Incremental convex hulls BEN LEVY| DISCRETE ALGORITHMIC GEOMETRY

Lecture contents  Convex-hulls  Basic definitions  Data-structure representation  Lower bounds  Geometric preliminaries  Offline algorithm  Semi-dynamic algorithm  Fully-dynamic algorithm

Basic definitions - affine-hull

Basic definitions – convex-hull

Basic definitions – Polytope

Representation of polytopes  A polytope is generally represented by incidence graph of its faces.  A node is stored for each face.  An arc for each pair of incident faces.  Faces are incident if their dimensions differ by one and if one is contained in the other.  In Figure 8.1 we see four points that generates an affine space of dimension three.

Representation of polytopes  A j-skeleton of a polytope is the subgraph of the incidence of dimension at most j.  Examples: The 1-skeleton of a polytope is simply made up of the vertices and edges of that polytope.  The 0-skeleton of a polytope is simply made up of the vertices of that polytope.  Two (d-1)-faces of a polytope are said to be adjacent if they are incident to a common (d-2)-face.  Thus, incidence graph encodes the adjacency graph.

Lower bounds

Geometric preliminaries

 Observation : red faces are those who are visible from P (dashed). blue faces are those who are not visible from P (shaded). purple faces are tangent to P (bolded).

Geometric preliminaries  Observation : red faces are those who are visible from P. blue faces are those who are not visible from P purple faces are tangent to P.

Geometric preliminaries

C H2 H1 P G

Geometric preliminaries C H2 H1 P G

Geometric preliminaries

 Lemma Let C be a polytope and P a point in general position. If C has n vertices and does not contain P, then the set of the proper faces of C that are purple with respect to P is isomorphic, for the incidence relationship, to the set of faces of a (d-1)-polytope whose number of vertices is at most n.  For example, for a 3-polytope, the number of new faces created are isomorphic to the set of purple faces.

Deterministic algorithm

Deterministic algorithm – incremental step

Deterministic algorithm – Cost  Phase 1- proportional in time to the number of facets created at the previous step.  Total cost of phase 1 over all the steps is thus dominated by the total number of facets created.  Cost of phase 2,3,4 is proportional to the: number of red or purple faces + number of faces incident to red face + number of faces incident to purple faces.  Total cost of phases 2,3,4 is proportional to number of changes undergone by the incidence graph.

Deterministic algorithm – Cost

Online convex-hull  A similar problem, but the points inserted are randomized.  To find a red facet, we use the influence graph and not lexicographic order of the inserted points.

Online convex-hull

Online convex-hull - The Influence Graph  Leafs of the influence graph are regions without conflicts.  All incident nodes with a node B are contained in node B.  BAD is region free from conflicts, while BAC is region with conflict with vertex D.  CBA is region free from conflicts.

Online convex-hull - The Influence Graph  The algorithm- Initial step:  We build incidence graph of d-simplex formed by the first d+1 points.  We build influence graph with a node for each of the regions that in bijection to the d-2 faces of this simplex (vertices in our example) See below figure.

Online convex-hull - The Influence Graph  The algorithm- Location step:  Detects all “red” regions.  Visit recursively all nodes in conflict with P, starting from the root.

Online convex-hull - The Influence Graph  The algorithm- Update step:  If there is a conflicting leaf which is free of conflict on S, it is in bijection with some d-2 purple face of conv(S), vertices in 2D case, and this face is contained in a red facet. For example, in the last figure, P is in bijection with vertex A and D and both of them contained in a d-1 polytope (AD edge), facet in our case.

Online convex-hull - The Influence Graph

Dynamic convex-hull  Similar to the on-line algorithm, but now we want to allow deletion of points, much more complex task. A fully dynamic algorithm must keep some information for all the points in the current set, be they vertices of the current convex-hull or not.  For points information we use augmented influence graph as discussed in chapter 6.

Dynamic convex-hull  Augmented influence graph:  The algorithm maintains a graph whose nodes correspond to regions defined over the current set of points.  After each deletion, we rebuild the graph into the state it would have been in, had the deleted point never been inserted.  The graph depends only on the insertion order of the points in the current set.

Dynamic convex-hull

 Dynamic convex-hull – Insertion step  Insertion the n-th point into the convex hull is carried out exactly as in the on-line algorithm in section 8.5 with a single difference.  While we are locating the object in the influence graph, each detected conflict is added to the conflict lists.

Dynamic convex-hull

 Dynamic convex-hull – Deletion step - Example  After deleting point D:

Dynamic convex-hull