Approximation Algorithms for Directed Weighted Spanners

In the pairwise weighted spanner problem, the input consists of an $n$-vertex-directed graph, where each edge is assigned a cost and a length. Given $k$ vertex pairs and a distance constraint for each pair, the goal is to find a minimum-cost subgraph in which the distance constraints are satisfied. This formulation captures many well-studied connectivity problems, including spanners, distance preservers, and Steiner forests. In the offline setting, we show: 1. An $\tilde{O}(n^{4/5 + \epsilon})$-approximation algorithm for pairwise weighted spanners. When the edges have unit costs and lengths, the best previous algorithm gives an $\tilde{O}(n^{3/5 + \epsilon})$-approximation, due to Chlamt\'a\v{c}, Dinitz, Kortsarz, and Laekhanukit (TALG, 2020). 2. An $\tilde{O}(n^{1/2+\epsilon})$-approximation algorithm for all-pair weighted distance preservers. When the edges have unit costs and arbitrary lengths, the best previous algorithm gives an $\tilde{O}(n^{1/2})$-approximation for all-pair spanners, due to Berman, Bhattacharyya, Makarychev, Raskhodnikova, and Yaroslavtsev (Information and Computation, 2013). In the online setting, we show: 1. An $\tilde{O}(k^{1/2 + \epsilon})$-competitive algorithm for pairwise weighted spanners. The state-of-the-art results are $\tilde{O}(n^{4/5})$-competitive when edges have unit costs and arbitrary lengths, and $\min\{\tilde{O}(k^{1/2 + \epsilon}), \tilde{O}(n^{2/3 + \epsilon})\}$-competitive when edges have unit costs and lengths, due to Grigorescu, Lin, and Quanrud (APPROX, 2021). 2. An $\tilde{O}(k^{\epsilon})$-competitive algorithm for single-source weighted spanners. Without distance constraints, this problem is equivalent to the directed Steiner tree problem. The best previous algorithm for online directed Steiner trees is $\tilde{O}(k^{\epsilon})$-competitive, due to Chakrabarty, Ene, Krishnaswamy, and Panigrahi (SICOMP, 2018)

Online Directed Spanners and Steiner Forests

We present online algorithms for directed spanners and Steiner forests. These problems fall under the unifying framework of online covering linear programming formulations, developed by Buchbinder and Naor (MOR, 34, 2009), based on primal-dual techniques. Our results include the following: For the pairwise spanner problem, in which the pairs of vertices to be spanned arrive online, we present an efficient randomized $\tilde{O}(n^{4/5})$-competitive algorithm for graphs with general lengths, where $n$ is the number of vertices. With uniform lengths, we give an efficient randomized $\tilde{O}(n^{2/3+\epsilon})$-competitive algorithm, and an efficient deterministic $\tilde{O}(k^{1/2+\epsilon})$-competitive algorithm, where $k$ is the number of terminal pairs. These are the first online algorithms for directed spanners. In the offline setting, the current best approximation ratio with uniform lengths is $\tilde{O}(n^{3/5 + \epsilon})$, due to Chlamtac, Dinitz, Kortsarz, and Laekhanukit (TALG 2020). For the directed Steiner forest problem with uniform costs, in which the pairs of vertices to be connected arrive online, we present an efficient randomized $\tilde{O}(n^{2/3 + \epsilon})$-competitive algorithm. The state-of-the-art online algorithm for general costs is due to Chakrabarty, Ene, Krishnaswamy, and Panigrahi (SICOMP 2018) and is $\tilde{O}(k^{1/2 + \epsilon})$-competitive. In the offline version, the current best approximation ratio with uniform costs is $\tilde{O}(n^{26/45 + \epsilon})$, due to Abboud and Bodwin (SODA 2018). A small modification of the online covering framework by Buchbinder and Naor implies a polynomial-time primal-dual approach with separation oracles, which a priori might perform exponentially many calls. We convert the online spanner problem and the online Steiner forest problem into online covering problems and round in a problem-specific fashion

The Maximum Binary Tree Problem

We introduce and investigate the approximability of the maximum binary tree problem (MBT) in directed and undirected graphs. The goal in MBT is to find a maximum-sized binary tree in a given graph. MBT is a natural variant of the well-studied longest path problem, since both can be viewed as finding a maximum-sized tree of bounded degree in a given graph. The connection to longest path motivates the study of MBT in directed acyclic graphs (DAGs), since the longest path problem is solvable efficiently in DAGs. In contrast, we show that MBT in DAGs is in fact hard: it has no efficient exp(-O(log n/ log log n))-approximation algorithm under the exponential time hypothesis, where n is the number of vertices in the input graph. In undirected graphs, we show that MBT has no efficient exp(-O(log^0.63 n))-approximation under the exponential time hypothesis. Our inapproximability results rely on self-improving reductions and structural properties of binary trees. We also show constant-factor inapproximability assuming P ? NP. In addition to inapproximability results, we present algorithmic results along two different flavors: (1) We design a randomized algorithm to verify if a given directed graph on n vertices contains a binary tree of size k in 2^k poly(n) time. (2) Motivated by the longest heapable subsequence problem, introduced by Byers, Heeringa, Mitzenmacher, and Zervas, ANALCO 2011, which is equivalent to MBT in permutation DAGs, we design efficient algorithms for MBT in bipartite permutation graphs

Fixed-Parameter Algorithms for Longest Heapable Subsequence and Maximum Binary Tree

