Agglomerative Clustering of Growing Squares

We study an agglomerative clustering problem motivated by interactive glyphs in geo-visualization. Consider a set of disjoint square glyphs on an interactive map. When the user zooms out, the glyphs grow in size relative to the map, possibly with different speeds. When two glyphs intersect, we wish to replace them by a new glyph that captures the information of the intersecting glyphs. We present a fully dynamic kinetic data structure that maintains a set of n disjoint growing squares. Our data structure uses $$O\bigl (n \log n \log \log n\bigr )$$ O ( n log n log log n ) space, supports queries in worst case $$O\bigl (\log ^2 n\bigr )$$ O ( log 2 n ) time, and updates in $$O\bigl (\log ^5 n\bigr )$$ O ( log 5 n ) amortized time. This leads to an $$O\bigl (n\,\alpha (n)\log ^5 n\bigr )$$ O ( n α ( n ) log 5 n ) time algorithm to solve the agglomerative clustering problem. This is a significant improvement over the current best $$O\bigl (n^2\bigr )$$ O ( n 2 ) time algorithms.

An Efficient Kinetic Range Query for One Dimensional Axis Parallel Segments

We present a kinetic data structure named Kinetic Interval Graph (KI-Graph) for performing efficient range search on moving one dimensional axis-parallel segments. This finds applications in Artificial Intelligence such as robotic motion. The structure requires O(n) storage. The time taken per update when a critical event occurs is O (1) thereby improving responsiveness when compared to the kinetic segment trees, while the overall updates across all segments at a time instance is at most n/2. Also, range query is performed efficiently in θ (k) time, where k segments are reported.

KINETIC COLLISION DETECTION FOR SIMPLE POLYGONS

We design a simple and elegant kinetic data structure for detecting collisions between polygonal (but not necessarily convex) objects in motion in the plane. Our structure is compact, maintaining an active set of certificates whose number is proportional to a minimum-size set of separating polygons 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 object vertices. It is difficult to characterize the efficiency of our structure for lack of a canonical definition of external events. Nevertheless we give an easy upper bound on the worst case number of certificate failures.