Fast evaluation of union-intersection expressions

We show how to represent sets in a linear space data structure such that expressions involving unions and intersections of sets can be computed in a worst-case efficient way. This problem has applications in e.g. information retrieval and database systems. We mainly consider the RAM model of computation, and sets of machine words, but also state our results in the I/O model. On a RAM with word size $w$, a special case of our result is that the intersection of $m$ (preprocessed) sets, containing $n$ elements in total, can be computed in expected time $O(n (\log w)^2 / w + km)$, where $k$ is the number of elements in the intersection. If the first of the two terms dominates, this is a factor $w^{1-o(1)}$ faster than the standard solution of merging sorted lists. We show a cell probe lower bound of time $\Omega(n/(w m \log m)+ (1-\tfrac{\log k}{w}) k)$, meaning that our upper bound is nearly optimal for small $m$. Our algorithm uses a novel combination of approximate set representations and word-level parallelism

Recursive Algorithms for Distributed Forests of Octrees

The forest-of-octrees approach to parallel adaptive mesh refinement and coarsening (AMR) has recently been demonstrated in the context of a number of large-scale PDE-based applications. Although linear octrees, which store only leaf octants, have an underlying tree structure by definition, it is not often exploited in previously published mesh-related algorithms. This is because the branches are not explicitly stored, and because the topological relationships in meshes, such as the adjacency between cells, introduce dependencies that do not respect the octree hierarchy. In this work we combine hierarchical and topological relationships between octree branches to design efficient recursive algorithms. We present three important algorithms with recursive implementations. The first is a parallel search for leaves matching any of a set of multiple search criteria. The second is a ghost layer construction algorithm that handles arbitrarily refined octrees that are not covered by previous algorithms, which require a 2:1 condition between neighboring leaves. The third is a universal mesh topology iterator. This iterator visits every cell in a domain partition, as well as every interface (face, edge and corner) between these cells. The iterator calculates the local topological information for every interface that it visits, taking into account the nonconforming interfaces that increase the complexity of describing the local topology. To demonstrate the utility of the topology iterator, we use it to compute the numbering and encoding of higher-order $C^0$ nodal basis functions. We analyze the complexity of the new recursive algorithms theoretically, and assess their performance, both in terms of single-processor efficiency and in terms of parallel scalability, demonstrating good weak and strong scaling up to 458k cores of the JUQUEEN supercomputer.Comment: 35 pages, 15 figures, 3 table

Fast Deterministic Selection

The Median of Medians (also known as BFPRT) algorithm, although a landmark theoretical achievement, is seldom used in practice because it and its variants are slower than simple approaches based on sampling. The main contribution of this paper is a fast linear-time deterministic selection algorithm QuickselectAdaptive based on a refined definition of MedianOfMedians. The algorithm's performance brings deterministic selection---along with its desirable properties of reproducible runs, predictable run times, and immunity to pathological inputs---in the range of practicality. We demonstrate results on independent and identically distributed random inputs and on normally-distributed inputs. Measurements show that QuickselectAdaptive is faster than state-of-the-art baselines.Comment: Pre-publication draf

Distributed top-k aggregation queries at large

Top-k query processing is a fundamental building block for efficient ranking in a large number of applications. Efficiency is a central issue, especially for distributed settings, when the data is spread across different nodes in a network. This paper introduces novel optimization methods for top-k aggregation queries in such distributed environments. The optimizations can be applied to all algorithms that fall into the frameworks of the prior TPUT and KLEE methods. The optimizations address three degrees of freedom: 1) hierarchically grouping input lists into top-k operator trees and optimizing the tree structure, 2) computing data-adaptive scan depths for different input sources, and 3) data-adaptive sampling of a small subset of input sources in scenarios with hundreds or thousands of query-relevant network nodes. All optimizations are based on a statistical cost model that utilizes local synopses, e.g., in the form of histograms, efficiently computed convolutions, and estimators based on order statistics. The paper presents comprehensive experiments, with three different real-life datasets and using the ns-2 network simulator for a packet-level simulation of a large Internet-style network

Adaptive Tag Selection for Image Annotation

Not all tags are relevant to an image, and the number of relevant tags is image-dependent. Although many methods have been proposed for image auto-annotation, the question of how to determine the number of tags to be selected per image remains open. The main challenge is that for a large tag vocabulary, there is often a lack of ground truth data for acquiring optimal cutoff thresholds per tag. In contrast to previous works that pre-specify the number of tags to be selected, we propose in this paper adaptive tag selection. The key insight is to divide the vocabulary into two disjoint subsets, namely a seen set consisting of tags having ground truth available for optimizing their thresholds and a novel set consisting of tags without any ground truth. Such a division allows us to estimate how many tags shall be selected from the novel set according to the tags that have been selected from the seen set. The effectiveness of the proposed method is justified by our participation in the ImageCLEF 2014 image annotation task. On a set of 2,065 test images with ground truth available for 207 tags, the benchmark evaluation shows that compared to the popular top-$k$ strategy which obtains an F-score of 0.122, adaptive tag selection achieves a higher F-score of 0.223. Moreover, by treating the underlying image annotation system as a black box, the new method can be used as an easy plug-in to boost the performance of existing systems

Compressed Representations of Permutations, and Applications

We explore various techniques to compress a permutation $\pi$ over n integers, taking advantage of ordered subsequences in $\pi$, while supporting its application $\pi$(i) and the application of its inverse $\pi^{-1}(i)$ in small time. Our compression schemes yield several interesting byproducts, in many cases matching, improving or extending the best existing results on applications such as the encoding of a permutation in order to support iterated applications $\pi^k(i)$ of it, of integer functions, and of inverted lists and suffix arrays