the parts it falls in. A guillotine ut l is alled perfe t if l does not interior interse t any obje t of S. A binary spa e partition (BSP) is a s heme for re ursively dividing a set of obje ts by guillotine uts until all obje ts are separated. This division pro ess an be naturally represented as a binary tree (BSP tree) where a node represents a region of the plane and stores the ut that splits the plane into two parts that its two hildren represent; ea h leaf of the BSP tree represents the nal partitioning of the plane and stores at most one fragment of an input obje t. Sin e a guillotine ut may split an obje t into two, the number of fragments in the end of the partition, equivalently the number of leaves in the BSP tree, may ex eed the number of input obje ts. Given a set of obje ts, the total number of the resulting fragments in the partition is alled the size of the BSP. A BSP of a set of n obje ts is alled perfe t if the size of the BSP is n. A box in dimension d is an interval fx : e x s; e; s 2 Rdg. A box in R2 is alled a isotheti re tangle. In this paper we simply all it a re tangle without onfusion. A partition of the box [0; 1℄d is a set of boxes su h that ea h point a 2 [0; 1℄d lies within some box and no two boxes have a ommon interior point. A partition in R2 is also alled a tiling. A pa king of a box B is a set of interior disjoint boxes inside B. Figure 1 shows a BSP of three re tangles and its orresponding BSP tree. In Figure 1, there are 3 re tangles A;B;E. The ut l1 uts E(C [D) into two fragments: C and D, then the ut l2 separates re tangles A and C, and the ut l3 separates B and D. The orresponding BSP tree is in Figure 1. The size of this BSP tree is 4. There exists a BSP tree with size 3 for this example. Sin e the introdu tion of BSPs, the orresponding data stru ture BSP trees in [4℄ have be ome one of the most popular data stru tures. They present a way to implement a { 2 { geometri divide and onquer strategy. BSPs have numerous appli ations su h as: graphi s (hidden surfa e removal, shadow generation), omputational geometry (ray-tra ing, visibility problems, solid geometry), roboti s (motion planning), spatial databases and approximation algorithms, see e.g. [1, 7℄. A binary spa e partition problem onsists in minimizing the total number of fragments generated by the partition. The binary spa e partition problem is a fundamental problem for appli ations. In this paper we fo us on re tangles in the plane be ause su h re tangles are very important in appli ations and more omplex obje ts are often repla ed by their bounding re tangles. They also arise in plane tiling problems ( onstru ting histograms in the plane), and as subproblems when higher dimensional hyper re tangles are proje ted onto a plane [1℄. In this paper, as in [2, 5, 6℄, any BSP will be auto-partition, i.e., the ut lines will be the extensions of the input segments. Be ause the obje ts are re tangles, any ut will be either verti al or horizontal. The BSP in Figure 1 is a auto-partition. In 1992, Paterson and Yao proved an upper bound of 12n for the size of BSP of a set of n re tangles [6℄, later improvements led to 4n in [2, 5℄, re ently Berman et. al. proved upper bounds of 3n for n re tangles and 2n for tiling ase [1℄. The urrent best known lower bounds were 3 2n o(n) for the tiling ase [1℄ and 9 4n o(n) for the pa king ase [3℄. In this paper we answer the open problem in [1℄ to lose the gap for the tiling ase and improve the lower bound from 9 4n o(n) to 7 3n o(n) for the pa king ase. 2 Lower bound for the size of BSP of re tangles in tiling ase Let us asso iate a set of re tangles S to a simple dire ted graph GS: let L = fl1; l2; : : : ; lmg be the set of the longest line segments of S. We de ne that li ut lj if, by extending one side of li, the segment lj is the rst that is ut into two parts. Ea h segment l an ut at most two segments (on either side of l). We de ne the simple dire ted graph GS = (V;E): V = L; (li; lj) 2 E if li uts lj. For example, in Figure 2, there are 7 longest line segments for A;B: l1; l2; : : : ; l7. Among them only line l4 uts l5. The y les of the graph GS an help to he k if a set of re tangles has a perfe t ut. Ea h y le of GS orresponds to a set of line segments and its bounded ell. For example, in Figure 3, the lines a; b; ; d orrespond to a length 4 y le in the graph. We all a re tangle whose four sides form a y le a join. A y le (or its orresponding ell) is alled tou hed if there is a guillotine ut interse ting the interior of the ell bounded by the lines of the y le. The terms join, ell, tou hed were rst de ned in [3℄. For example, in Figure 4, lines l1; l2; l3; l4 form a y le C. The line l5 interse ts the ell of the y le, so C is tou hed. Lemma 1. Given a set of re tangles S, there is no perfe t BSP, if and only if there is an untou hed y le.