More Dynamic Data Structures for Geometric Set Cover with Sublinear Update Time

We study geometric set cover problems in dynamic settings, allowing insertions and deletions of points and objects. We present the first dynamic data structure that can maintain an $O(1)$-approximation in sublinear update time for set cover for axis-aligned squares in 2D. More precisely, we obtain randomized update time $O(n^{2/3+delta})$ for an arbitrarily small constant $delta>0$. Previously, a dynamic geometric set cover data structure with sublinear update time was known only for unit squares by Agarwal, Chang, Suri, Xiao, and Xue [SoCG 2020]. If only an approximate size of the solution is needed, then we can also obtain sublinear amortized update time for disks in 2D and halfspaces in 3D. As a byproduct, our techniques for dynamic set cover also yield an optimal randomized $O(nlog n)$-time algorithm for static set cover for 2D disks and 3D halfspaces, improving our earlier $O(nlog n(loglog n)^{O(1)})$ result [SoCG 2020].

Faster Approximation Algorithms for Geometric Set Cover

We improve the running times of $O(1)$-approximation algorithms for the set cover problem in geometric settings, specifically, covering points by disks in the plane, or covering points by halfspaces in three dimensions. In the unweighted case, Agarwal and Pan [SoCG 2014] gave a randomized $O(nlog^4 n)$-time, $O(1)$-approximation algorithm, by using variants of the multiplicative weight update (MWU) method combined with geometric data structures. We simplify the data structure requirement in one of their methods and obtain a deterministic $O(nlog^3 nloglog n)$-time algorithm. With further new ideas, we obtain a still faster randomized $O(nlog n(loglog n)^{O(1)})$-time algorithm. For the weighted problem, we also give a randomized $O(nlog^4nloglog n)$-time, $O(1)$-approximation algorithm, by simple modifications to the MWU method and the quasi-uniform sampling technique.

Further Results on Colored Range Searching

We present a number of new results about range searching for colored (or categorical) data: 1. For a set of $n$ colored points in three dimensions, we describe randomized data structures with $O(nmathop{rm polylog}n)$ space that can report the distinct colors in any query orthogonal range (axis-aligned box) in $O(kmathop{rm polyloglog} n)$ expected time, where $k$ is the number of distinct colors in the range, assuming that coordinates are in ${1,ldots,n}$. Previous data structures require $O(frac{log n}{loglog n} + k)$ query time. Our result also implies improvements in higher constant dimensions. 2. Our data structures can be adapted to halfspace ranges in three dimensions (or circular ranges in two dimensions), achieving $O(klog n)$ expected query time. Previous data structures require $O(klog^2n)$ query time. 3. For a set of $n$ colored points in two dimensions, we describe a data structure with $O(nmathop{rm polylog}n)$ space that can answer colored type-2 range counting queries: report the number of occurrences of every distinct color in a query orthogonal range. The query time is $O(frac{log n}{loglog n} + kloglog n)$, where $k$ is the number of distinct colors in the range. Naively performing $k$ uncolored range counting queries would require $O(kfrac{log n}{loglog n})$ time. Our data structures are designed using a variety of techniques, including colored variants of randomized incremental construction (which may be of independent interest), colored variants of shallow cuttings, and bit-packing tricks.

Approximating Text-to-Pattern Hamming Distances

We revisit a fundamental problem in string matching: given a pattern of length m and a text of length n, both over an alphabet of size $sigma$, compute the Hamming distance between the pattern and the text at every location. Several $(1+epsilon)$-approximation algorithms have been proposed in the literature, with running time of the form $O(epsilon^{-O(1)}nlog nlog m)$, all using fast Fourier transform (FFT). We describe a simple $(1+epsilon)$-approximation algorithm that is faster and does not need FFT. Combining our approach with additional ideas leads to numerous new results: - We obtain the first linear-time approximation algorithm; the running time is $O(epsilon^{-2}n)$. - We obtain a faster exact algorithm computing all Hamming distances up to a given threshold k; its running time improves previous results by logarithmic factors and is linear if $klesqrt m$. - We obtain approximation algorithms with better $epsilon$-dependence using rectangular matrix multiplication. The time-bound is $~O(n)$ when the pattern is sufficiently long: $mge epsilon^{-28}$. Previous algorithms require $~O(epsilon^{-1}n)$ time. - When k is not too small, we obtain a truly sublinear-time algorithm to find all locations with Hamming distance approximately (up to a constant factor) less than k, in $O((n/k^{Omega(1)}+occ)n^{o(1)})$ time, where occ is the output size. The algorithm leads to a property tester, returning true if an exact match exists and false if the Hamming distance is more than $delta m$ at every location, running in $~O(delta^{-1/3}n^{2/3}+delta^{-1}n/m)$ time. - We obtain a streaming algorithm to report all locations with Hamming distance approximately less than k, using $~O(epsilon^{-2}sqrt k)$ space. Previously, streaming algorithms were known for the exact problem with ~O(k) space or for the approximate problem with $~O(epsilon^{-O(1)}sqrt m)$ space.

Stabbing Rectangles by Line Segments - How Decomposition Reduces the Shallow-Cell Complexity

We initiate the study of the following natural geometric optimization problem. The input is a set of axis-aligned rectangles in the plane. The objective is to find a set of horizontal line segments of minimum total length so that every rectangle is stabbed by some line segment. A line segment stabs a rectangle if it intersects its left and its right boundary. The problem, which we call Stabbing, can be motivated by a resource allocation problem and has applications in geometric network design. To the best of our knowledge, only special cases of this problem have been considered so far. Stabbing is a weighted geometric set cover problem, which we show to be NP-hard. A constrained variant of Stabbing turns out to be even APX-hard. While for general set cover the best possible approximation ratio is $Theta(log n)$, it is an important field in geometric approximation algorithms to obtain better ratios for geometric set cover problems. Chan et al. [SODA12] generalize earlier results by Varadarajan [STOC10] to obtain sub-logarithmic performances for a broad class of weighted geometric set cover instances that are characterized by having low shallow-cell complexity. The shallow-cell complexity of Stabbing instances, however, can be high so that a direct application of the framework of Chan et al. gives only logarithmic bounds. We still achieve a constant-factor approximation by decomposing general instances into what we call laminar instances that have low enough complexity. Our decomposition technique yields constant-factor approximations also for the variant where rectangles can be stabbed by horizontal and vertical segments and for two further geometric set cover problems.