Kinetic maintenance of context-sensitive hierarchical representations for disjoint simple polygons

We describe how to construct and kinetically maintain a tessellation of the free space between a collection of <i>k</i> disjoint simple polygonal objects with a total of <i>N</i> vertices, <i>R</i> of which are reflex. Our linear size tessellation consists of pseudo-triangles and has the following properties: (<i>i</i>) it contains disjoint outer hierarchical representations of all objects where the size of the outer boundary of these representations is proportional to a minimum link separator for the objects, and (<i>ii</i>) any line segment in the free space intersects at most <i>O</i>((<i>k</i> + log <i>R</i>) log <i>N</i>) pseudo-triangles (each of constant size).We maintain our tessellation by using the Kinetic Data Structure (KDS) framework. Our structure is compact, maintaining an active set of certificates whose number is linear in the size of a minimum link subdivision for the objects. It is also responsive; on the failure of a certificate invariants can be restored in time logarithmic in the total number of vertices. While its efficiency is difficult to establish precisely, it is shown that at most <i>O</i>(<i>k</i> + κ<inf>max</inf>log <i>R</i>)log <i>N</i> events happen during straight line motion of one object <i>A</i> in the context of <i>k</i> (fixed) others, where κ<inf>max</inf> denotes the maximum size of the minimum link polygon separating object <i>A</i> from the rest, during the motion.Furthermore, ray shooting queries (that use point location) can be answered in <i>O</i>((<i>k</i> + log <i>R</i>) log <i>N</i>) time for rays with arbitrary direction.

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