A heapable sequence is a sequence of numbers that can be arranged in a min-heap data structure. Finding a longest heapable subsequence of a given sequence was proposed by Byers, Heeringa, Mitzenmacher, and Zervas (ANALCO 2011) as a generalization of the well-studied longest increasing subsequence problem and its complexity still remains open. An equivalent formulation of the longest heapable subsequence problem is that of finding a maximum-sized binary tree in a given permutation directed acyclic graph (permutation DAG). In this work, we study parameterized algorithms for both longest heapable subsequence and maximum-sized binary tree. We introduce alphabet size as a new parameter in the study of computational problems in permutation DAGs and show that this parameter with respect to a fixed topological ordering admits a complete characterization and a polynomial time algorithm. We believe that this parameter is likely to be useful in the context of optimization problems defined over permutation DAGs

Influence of Y-doped induced defects on the optical and magnetic properties of ZnO nanorod arrays prepared by low-temperature hydrothermal process

One-dimensional pure zinc oxide (ZnO) and Y-doped ZnO nanorod arrays have been successfully fabricated on the silicon substrate for comparison by a simple hydrothermal process at the low temperature of 90°C. The Y-doped nanorods exhibit the same c-axis-oriented wurtzite hexagonal structure as pure ZnO nanorods. Based on the results of photoluminescence, an enhancement of defect-induced green-yellow visible emission is observed for the Y-doped ZnO nanorods. The decrease of E(2)(H) mode intensity and increase of E(1)(LO) mode intensity examined by the Raman spectrum also indicate the increase of defects for the Y-doped ZnO nanorods. As compared to pure ZnO nanorods, Y-doped ZnO nanorods show a remarked increase of saturation magnetization. The combination of visible photoluminescence and ferromagnetism measurement results indicates the increase of oxygen defects due to the Y doping which plays a crucial role in the optical and magnetic performances of the ZnO nanorods

Comparison of Prevention of NSAID-Induced Gastrointestinal Complications by Rebamipide and Misoprostol: A Randomized, Multicenter, Controlled Trial—STORM STUDY

Nonsteroidal anti-inflammatory drugs (NSAIDs) have gastrointestinal side effects such as dyspepsia, peptic ulcer, hemorrhage, and perforation. Misoprostol and PPIs have been used to prevent NSAID-induced gastroduodenal injury. Rebamipide increases gastric mucus and stimulates the production of endogenous prostaglandins. The prophylactic effect of rebamipide on NSAID-induced gastrointestinal complications is unknown. The aim of this study was to compare NSAID-induced gastrointestinal complications in rebamipide- and misoprostol-treated groups. Patients were randomized to two groups and took a conventional NSAID plus rebamipide or misoprostol for 12 weeks. Gastric mucosal damage was evaluated by endoscopy at screening and the end of the study. The prevalences of active gastric ulcer were 7/176 (3.9%) in the rebamipide group and 3/156 (1.9%) in the misoprostol group. The prevalences of peptic ulcer were 8/176 (4.5%) in the rebamipide group and 7/156 (4.4%) in the misoprostol group. The cumulative incidences of peptic ulcer in the high-risk subgroup were 6/151 (4.0%) for rebamipide and 6/154 (3.9%) for misoprostol. In conclusion, rebamipide prevented NSAID-induced peptic ulcer as effectively as misoprostol in patients on long-term NSAID therapy. Rebamipide may be a useful therapeutic option for the prevention of NSAID-induced gastrointestinal ulcer because of its therapeutic effect and safety

Integration of hybridization-based markers (overgos) into physical maps for comparative and evolutionary explorations in the genus Oryza and in Sorghum

BACKGROUND: With the completion of the genome sequence for rice (Oryza sativa L.), the focus of rice genomics research has shifted to the comparison of the rice genome with genomes of other species for gene cloning, breeding, and evolutionary studies. The genus Oryza includes 23 species that shared a common ancestor 8–10 million years ago making this an ideal model for investigations into the processes underlying domestication, as many of the Oryza species are still undergoing domestication. This study integrates high-throughput, hybridization-based markers with BAC end sequence and fingerprint data to construct physical maps of rice chromosome 1 orthologues in two wild Oryza species. Similar studies were undertaken in Sorghum bicolor, a species which diverged from cultivated rice 40–50 million years ago. RESULTS: Overgo markers, in conjunction with fingerprint and BAC end sequence data, were used to build sequence-ready BAC contigs for two wild Oryza species. The markers drove contig merges to construct physical maps syntenic to rice chromosome 1 in the wild species and provided evidence for at least one rearrangement on chromosome 1 of the O. sativa versus Oryza officinalis comparative map. When rice overgos were aligned to available S. bicolor sequence, 29% of the overgos aligned with three or fewer mismatches; of these, 41% gave positive hybridization signals. Overgo hybridization patterns supported colinearity of loci in regions of sorghum chromosome 3 and rice chromosome 1 and suggested that a possible genomic inversion occurred in this syntenic region in one of the two genomes after the divergence of S. bicolor and O. sativa. CONCLUSION: The results of this study emphasize the importance of identifying conserved sequences in the reference sequence when designing overgo probes in order for those probes to hybridize successfully in distantly related species. As interspecific markers, overgos can be used successfully to construct physical maps in species which diverged less than 8 million years ago, and can be used in a more limited fashion to examine colinearity among species which diverged as much as 40 million years ago. Additionally, overgos are able to provide evidence of genomic rearrangements in comparative physical mapping studies