We define a natural class of range query problems, and prove that all problems within this class have the same time complexity (up to polylogarithmic factors). The equivalence is very general, and even applies to online algorithms. This allows us to obtain new improved algorithms for all of the problems in the class.We then focus on the special case of the problems when the queries are offline and the number of queries is linear. We show that our range query problems are runtime-equivalent (up to polylogarithmic factors) to counting for each edge e in an m-edge graph the number of triangles through e. This natural triangle problem can be solved using the best known triangle counting algorithm,running time is known to be tight (within m o(1) factors) under the 3SUM Hypothesis. In this case, our equivalence settles the complexity of the range query problems. Our problems constitute the first equivalence class with this peculiar running time bound.To better understand the complexity of these problems, we also provide a deeper insight into the family of triangle problems, in particular showing black-box reductions between triangle listing and per-edge triangle detection and counting. As a byproduct of our reductions, we obtain a simple triangle listing algorithm matching the state-of-the-art for all regimes of the number of triangles. We also give some not necessarily tight, but still surprising reductions from variants of matrix products, such as the (min, max)-product. * Partially supported by the National Science Center, Poland under grants 2017/27/N/ST6/01334 and 2018/28/T/ST6/00305. O m 2ω/(ω+1) time by the Alon-Yuster-Zwick [4] algorithm.Definition 1 (EdgeTriangleCounting). Given an undirected graph G = (V, E), with n nodes and m edges, compute for every edge e ∈ E the number of triangles in G which contain e.
Machine-learned predictors, although achieving very good results for inputs resembling training data, cannot possibly provide perfect predictions in all situations. Still, decision-making systems that are based on such predictors need not only benefit from good predictions, but should also achieve a decent performance when the predictions are inadequate. In this paper, we propose a prediction setup for arbitrary metrical task systems (MTS) (e.g., caching , k -server and convex body chasing ) and online matching on the line . We utilize results from the theory of online algorithms to show how to make the setup robust. Specifically for caching, we present an algorithm whose performance, as a function of the prediction error, is exponentially better than what is achievable for general MTS. Finally, we present an empirical evaluation of our methods on real world datasets, which suggests practicality.
The clustering coefficient and the transitivity ratio are concepts often used in network analysis, which creates a need for fast practical algorithms for counting triangles in large graphs. Previous research in this area focused on sequential algorithms, MapReduce parallelization, and fast approximations. In this paper we propose a parallel triangle counting algorithm for CUDA GPU. We describe the implementation details necessary to achieve high performance and present the experimental evaluation of our approach. Our algorithm achieves 8 to 15 times speedup over the CPU implementation and is capable of finding 3.8 billion triangles in an 89 million edges graph in less than 10 seconds on the Nvidia Tesla C2050 GPU.Comment: 2016 IEEE International Parallel and Distributed Processing Symposium Workshops (IPDPSW
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