Optimum Partition Parameter of Divide-and-Conquer Algorithm for Solving Closest-Pair Problem

Divide and Conquer is a well known algorithmic procedure for solving many kinds of problem. In this procedure, the problem is partitioned into two parts until the problem is trivially solvable. Finding the distance of the closest pair is an interesting topic in computer science. With divide and conquer algorithm we can solve closest pair problem. Here also the problem is partitioned into two parts until the problem is trivially solvable. But it is theoretically and practically observed that sometimes partitioning the problem space into more than two parts can give better performances. In this paper, a new proposal is given that dividing the problem space into (n) number of parts can give better result while divide and conquer algorithm is used for solving the closest pair of point's problem.Comment: arXiv admin note: substantial text overlap with arXiv:1010.590

Load-Balancing for Parallel Delaunay Triangulations

Computing the Delaunay triangulation (DT) of a given point set in $\mathbb{R}^D$ is one of the fundamental operations in computational geometry. Recently, Funke and Sanders (2017) presented a divide-and-conquer DT algorithm that merges two partial triangulations by re-triangulating a small subset of their vertices - the border vertices - and combining the three triangulations efficiently via parallel hash table lookups. The input point division should therefore yield roughly equal-sized partitions for good load-balancing and also result in a small number of border vertices for fast merging. In this paper, we present a novel divide-step based on partitioning the triangulation of a small sample of the input points. In experiments on synthetic and real-world data sets, we achieve nearly perfectly balanced partitions and small border triangulations. This almost cuts running time in half compared to non-data-sensitive division schemes on inputs exhibiting an exploitable underlying structure.Comment: Short version submitted to EuroPar 201

Empirical Evaluation of the Parallel Distribution Sweeping Framework on Multicore Architectures

In this paper, we perform an empirical evaluation of the Parallel External Memory (PEM) model in the context of geometric problems. In particular, we implement the parallel distribution sweeping framework of Ajwani, Sitchinava and Zeh to solve batched 1-dimensional stabbing max problem. While modern processors consist of sophisticated memory systems (multiple levels of caches, set associativity, TLB, prefetching), we empirically show that algorithms designed in simple models, that focus on minimizing the I/O transfers between shared memory and single level cache, can lead to efficient software on current multicore architectures. Our implementation exhibits significantly fewer accesses to slow DRAM and, therefore, outperforms traditional approaches based on plane sweep and two-way divide and conquer.Comment: Longer version of ESA'13 pape

Distributed Robust Learning

We propose a framework for distributed robust statistical learning on {\em big contaminated data}. The Distributed Robust Learning (DRL) framework can reduce the computational time of traditional robust learning methods by several orders of magnitude. We analyze the robustness property of DRL, showing that DRL not only preserves the robustness of the base robust learning method, but also tolerates contaminations on a constant fraction of results from computing nodes (node failures). More precisely, even in presence of the most adversarial outlier distribution over computing nodes, DRL still achieves a breakdown point of at least $\lambda^*/2$, where $\lambda^*$ is the break down point of corresponding centralized algorithm. This is in stark contrast with naive division-and-averaging implementation, which may reduce the breakdown point by a factor of $k$ when $k$ computing nodes are used. We then specialize the DRL framework for two concrete cases: distributed robust principal component analysis and distributed robust regression. We demonstrate the efficiency and the robustness advantages of DRL through comprehensive simulations and predicting image tags on a large-scale image set.Comment: 18 pages, 2 figure

Eigenvector Synchronization, Graph Rigidity and the Molecule Problem

The graph realization problem has received a great deal of attention in recent years, due to its importance in applications such as wireless sensor networks and structural biology. In this paper, we extend on previous work and propose the 3D-ASAP algorithm, for the graph realization problem in $\mathbb{R}^3$, given a sparse and noisy set of distance measurements. 3D-ASAP is a divide and conquer, non-incremental and non-iterative algorithm, which integrates local distance information into a global structure determination. Our approach starts with identifying, for every node, a subgraph of its 1-hop neighborhood graph, which can be accurately embedded in its own coordinate system. In the noise-free case, the computed coordinates of the sensors in each patch must agree with their global positioning up to some unknown rigid motion, that is, up to translation, rotation and possibly reflection. In other words, to every patch there corresponds an element of the Euclidean group Euc(3) of rigid transformations in $\mathbb{R}^3$, and the goal is to estimate the group elements that will properly align all the patches in a globally consistent way. Furthermore, 3D-ASAP successfully incorporates information specific to the molecule problem in structural biology, in particular information on known substructures and their orientation. In addition, we also propose 3D-SP-ASAP, a faster version of 3D-ASAP, which uses a spectral partitioning algorithm as a preprocessing step for dividing the initial graph into smaller subgraphs. Our extensive numerical simulations show that 3D-ASAP and 3D-SP-ASAP are very robust to high levels of noise in the measured distances and to sparse connectivity in the measurement graph, and compare favorably to similar state-of-the art localization algorithms.Comment: 49 pages, 8 figure