Agglomerative clustering of growing squares

Abstract

\u3cp\u3eWe 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(n(log n log log n)\u3csup\u3e2\u3c/sup\u3e) space, supports queries in worst case O(log\u3csup\u3e3\u3c/sup\u3e n) time, and updates in O(log\u3csup\u3e7 \u3c/sup\u3en) amortized time. This leads to an O(n α(n) log\u3csup\u3e7 \u3c/sup\u3en) time algorithm (where α is the inverse Ackermann function) to solve the agglomerative clustering problem, which is a significant improvement over the straightforward O(n\u3csup\u3e2\u3c/sup\u3e log n) time algorithm.\u3c/p\u3