Geometric Spanners in the MapReduce Model

Journal of Computational Geometry, 2012

A ball graph is an intersection graph of a set of balls with arbitrary radii. Given a real number t > 1, we say that a subgraph G ′ of a graph G is a t-spanner of G, if for every pair of vertices u, v in G, there exists a path in G ′ of length at most t times the distance between u and v in G. In this paper, we consider the problem of efficiently constructing sparse spanners of ball graphs which supports fast shortest path distance queries. We present the first algorithm for constructing spanners of ball graphs. For a disk graph in R 2 , we construct a (1 + ǫ)-spanner for any ǫ > 0 with O(nǫ −2) edges in O(n 4/3+δ ǫ −4/3 log 2/3 S) time, using an efficient partitioning of the plane into squares and solving intersection problems. Here δ is any positive constant, and S is the ratio between the largest and smallest radius. For the special case when the disks all have unit size, we show that the complexity of constructing a (1+ ǫ)-spanner is almost equal to the complexity of constructing a Euclidean minimum spanning tree. The algorithm extends naturally to other "disk-like" objects, also in higher dimensions. The algorithm uses an efficient subdivision of space to construct a sparse graph having many of the same distance properties as the input ball graph. Additionally, the constructed spanners have a small vertex separator decomposition (hereditary). In dimension 2, the disk graph spanner has an O(√ nǫ −3/2 + ǫ −3 log S) separator. The presence of a small separator is then exploited to obtain very efficient data structures for approximate distance queries. The results on geometric graph separators might be of independent interest. For example, since complete Euclidean graphs are just a special case of (unit) ball graphs, our results also provide a new approach for constructing spanners with small separators in these graphs.

Discrete & Computational Geometry, 1994

Given an undirected edge-weighted graph G = (V, E), a subgraph G' = (IT, E') is a t-spanner of G if, for every u, v ~ V, the weighted distance between u and v in G' is at most t times the weighted distance between u and v in G. We consider the problem of approximating the distances among points of a Euclidean metric space: given a finite set V of points in ~a, we want to construct a sparse t-spanner of the complete weighted graph induced by V. The weight of an edge in these graphs is the Euclidean distance between the endpoints of the edge. We show by a simple greedy argument that, for any t > 1 and any V c R a, a t-spanner G of V exists such that G has degree bounded by a function of d and r The analysis of our bounded degree spanners improves over previously known upper bounds on the minimum number of edges of Euclidean t-spanners, even compared with spanners of bounded average degree. Our results answer two open problems, one proposed by Vaidya and the other by Keil and Gutwin. The main result of the paper concerns the case of dimension d = 2. It is fairly easy to see that, for some t (t > 7.6), t-spanners of maximum degree 6 exist for any set of points in the Euclidean plane, but it was not known that degree 5 would suffice. We prove that for some (fixed) t, t-spanners of degree 5 exist for any set of points in the plane. We do not know if 5 is the best possible upper bound on the degree. * This research was supported by Conselho Nacional de Desenvolvimento Cientifico e Tecnol6gico, Proc 203039/87.4 (Brazil). 214 J. Soares shortest path between x and y. We say that a subgraph G' = (V, E') (with the same weights on E') is a t-spanner of G if, for every x, y 6 V, dG,(x, y) < t" da(x, y). The number t is a measure of how well G' approximates G with respect to the distances. The construction of t-spanners has received recent attention in several works: [2], [3], [5], [8], [9], [11], and [18], among others. Given a set V ~ •a the complete Euclidean graph on V is the complete graph on V where each edge weight is the Euclidean distance I[x-Y]I. In this paper we consider the problem of constructing bounded degree spanners of complete Euclidean graphs. For brevity we write t-spanner of V instead of t-spanner of the complete Euclidean graph on V. Let A(G) denote the maximum degree of a graph G. Dobkin et al. [5] mention that Feder and others had shown that, for some fixed t and for any set V of points in the Euclidean plane, a t-spanner G of V exists such that A(G) < 7. Then they ask what would be the minimum A for which such a result is possible? This paper has a partial answer to this question. Our main result (Section 4) is that, for some fixed t, t-spanners with A < 5 exist. Nisan [10] has proved the same for A < 6. Section 2 contains the basic algorithm used to construct bounded degree t-spanners. Although the algorithm has been used before by Althrfer et al. [1] and Soares [16] to construct t-spanners for arbitrary graphs, it was not known that the algorithm also constructs bounded degree spanners for complete Euclidean graphs. Section 3 contains a brief analysis of the problem when V is in d-dimensional Euclidean space. We show that, for any t > 1 and any V c ~d, a t-spanner G of V exists where A(G) is bounded by a function that depends only on d and t. This answers a question proposed by Keil and Gutwin in [8]. This bound on the maximum degree implies an improvement on the previously known upper bounds on the number of edges sufficient to build Euclidean spanners. Then we show that, for each dimension d, the least A(G) for which our algorithm constructs Od(1)spanners coincides with the kissing number in dimension d. (Od(1) denotes some function of d, i.e., a constant for each d.) Section 4 contains our main result, the construction of O(1)-spanners of degree 5 for any set of points in the Euclidean plane.

Theoretical Computer Science, 2011

A geometric network is a distributed network where each processor is identified by two numbers, representing the coordinates of the point in the plane where the processor is located. The edges of the network correspond to straight line segments such that no two of them intersect. In this paper we introduce the study of distributed computing in geometric networks. We study several computational geometry problems from the distributed computing point of view, such as finding convex hulls of geometric networks and identification of the external face. In particular, we obtain a O(n log 2 n) message complexity algorithm to find the convex hull of a planar geometric graph, and a O(n log n) algorithm to identify the external face of a geometric graph. We also prove that the message complexity of leader election in an asynchronous geometric ring of n processors is Ω(n log n).

