FRESH: Fréchet similarity with hashing

This paper studies the r-range search problem for curves under the continuous Fréchet distance: given a dataset S of n polygonal curves and a threshold >0 , construct a data structure that, for any query curve q, efficiently returns all entries in S with distance at most r from q. We propose FRESH, an approximate and randomized approach for r-range search, that leverages on a locality sensitive hashing scheme for detecting candidate near neighbors of the query curve, and on a subsequent pruning step based on a cascade of curve simplifications. We experimentally compare FRESH to exact and deterministic solutions, and we show that high performance can be reached by suitably relaxing precision and recall

Skyline Computation with Noisy Comparisons

Given a set of $n$ points in a $d$-dimensional space, we seek to compute the skyline, i.e., those points that are not strictly dominated by any other point, using few comparisons between elements. We adopt the noisy comparison model [FRPU94] where comparisons fail with constant probability and confidence can be increased through independent repetitions of a comparison. In this model motivated by Crowdsourcing applications, Groz & Milo [GM15] show three bounds on the query complexity for the skyline problem. We improve significantly on that state of the art and provide two output-sensitive algorithms computing the skyline with respective query complexity $O(nd\log (dk/\delta))$ and $O(ndk\log (k/\delta))$ where $k$ is the size of the skyline and $\delta$ the expected probability that our algorithm fails to return the correct answer. These results are tight for low dimensions

Deterministic Rectangle Enclosure and Offline Dominance Reporting on the RAM

We revisit a classical problem in computational geometry that has been studied since the 1980s: in the rectangle enclosure problem we want to report all k enclosing pairs of n input rectangles in 2D. We present the first deterministic algorithm that takes O(n log n + k) worst-case time and O(n) space in the word-RAM model. This improves previous deterministic algorithms with O((n log n+k) log log n) running time. We achieve the result by derandomizing the algorithm of Chan, Larsen and Paatrascu [SoCG'11] that attains the same time complexity but in expectation. The 2D rectangle enclosure problem is related to the offline dominance range reporting problem in 4D, and our result leads to the currently fastest deterministic algorithm for offline dominance reporting in any constant dimension d >= 4. A key tool behind Chan et al.'s previous randomized algorithm is shallow cuttings for 3D dominance ranges. Recently, Afshani and Tsakalidis [SODA'14] obtained a deterministic O(n log n)-time algorithm to construct such cuttings. We first present an improved deterministic construction algorithm that runs in O(n log log n) time in the word-RAM; this result is of independent interest. Many additional ideas are then incorporated, including a linear-time algorithm for merging shallow cuttings and an algorithm for an offline tree point location problem