We consider the classic k-center problem in the constant dimensional Euclidean space under a parallel setting, where S is an element of (0,1) is an arbitrary constant. As a central clustering problem, the k-center problem has been studied extensively. Still, until very recently, all parallel MPC algorithms have been requiring Omega(k) or even Omega (kns) local space per machine. While this setting covers the case of small values of k, for a large number of clusters these algorithms require large local memory, making them poorly scalable. The case of large k, k >= Omega(nb), has been considered recently for the low-local-space MPC model by Bateni et al. [2021], who gave an O ( log log n)-round MPC algorithm that produces k(1 + 0(1)) centers whose cost has multiplicative approximation of O (log log log n). In this article, we extend the algorithm of Bateni et al. and design a lowlocal-space MPC algorithm that in O( log log n) rounds returns a clustering with k(1 + 0(1)) clusters that is CCS Concepts: center dot Theory of computation -> Massively parallel algorithms; Facility location and clustering;
For the online transportation problem with \(m\) server sites, it has long been known that the competitive ratio of any deterministic algorithm is at least \(2m-1\) . Kalyanasundaram and Pruhs conjectured in 1998 that a deterministic \((2m-1)\) -competitive algorithm exists for this problem, a conjecture that has remained open for over two decades. In this paper, we propose a new deterministic algorithm for the online transportation problem and show that it achieves a competitive ratio of at most \(8m-13\) . This is the first \(O(m)\) -competitive deterministic algorithm, coming close to the lower bound of \(2m-1\) within a constant factor.
We study the Maximum Independent Set of Rectangles (MISR) problem, where we are given a set of axis-parallel rectangles in the plane and the goal is to select a subset of non-overlapping rectangles of maximum cardinality. In a recent breakthrough, Mitchell (FOCS’21) obtained the first constant-factor approximation algorithm for MISR. His algorithm achieves an approximation ratio of 10, and it is based on a dynamic program that intuitively recursively partitions the input plane into special polygons called Corner-Clipped Rectangles (CCRs) , without intersecting certain special horizontal line segments called fences . In this article, we present a \((2+\varepsilon)\) -approximation algorithm for MISR, which is also based on a recursive partitioning scheme. First, we use a partition into a class of axis-parallel polygons with constant complexity each that are more general than CCRs. This allows us to provide an arguably simpler analysis and, at the same time, already improves the approximation ratio to 4. Then, using a more elaborate charging scheme and a recursive partitioning into general axis-parallel polygons with constant complexity, we improve our approximation ratio to \(2+\varepsilon\) . In particular, we construct a recursive partitioning based on more general fences which can be sequences of up to \(O(1/\varepsilon)\) line segments each. This partitioning routine and our other new ideas may be useful for future work toward a PTAS for MISR.
Compressed suffix arrays index large repetitive collections and are widely used in bioinformatics, version control, and natural language processing. The \( r \) -index and its variants combine the run-length Burrows–Wheeler transform (BWT) with a sample of the suffix array to achieve space proportional to the number of equal-symbol runs. While effective for near-identical strings, their size grows quickly when edited copies are added to the collection, as the number of BWT runs is sensitive to variation. Existing approaches either apply extra compression at the cost of slower queries or prioritize speed at the expense of space, limiting practical tradeoffs at large scale. We introduce variable-length blocking (VLB), an adaptive encoding for BWT-based compressed suffix arrays. Our technique adapts the amount of indexing information to the local run structure: compressible regions retain less auxiliary data, while incompressible areas store more. Specifically, we recursively partition the BWT until each block contains at most \( w \) runs (a parameter), and organize the blocks into a tree. Accessing a position requires descending a root-to-leaf path followed by a short run scan. Compressible regions are placed near the root, while incompressible regions lie deeper and are augmented with additional indexing data to speed up access. This strategy balances space and query speed by reallocating bits saved in compressible areas to accelerate access to incompressible ones. The VLB-tree also improves memory locality by co-locating related BWT and suffix array information. Backward search relies on rank and successor queries over the BWT. We introduce a sampling technique that relaxes global correctness by guaranteeing that these queries are correct only along valid backward-search states. This restriction reduces space usage without affecting the speed or correctness of pattern matching. We further extend the concepts of the VLB-tree to encode suffix array samples and then to the subsampled \( r \) -index ( \(sr\) -index), thus producing a small fully functional compressed suffix array. Experiments show that VLB-tree variants outperform the \( r \) -index and \(sr\) -index in query time, while retaining space close to that of the \(sr\) -index. The move data structure, a recent encoding, is faster than the VLB-tree but uses considerably more space, yielding a clear space-time tradeoff.
Cut arcs , or strong bridges , are one of the most fundamental reachability notions in directed graphs. Specifically, in a strongly connected graph \(G=(V,E)\) ( \(|V|=n\) , \(|E|=m\) ), a cut arc is an arc \(e\in E\) for which there exist \(u,v\in V\) , such that all \(u\) - \(v\) walks contain \(e\) . In this paper we generalise this notion to cut paths , that is, walks \(W\) for which there exist \(u,v\in V\) , such that all \(u\) - \(v\) walks contain \(W\) as subwalk. We first prove various properties of cut paths and define their remainder structure , which we use to present a simple \(O(m)\) -time verification algorithm for a cut path. We further show that a graph contains at most \(O(n)\) maximal cut paths of length at most \(O(n)\) each, and present an optimal \(O(n^{2})\) enumeration algorithm for maximal cut paths. We apply cut paths and their remainder structure to improve several reachability problems from bioinformatics, as follows. A walk is called safe if it is a subwalk of every node-covering closed walk of a strongly connected graph. Multi-safety is defined analogously, by considering node-covering sets of closed walks instead. Cut paths provide simple \(O(m)\) -time algorithms verifying if a walk is safe or multi-safe. Further, by simultaneous computation of remainder structures of all subwalks of a cut path in linear time, we can identify all maximal multi-safe walks in \(O(mn)\) time. This improves over the state-of-the-art algorithm running in time \(O(m^{2}+n^{3}\log n)\) .