Proceedings of the 26th ACM international conference on Supercomputing - ICS '12, 2012

This work explores fundamental modeling and algorithmic issues arising in the well-established MapReduce framework. First, we formally specify a computational model for MapReduce which captures the functional flavor of the paradigm by allowing for a flexible use of parallelism. Indeed, the model diverges from a traditional processor-centric view by featuring parameters which embody only global and local memory constraints, thus favoring a more data-centric view. Second, we apply the model to the fundamental computation task of matrix multiplication presenting upper and lower bounds for both dense and sparse matrix multiplication, which highlight interesting tradeoffs between space and round complexity. Finally, building on the matrix multiplication results, we derive further space-round tradeoffs on matrix inversion and matching.

Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, 2010

In recent years the MapReduce framework has emerged as one of the most widely used parallel computing platforms for processing data on terabyte and petabyte scales. Used daily at companies such as Yahoo!, Google, Amazon, and Facebook, and adopted more recently by several universities, it allows for easy parallelization of data intensive computations over many machines. One key feature of MapReduce that differentiates it from previous models of parallel computation is that it interleaves sequential and parallel computation. We propose a model of efficient computation using the MapReduce paradigm. Since MapReduce is designed for computations over massive data sets, our model limits the number of machines and the memory per machine to be substantially sublinear in the size of the input. On the other hand, we place very loose restrictions on the computational power of of any individual machineour model allows each machine to perform sequential computations in time polynomial in the size of the original input. We compare MapReduce to the PRAM model of computation. We prove a simulation lemma showing that a large class of PRAM algorithms can be efficiently simulated via MapReduce. The strength of MapReduce, however, lies in the fact that it uses both sequential and parallel computation. We demonstrate how algorithms can take advantage of this fact to compute an MST of a dense graph in only two rounds, as opposed to Ω(log(n)) rounds needed in the standard PRAM model. We show how to evaluate a wide class of functions using the MapReduce framework. We conclude by applying this result to show how to compute some basic algorithmic problems such as undirected s-t connectivity in the MapReduce framework.

2012

The MapReduce framework, originally proposed by Google [8], and its open source implementation, Hadoop [27], are nowadays considered the standard frameworks, both in industry and academia, to deal with petabyte scale datasets. In this paper we describe a two-rounds MapReduce approach to biconnectivity in undirected graphs, that is the computation of the set of articulation points, the set of bridges and the set of biconnected components of a graph G. We recall that an articulation point (resp. a bridge) is a vertex (resp. an edge) whose removal increases the number of connected components. A biconnected component is a maximal biconnected subgraph, i.e., it does not include neither articulation points nor bridges.

Proceedings of the twenty-seventh ACM symposium on Principles of distributed computing - PODC '08, 2008

The paper presents a deterministic distributed algorithm that, given k 1, constructs in k rounds a (2k−1, 0)-spanner of O(kn 1+1/k ) edges for every n-node unweighted graph. (If n is not available to the nodes, then our algorithm executes in 3k − 2 rounds, and still returns a (2k − 1, 0)-spanner with O(kn 1+1/k ) edges.) Previous distributed solutions achieving such optimal stretch-size trade-off either make use of randomization providing performance guarantees in expectation only, or perform in log Ω(1) n rounds, and all require a priori knowledge of n. Based on this algorithm, we propose a second deterministic distributed algorithm that, for every > 0, constructs a (1 + , 2)-spanner of O( −1 n 3/2 ) edges in O( −1 ) rounds, without any prior knowledge on the graph.

Algorithms-ESA …, 2009

2019 IEEE International Conference on Big Data (Big Data), 2019

Motivated by the needs of a scalable out-of-core surface reconstruction algorithm available on the cloud, this paper addresses the computation of distributed Delaunay triangulations of massive point sets. The proposed algorithm takes as input a point cloud and first partitions it across multiple processing elements into tiles of relatively homogeneous point sizes. The distributed computation and communication between processing elements is orchestrated so that each one discovers the Delaunay neighbors of its input points within the theoretical overall Delaunay triangulation of all points and computes locally a partial view of this triangulation. This approach prevents memory limitations by never materializing the global triangulation. This efficiency is due to our proposed uncentralized model to represent, manage and locally construct the triangulation corresponding to each tile. The point set is first partitioned into non-overlapping tiles, then we construct within each tile the Delaunay triangulation of the local points and a minimal set of replicated foreign points in order to capture the simplices spanning multiple tiles. Inspired by the star splaying approach for Delaunay triangulation computation/repair, communication is limited to exchanging points of potential Delaunay neighbors across tiles. Therefore, our method is guaranteed to reconstruct, within each tile, a triangulation that contains the star of its local points, as though it were computed within the Delaunay triangulation of all points. The proposed algorithm is implemented with Spark for the scheduling and C ++ for the geometric computations. This allows both an optimal scheduling on multiple machines and efficient low-level computation. The results show the efficiency of our algorithm in terms of speedup and strong scaling on a classical Spark configuration with both synthetic and real use case datasets.

Journal of Parallel and Distributed Computing, 1986

This paper presents new algorithms for solving some geometric problems on a shared memory parallel computer, where concurrent reads are allowed but no two processors can simultaneously attempt to write in the same memory location. The algorithms are quite different from known sequential algorithms, and are based on the use of a new parallel divide-and-conquer technique. One of our results is an 0 (log n) time, O(n) processor algorithm for the convex hull problem. Another result is an O(log n log log n) time, O(n) processor algorithm for the problem of selecting a closest pair of points among n inpUt pOint.3.