Data Structures for Halfplane Proximity Queries and Incremental Voronoi Diagrams

We consider preprocessing a set $S$ of $n$ points in convex position in the plane into a data structure supporting queries of the following form: given a point $q$ and a directed line $\ell$ in the plane, report the point of $S$ that is farthest from (or, alternatively, nearest to) the point $q$ among all points to the left of line $\ell$. We present two data structures for this problem. The first data structure uses $O(n^{1+\varepsilon})$ space and preprocessing time, and answers queries in $O(2^{1/\varepsilon} \log n)$ time, for any $0 < \varepsilon < 1$. The second data structure uses $O(n \log^3 n)$ space and polynomial preprocessing time, and answers queries in $O(\log n)$ time. These are the first solutions to the problem with $O(\log n)$ query time and $o(n^2)$ space. The second data structure uses a new representation of nearest- and farthest-point Voronoi diagrams of points in convex position. This representation supports the insertion of new points in clockwise order using only $O(\log n)$ amortized pointer changes, in addition to $O(\log n)$-time point-location queries, even though every such update may make $\Theta(n)$ combinatorial changes to the Voronoi diagram. This data structure is the first demonstration that deterministically and incrementally constructed Voronoi diagrams can be maintained in $o(n)$ amortized pointer changes per operation while keeping $O(\log n)$-time point-location queries.Comment: 17 pages, 6 figures. Various small improvements. To appear in Algorithmic

The Skip Quadtree: A Simple Dynamic Data Structure for Multidimensional Data

We present a new multi-dimensional data structure, which we call the skip quadtree (for point data in R^2) or the skip octree (for point data in R^d, with constant d>2). Our data structure combines the best features of two well-known data structures, in that it has the well-defined "box"-shaped regions of region quadtrees and the logarithmic-height search and update hierarchical structure of skip lists. Indeed, the bottom level of our structure is exactly a region quadtree (or octree for higher dimensional data). We describe efficient algorithms for inserting and deleting points in a skip quadtree, as well as fast methods for performing point location and approximate range queries.Comment: 12 pages, 3 figures. A preliminary version of this paper appeared in the 21st ACM Symp. Comp. Geom., Pisa, 2005, pp. 296-30

HBST: A Hamming Distance embedding Binary Search Tree for Visual Place Recognition

Reliable and efficient Visual Place Recognition is a major building block of modern SLAM systems. Leveraging on our prior work, in this paper we present a Hamming Distance embedding Binary Search Tree (HBST) approach for binary Descriptor Matching and Image Retrieval. HBST allows for descriptor Search and Insertion in logarithmic time by exploiting particular properties of binary Feature descriptors. We support the idea behind our search structure with a thorough analysis on the exploited descriptor properties and their effects on completeness and complexity of search and insertion. To validate our claims we conducted comparative experiments for HBST and several state-of-the-art methods on a broad range of publicly available datasets. HBST is available as a compact open-source C++ header-only library.Comment: Submitted to IEEE Robotics and Automation Letters (RA-L) 2018 with International Conference on Intelligent Robots and Systems (IROS) 2018 option, 8 pages, 10 figure

Representing and retrieving regions using binary partition trees

This paper discusses the interest of Binary Partition Trees for image and region representation in the context of indexing and similarity based retrieval. Binary Partition Trees concentrate in a compact and structured way the set of regions that compose an image. Since the tree is able to represent images in a multiresolution way, only simple descriptors need to be attached to the nodes. Moreover, this representation is used for similarity based region retrieval.Peer ReviewedPostprint (published version

Geometry Helps to Compare Persistence Diagrams

Exploiting geometric structure to improve the asymptotic complexity of discrete assignment problems is a well-studied subject. In contrast, the practical advantages of using geometry for such problems have not been explored. We implement geometric variants of the Hopcroft--Karp algorithm for bottleneck matching (based on previous work by Efrat el al.) and of the auction algorithm by Bertsekas for Wasserstein distance computation. Both implementations use k-d trees to replace a linear scan with a geometric proximity query. Our interest in this problem stems from the desire to compute distances between persistence diagrams, a problem that comes up frequently in topological data analysis. We show that our geometric matching algorithms lead to a substantial performance gain, both in running time and in memory consumption, over their purely combinatorial counterparts. Moreover, our implementation significantly outperforms the only other implementation available for comparing persistence diagrams.Comment: 20 pages, 10 figures; extended version of paper published in ALENEX 201