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On Regular Vertices of the Union of Planar Convex Objects Esther Ezra Janos Pach Micha Sharir. Intersection vertices of the union. Union of Planar objects. Input: A collection of n compact connected sets in the plane, with a simple boundary. Vertices of the union:
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On Regular Vertices of the Union of Planar Convex ObjectsEstherEzra Janos PachMicha Sharir
Intersection vertices of the union. Union of Planar objects Input: A collection of ncompact connectedsets in the plane, with a simple boundary. Vertices of the union: The intersection vertices that appear on the boundary of their union are all boundary intersections that are not contained in the interior any set. Each boundary sets intersect in at most a constant number of points.
The number of vertices of the union A grid lower bound construction: (n2) vertices. Assume that the boundary of each set is simple. In the worst case, the number of these vertices is proportional to the complexity of the entire arrangement!
Regular vertices Suppose now we only would like to bound the number of regular vertices of the union. Intersection points of two boundaries that intersect twice. Irregular Regular
Lower bound construction Is it possible to construct (n2) regular vertices on the boundary of the union? The grid construction can yield up to (n) regular vertices. A super-linear lower bound construction seems non-trivial!
Lower bound constructionVs. incidences (n4/3) lower bound construction: • Construct a set of n points and n lines with (n4/3) incidences. • Map each line to a long and thin rectangle, and each point to a small disk. A collection of rectangles and disks with (n4/3) regular vertices on the union boundary. Each point-line incidence is now a regular vertex of the union.
Previous results The case of pseudodisks (sets with a non-simple boundary) : The boundaries of any pair of sets intersect at most twice. All vertices on the union boundary are regular. R 6n-12 = O(n). [Kedem et al. 1986] Note! In the(n4/3) lower bound construction each pair of boundary sets intersect at most 4 times. They are not pseudodisks! Number of regular vertices of the union.
Previous results The general case: • The sets are convex with a non-simple boundary.A bound that depends on the number of irregular vertices of the union.R 2I +6n-12[Pach & Sharir 1999]. • The sets are convex with a simple boundary.R = O*(n3/2)[Aronov et al. 2001]. • The sets are not necessarily convex with a simple boundary.R = O(n2-), for some > 0 [Aronov et al. 2001]. Generalizing the result of [Kedem et al. 1986]. Number of irregular vertices of the union. O(n3/2+), for any > 0.
Our result The sets are convex with a simple boundary: R = O*(n4/3) . Improve the bound R = O*(n3/2) of [Aronov et al. 2001]. Almost tight!
From regular vertices toboundary touchings The sets remain convex with a simple boundary after the transformation. Apply a transformation to the sets that transforms regular vertices of the union to boundary touchings: C C C’ C’ Before the transformation: C and C’ intersect regularly on the boundary of the union. After the transformation: C and C’ touch at a single point on the boundary of the union.
Spines The spineof a set C is the segment that connects the leftmost and the rightmost points of C. Crucial property: After applying the transformation, if C, C’ create a regular vertex on the union boundary, the spines , ’of C, C’ are disjoint. C ’ C’ C
Restricted setting Input: Two collections A, B of convex sets with a simple boundary, s.t, all the spines of A lie below all the spines of B. Observation: Each red-blue regular vertex v on the boundary of the union must lie either on the upper envelope of the sets in A, or on the lower envelope of the sets in B. ’ C’ v C1 C2 C Otherwise, v is contained in the interior of C1 C2 .
Restricted setting Suppose now that v lies on the upper envelope EA+ of A. Enumerate the arcs of EA+ as 1, …, m . 1 2 m … Let A* = {1 ,…, m }, m O(|A|) . Due to upper envelope properties.
A more restricted setting Input: Two collections A1* A*, B1 B, s.t, for each A1* and C B1, lies fully below the spine C of C. Observation: Each red-blue regular vertex v on the boundary of the union must lie on the lower envelope of the sets in B1. C C’ v Otherwise, v is contained in the interior of C’ .
The more restricted setting Each red-blue regular vertex v on the boundary of the union, created by A1* andB1, lies in the sandwich region between the arcs of A1* and the lower envelope ofB1 . The overall number of these regular vertices is O( |A1*| + |B1| ) . Sandwich region property.
The bi-clique decomposition Let G = (V, E) be the graph with: V = input sets, E = (C,C’) , s.t., C, C’ create a regular vertices on the boundary of the union of the input sets. The final goal: Decompose G into complete bipartite graphs A1*B1as above, with overall storage O*(n4/3).
The decomposition Decomposition is performed in two step: • Complete bipartite graphs AB, s.t, all the spines of A lie below all the spines of B.Overall decomposition size: O*(n4/3). • Complete bipartite graphs A1*B1, A1* A*, B1 B ,s.t, for each A1* and C B1, lies fully below the spine C of C. Overall decomposition size: O*(n4/3). Using standard range-search technique. More involved.
