Optimal Random Matchings, Tours, and Spanning Trees in Hierarchically Separated Trees

We derive tight bounds on the expected weights of several combinatorial optimization problems for random point sets of size $n$ distributed among the leaves of a balanced hierarchically separated tree. We consider {\it monochromatic} and {\it bichromatic} versions of the minimum matching, minimum spanning tree, and traveling salesman problems. We also present tight concentration results for the monochromatic problems.Comment: 24 pages, to appear in TC

Dado2 llc User's Manual

llc is an ex.tension of C for hierarchically parallel processing on distributed-memory parallel processors. The language has been implemented on Dado2, a massively parallel tree-structured MIMD multicomputer. This manual explains the features of the language as it is implemented on Dado2, its compilation, and execution. The manual complements the Ilc tutorial, the Ilc reference manual, and the llc report

Dado2 llc User's Manual

llc is an ex.tension of C for hierarchically parallel processing on distributed-memory parallel processors. The language has been implemented on Dado2, a massively parallel tree-structured MIMD multicomputer. This manual explains the features of the language as it is implemented on Dado2, its compilation, and execution. The manual complements the Ilc tutorial, the Ilc reference manual, and the llc report

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

Dynamic Analysis of the Arrow Distributed Directory Protocol in General Networks

The Arrow protocol is a simple and elegant protocol to coordinate exclusive access to a shared object in a network. The protocol solves the underlying distributed queueing problem by using path reversal on a pre-computed spanning tree (or any other tree topology simulated on top of the given network). It is known that the Arrow protocol solves the problem with a competitive ratio of O(log D) on trees of diameter D. This implies a distributed queueing algorithm with competitive ratio O(s log D) for general networks with a spanning tree of diameter D and stretch s. In this work we show that when running the Arrow protocol on top of the well-known probabilistic tree embedding of Fakcharoenphol, Rao, and Talwar [STOC\u2703], we obtain a randomized distributed online queueing algorithm with expected competitive ratio O(log n) against an oblivious adversary even on general n-node network topologies. The result holds even if the queueing requests occur in an arbitrarily dynamic and concurrent fashion and even if communication is asynchronous. The main technical result of the paper shows that the competitive ratio of the Arrow protocol is constant on a special family of tree topologies, known as hierarchically well separated trees

Low Diameter Graph Decompositions by Approximate Distance Computation

In many models for large-scale computation, decomposition of the problem is key to efficient algorithms. For distance-related graph problems, it is often crucial that such a decomposition results in clusters of small diameter, while the probability that an edge is cut by the decomposition scales linearly with the length of the edge. There is a large body of literature on low diameter graph decomposition with small edge cutting probabilities, with all existing techniques heavily building on single source shortest paths (SSSP) computations. Unfortunately, in many theoretical models for large-scale computations, the SSSP task constitutes a complexity bottleneck. Therefore, it is desirable to replace exact SSSP computations with approximate ones. However this imposes a fundamental challenge since the existing constructions of low diameter graph decomposition with small edge cutting probabilities inherently rely on the subtractive form of the triangle inequality, which fails to hold under distance approximation. The current paper overcomes this obstacle by developing a technique termed blurry ball growing. By combining this technique with a clever algorithmic idea of Miller et al. (SPAA 2013), we obtain a construction of low diameter decompositions with small edge cutting probabilities which replaces exact SSSP computations by (a small number of) approximate ones. The utility of our approach is showcased by deriving efficient algorithms that work in the CONGEST, PRAM, and semi-streaming models of computation. As an application, we obtain metric tree embedding algorithms in the vein of Bartal (FOCS 1996) whose computational complexities in these models are optimal up to polylogarithmic factors. Our embeddings have the additional useful property that the tree can be mapped back to the original graph such that each edge is "used" only logaritmically many times, which is of interest for capacitated problems and simulating CONGEST algorithms on the tree into which the graph is embedded

An efficient multi-core implementation of a novel HSS-structured multifrontal solver using randomized sampling

We present a sparse linear system solver that is based on a multifrontal variant of Gaussian elimination, and exploits low-rank approximation of the resulting dense frontal matrices. We use hierarchically semiseparable (HSS) matrices, which have low-rank off-diagonal blocks, to approximate the frontal matrices. For HSS matrix construction, a randomized sampling algorithm is used together with interpolative decompositions. The combination of the randomized compression with a fast ULV HSS factorization leads to a solver with lower computational complexity than the standard multifrontal method for many applications, resulting in speedups up to 7 fold for problems in our test suite. The implementation targets many-core systems by using task parallelism with dynamic runtime scheduling. Numerical experiments show performance improvements over state-of-the-art sparse direct solvers. The implementation achieves high performance and good scalability on a range of modern shared memory parallel systems, including the Intel Xeon Phi (MIC). The code is part of a software package called STRUMPACK -- STRUctured Matrices PACKage, which also has a distributed memory component for dense rank-structured matrices

Visualization with hierarchically structured trees for an explanation reasoning system

This work is concerned with an application of drawing hierarchically structured trees. The tree drawing is applied to an explanation reasoning system. The reasoning is based on synthetic abduction (hypothesis) that gets a case from a rule and a result. In other words, the system searches a proper environment to get a desired result. In order that the system may be reliably related to the amount of rules which are used to get the answer, we visualize a process of reasoning to show how rules have concern with the process. Since the process of reasoning in the system makes a hierarchically structured tree, the visualization of reasoning is a drawing of a hierarchically structured tree. We propose a method of visualization that is applicable to the explanation reasoning system.</p