Point Location in Incremental Planar Subdivisions

We study the point location problem in incremental (possibly disconnected) planar subdivisions, that is, dynamic subdivisions allowing insertions of edges and vertices only. Specifically, we present an O(n log n)-space data structure for this problem that supports queries in O(log^2 n) time and updates in O(log n log log n) amortized time. This is the first result that achieves polylogarithmic query and update times simultaneously in incremental planar subdivisions. Its update time is significantly faster than the update time of the best known data structure for fully-dynamic (possibly disconnected) planar subdivisions

Shortest-Path Queries in Geometric Networks

A Euclidean t-spanner for a point set V ? ?^d is a graph such that, for any two points p and q in V, the distance between p and q in the graph is at most t times the Euclidean distance between p and q. Gudmundsson et al. [TALG 2008] presented a data structure for answering ?-approximate distance queries in a Euclidean spanner in constant time, but it seems unlikely that one can report the path itself using this data structure. In this paper, we present a data structure of size O(nlog n) that answers ?-approximate shortest-path queries in time linear in the size of the output

Algorithms for Computing Maximum Cliques in Hyperbolic Random Graphs

In this paper, we study the maximum clique problem on hyperbolic random graphs. A hyperbolic random graph is a mathematical model for analyzing scale-free networks since it effectively explains the power-law degree distribution of scale-free networks. We propose a simple algorithm for finding a maximum clique in hyperbolic random graph. We first analyze the running time of our algorithm theoretically. We can compute a maximum clique on a hyperbolic random graph G in O(m + n^{4.5(1-?)}) expected time if a geometric representation is given or in O(m + n^{6(1-?)}) expected time if a geometric representation is not given, where n and m denote the numbers of vertices and edges of G, respectively, and ? denotes a parameter controlling the power-law exponent of the degree distribution of G. Also, we implemented and evaluated our algorithm empirically. Our algorithm outperforms the previous algorithm [BFK18] practically and theoretically. Beyond the hyperbolic random graphs, we have experiment on real-world networks. For most of instances, we get large cliques close to the optimum solutions efficiently

Algorithms for Computing Maximum Cliques in Hyperbolic Random Graphs

In this paper, we study the maximum clique problem on hyperbolic random graphs. A hyperbolic random graph is a mathematical model for analyzing scale-free networks since it effectively explains the power-law degree distribution of scale-free networks. We propose a simple algorithm for finding a maximum clique in hyperbolic random graph. We first analyze the running time of our algorithm theoretically. We can compute a maximum clique on a hyperbolic random graph $G$ in $O(m + n^{4.5(1-\alpha)})$ expected time if a geometric representation is given or in $O(m + n^{6(1-\alpha)})$ expected time if a geometric representation is not given, where $n$ and $m$ denote the numbers of vertices and edges of $G$, respectively, and $\alpha$ denotes a parameter controlling the power-law exponent of the degree distribution of $G$. Also, we implemented and evaluated our algorithm empirically. Our algorithm outperforms the previous algorithm [BFK18] practically and theoretically. Beyond the hyperbolic random graphs, we have experiment on real-world networks. For most of instances, we get large cliques close to the optimum solutions efficiently.Comment: Accepted in ESA 202

Parameterized Algorithm for the Disjoint Path Problem on Planar Graphs: Exponential in k2k^2 and Linear in nn

In this paper, we study the \textsf{Planar Disjoint Paths} problem: Given an undirected planar graph $G$ with $n$ vertices and a set $T$ of $k$ pairs $(s_i,t_i)_{i=1}^k$ of vertices, the goal is to find a set $\mathcal P$ of $k$ pairwise vertex-disjoint paths connecting $s_i$ and $t_i$ for all indices $i\in\{1,\ldots,k\}$. We present a $2^{O(k^2)}n$-time algorithm for the \textsf{Planar Disjoint Paths} problem. This improves the two previously best-known algorithms: $2^{2^{O(k)}}n$-time algorithm [Discrete Applied Mathematics 1995] and $2^{O(k^2)}n^6$-time algorithm [STOC 2020].Comment: SODA 202

Linear-Time Approximation Scheme for k-Means Clustering of Axis-Parallel Affine Subspaces

In this paper, we present a linear-time approximation scheme for k-means clustering of incomplete data points in d-dimensional Euclidean space. An incomplete data point with ∆ > 0 unspecified entries is represented as an axis-parallel affine subspace of dimension ∆. The distance between two incomplete data points is defined as the Euclidean distance between two closest points in the axis-parallel affine subspaces corresponding to the data points. We present an algorithm for k-means clustering of axis-parallel affine subspaces of dimension ∆ that yields an (1 + ϵ)-approximate solution in O(nd) time. The constants hidden behind O(·) depend only on ∆, ϵ and k. This improves the O(n1

Feedback Vertex Set on Geometric Intersection Graphs

In this paper, we present an algorithm for computing a feedback vertex set of a unit disk graph of size k, if it exists, which runs in time 2^O(?k)(n+m), where n and m denote the numbers of vertices and edges, respectively. This improves the 2^O(?klog k) n^O(1)-time algorithm for this problem on unit disk graphs by Fomin et al. [ICALP 2017]. Moreover, our algorithm is optimal assuming the exponential-time hypothesis. Also, our algorithm can be extended to handle geometric intersection graphs of similarly sized fat objects without increasing the running time

Dynamic Geodesic Convex Hulls in Dynamic Simple Polygons

We consider the geodesic convex hulls of points in a simple polygonal region in the presence of non-crossing line segments (barriers) that subdivide the region into simply connected faces. We present an algorithm together with data structures for maintaining the geodesic convex hull of points in each face in a sublinear update time under the fully-dynamic setting where both input points and barriers change by insertions and deletions. The algorithm processes a mixed update sequence of insertions and deletions of points and barriers. Each update takes O(n^2/3 log^2 n) time with high probability, where n is the total number of the points and barriers at the moment. Our data structures support basic queries on the geodesic convex hull, each of which takes O(polylog n) time. In addition, we present an algorithm together with data structures for geodesic triangle counting queries under the fully-dynamic setting. With high probability, each update takes O(n^2/3 log n) time, and each query takes O(n^2/3 log n) time