45840953
Dynamic Cluster Formation Using Level Set Methods
Density-based clustering has the advantages for (i) allowing arbitrary shape of cluster and (ii) not requiring the number of clusters as input. However, when clusters touch each other, both the cluster centers and cluster boundaries (as the peaks and valleys of the density distribution) become fuzzy and difficult to determine. In higher dimension, the boundaries become wiggly and over-fitting often occurs. We introduce the notion of cluster intensity function (CIF) which captures the important characteristics of clusters. When clusters are well-separated, CIFs are similar to density functions. But as clusters touch each other, CIFs still clearly reveal cluster centers, cluster boundaries, and, degree of membership of each data point to the cluster that it belongs. Clustering through bump hunting and valley seeking based on these functions are more robust than that based on kernel density functions which are often oscillatory or over-smoothed. These problems of kernel density estimation are resolved using Level Set Methods and related techniques. Comparisons with two existing density-based methods, valley seeking and DBSCAN, are presented to illustrate the advantages of our approach.
13840611
Dynamic cluster formation using level set methods
Density-based clustering has the advantages for: 1) allowing arbitrary shape of cluster and 2) not requiring the number of clusters as input. However, when clusters touch each other, both the cluster centers and cluster boundaries (as the peaks and valleys of the density distribution) become fuzzy and difficult to determine. We introduce the notion of cluster intensity function (CIF) which captures the important characteristics of clusters. When clusters are well-separated, CIFs are similar to density functions. But, when clusters become closed to each other, CIFs still clearly reveal cluster centers, cluster boundaries, and degree of membership of each data point to the cluster that it belongs. Clustering through bump hunting and valley seeking based on these functions are more robust than that based on density functions obtained by kernel density estimation, which are often oscillatory or oversmoothed. These problems of kernel density estimation are resolved using level set methods and related techniques. Comparisons with two existing density-based methods, valley seeking and DBSCAN, are presented which illustrate the advantages of our approach.
15741927
Dynamic subgraph connectivity with geometric applications
(MATH) Inspired by dynamic connectivity applications in computational geometry, we consider a problem we call dynamic subgraph connectivity: design a data structure for an undirected graph $G=(V,E)$ and a subset of vertices $S\on V$, to support insertions and deletions in~$S$ and connectivity queries (are two vertices connected\@?) in the subgraph induced by~$S$. We develop the first sublinear, fully dynamic method for this problem for general sparse graphs, using an elegant combination of several simple ideas. Our method requires linear space, $\OO (|E|^{4\w/(3\w+3)})=O(|E|^{0.94})$ amortized update time, and $\OO(|E|^{1/3})$ query time, where $\w$ is the matrix multiplication exponent and $\OO$ hides polylogarithmic factors.
111153
Low-Dimensional Linear Programming with Violations
Two decades ago, Megiddo and Dyer showed that linear programming (LP) in two and three dimensions (and subsequently any constant number of dimensions) can be solved in linear time. In this paper, we consider the LP problem with at most k violations, i.e., finding a point inside all but at most k halfspaces, given a set of n halfspaces. We present a simple algorithm in two dimensions that runs in O((n+k2)log n) expected time; this is faster than earlier algorithms by Everett, Robert, and van Kreveld (1993) and Matousek (1994) for many values of k and is probably near-optimal. An extension of our algorithm in three dimensions runs in near O(n+k11/4n1/4) expected time. Interestingly, the idea is based on concave-chain decompositions (or covers) of the $(\le k)$-level, previously used in proving combinatorial k-level bounds. 
Applications in the plane include improved algorithms for finding a line that misclassifies the fewest among a set of bichromatic points, and finding the smallest circle enclosing all but k points. We also discuss related problems of finding local minima in levels.
62791932
All-pairs shortest paths for unweighted undirected graphs in o(mn) time
We revisit the all-pairs-shortest-paths problem for an unweighted undirected graph with <i>n</i> vertices and <i>m</i> edges. We present new algorithms with the following running times:[EQUATION]These represent the best time bounds known for the problem for all <i>m</i> ≪ <i>n</i><sup>1.376</sup>. We also obtain a similar type of result for the diameter problem for unweighted directed graphs.
30394025
Dynamic Subgraph Connectivity with Geometric Applications
Inspired by dynamic connectivity applications in computational geometry, we consider a problem we call dynamic subgraph connectivity: design a data structure for an undirected graph $G=(V,E)$ and a subset of vertices $S \subseteq V$ to support insertions/deletions in $S$ and connectivity queries (are two vertices connected?) in the subgraph induced by $S$. We develop the first sublinear, fully dynamic method for this problem for general sparse graphs, using a combination of several simple ideas. Our method requires $\widetilde O(|E|^{4\omega/(3\omega+3)})=O(|E|^{0.94})$ amortized update time, and $\widetilde O(|E|^{1/3})$ query time, after $\widetilde O(|E|^{(5\omega+1)/(3\omega+3)})$ preprocessing time, where o is the matrix multiplication exponent and $\widetilde O$ hides polylogarithmic factors.
15654193
A dynamic data structure for 3-d convex hulls and 2-d nearest neighbor queries
We present a fully dynamic randomized data structure that can answer queries about the convex hull of a set of <i>n</i> points in three dimensions, where insertions take <i>O</i>(log<sup>3</sup> <i>n</i>) expected amortized time, deletions take <i>O</i>(log<sup>6</sup> <i>n</i>) expected amortized time, and extreme-point queries take <i>O</i>(log<sup>2</sup> <i>n</i>) worst-case time. This is the first method that guarantees polylogarithmic update and query cost for arbitrary sequences of insertions and deletions, and improves the previous <i>O</i>(<i>n</i><sup>ε</sup>)-time method by Agarwal and Matoušek a decade ago. As a consequence, we obtain similar results for nearest neighbor queries in two dimensions and improved results for numerous fundamental geometric problems (such as levels in three dimensions and dynamic Euclidean minimum spanning trees in the plane).