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 $O(n (\log n \log\log n)^2)$ space, supports queries in worst case $O(\log^3 n)$ time, and updates in $O(\log^7 n)$ amortized time. This leads to an $O(n\alpha(n)\log^7 n)$ time algorithm to solve the agglomerative clustering problem. This is a significant improvement over the current best $O(n^2)$ time algorithms.Comment: 14 pages, 7 figure

Short Plane Supports for Spatial Hypergraphs

A graph $G=(V,E)$ is a support of a hypergraph $H=(V,S)$ if every hyperedge induces a connected subgraph in $G$. Supports are used for certain types of hypergraph visualizations. In this paper we consider visualizing spatial hypergraphs, where each vertex has a fixed location in the plane. This is the case, e.g., when modeling set systems of geospatial locations as hypergraphs. By applying established aesthetic quality criteria we are interested in finding supports that yield plane straight-line drawings with minimum total edge length on the input point set $V$. We first show, from a theoretical point of view, that the problem is NP-hard already under rather mild conditions as well as a negative approximability results. Therefore, the main focus of the paper lies on practical heuristic algorithms as well as an exact, ILP-based approach for computing short plane supports. We report results from computational experiments that investigate the effect of requiring planarity and acyclicity on the resulting support length. Further, we evaluate the performance and trade-offs between solution quality and speed of several heuristics relative to each other and compared to optimal solutions.Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018

Competitive Searching for a Line on a Line Arrangement

We discuss the problem of searching for an unknown line on a known or unknown line arrangement by a searcher S, and show that a search strategy exists that finds the line competitively, that is, with detour factor at most a constant when compared to the situation where S has all knowledge. In the case where S knows all lines but not which one is sought, the strategy is 79-competitive. We also show that it may be necessary to travel on Omega(n) lines to realize a constant competitive ratio. In the case where initially, S does not know any line, but learns about the ones it encounters during the search, we give a 414.2-competitive search strategy

Ruler of the Plane - Games of Geometry (Multimedia Contribution)

Ruler of the Plane is a set of games illustrating concepts from combinatorial and computational geometry. The games are based on the art gallery problem, ham-sandwich cuts, the Voronoi game, and geometric network connectivity problems like the Euclidean minimum spanning tree and traveling salesperson problem

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(nlog nlog log n) space, supports queries in worst case O(log 2n) time, and updates in O(log 5n) amortized time. This leads to an O(nα(n)log5n) time algorithm to solve the agglomerative clustering problem. This is a significant improvement over the current best O(n2) time algorithms

A practical algorithm for spatial agglomerative clustering

\u3cp\u3e We study an agglomerative clustering problem motivated by visualizing disjoint glyphs (represented by geometric shapes) centered at specific locations on a geographic map. As we zoom out, the glyphs grow and start to overlap. We replace overlapping glyphs by one larger merged glyph to maintain disjointness. Our goal is to compute the resulting hierarchical clustering efficiently in practice. A straightforward algorithm for such spatial agglomerative clustering runs in On \u3csup\u3e2\u3c/sup\u3e log n time, where n is the number of glyphs. This is not efficient enough for many real-world datasets which contain up to tens or hundreds of thousands of glyphs. Recently the theoretical upper bound was improved to Onα(n) log \u3csup\u3e7\u3c/sup\u3e n time [10], where α(n) is the extremely slow growing inverse Ackermann function. Although this new algorithm is asymptotically much faster than the naive algorithm, from a practical point of view, it does not perform better for n ≤ 10 \u3csup\u3e6\u3c/sup\u3e . In this paper we present a new agglomerative clustering algorithm which works efficiently in practice. Our algorithm relies on the use of quadtrees to speed up spatial computations. Interestingly, even in non-pathological datasets we can encounter large glyphs that intersect many quadtree cells and that are involved in many clustering events. We therefore devise a special strategy to handle such large glyphs. We test our algorithm on several synthetic and real-world datasets and show that it performs well in practice. \u3c/p\u3

Short plane support trees for hypergraphs

In many domains, the aggregation or classification of data elements leads to various intersecting sets. To allow for intuitive exploration and analysis of such data, set visualization aims to represent the elements and sets graphically. In more theoretical literature, such set systems are often referred to as hypergraphs. A support graph is a notion for drawing such a hypergraph, understood as a regular graph spanning the same vertices (elements), in which each hyperedge (set) induces a connected subgraph. In this paper, we investigate finding a support graph of a hypergraph with fixed vertex locations under various constraints. We focus on enforcing planarity using a straight-line embedding, while minimizing the total length of the edges of the support graph, and consider the effect of the additional requirement that the support graph is acyclic

Short plane supports for spatial hypergraphs

\u3cp\u3eA graph G = (V,E) is a support of a hypergraph H = (V,S) if every hyperedge induces a connected subgraph in G. Supports are used for certain types of hypergraph visualizations. In this paper we consider visualizing spatial hypergraphs, where each vertex has a fixed location in the plane. This is the case, e.g., when modeling set systems of geospatial locations as hypergraphs. By applying established aesthetic quality criteria we are interested in finding supports that yield plane straight-line drawings with minimum total edge length on the input point set V. We first show, from a theoretical point of view, that the problem is NP-hard already under rather mild conditions as well as a negative approximability results. Therefore, the main focus of the paper lies on practical heuristic algorithms as well as an exact, ILP-based approach for computing short plane supports. We report results from computational experiments that investigate the effect of requiring planarity and acyclicity on the resulting support length. Further, we evaluate the performance and trade-offs between solution quality and speed of several heuristics relative to each other and compared to optimal solutions.\u3c/p\u3

Short plane support trees for hypergraphs

In many domains, the aggregation or classification of data elements leads to various intersecting sets. To allow for intuitive exploration and analysis of such data, set visualization aims to represent the elements and sets graphically. In more theoretical literature, such set systems are often referred to as hypergraphs. A support graph is a notion for drawing such a hypergraph, understood as a regular graph spanning the same vertices (elements), in which each hyperedge (set) induces a connected subgraph.\u3cbr/\u3e\u3cbr/\u3eIn this paper, we investigate finding a support graph of a hypergraph with fixed vertex locations under various constraints. We focus on enforcing planarity using a straight-line embedding, while minimizing the total length of the edges of the support graph, and consider the effect of the additional requirement that the support graph is acyclic