Sharif Digital Repository

A geometric spanner on a point set is a sparse graph that approximates the Euclidean distances between all pairs of points in the point set. Here, we intend to construct a geometric spanner for a massive point set, using a distributed algorithm on parallel machines. In particular, we use the MapReduce model of computation to construct spanners in several rounds with inter-communications in between. An algorithm in this model is called efficient if it uses a sublinear number of machines and runs in a polylogarithmic number of rounds. In this paper, we propose an efficient MapReduce algorithm for constructing a geometric spanner in a constant number of rounds, using linear amount of... 

We focus on two metric clusterings namely r-gather and (r, ?)-gather. The objective of r-gather is to minimize the radius of clustering, such that each cluster has at least r points. (r, ?)-gather is a version of r-gather with the extra condition that at most n? points can be left unclustered (outliers). MapReduce is a model used for processing big data. In each round, it distributes data to multiple servers, then simultaneously processes each server's data. We prove a lower bound 2 on the approximation factor of metric r-gather in the MapReduce model, even if an optimal algorithm for r-gather exists. Then, we give a (4+ δ)-approximation algorithm for r-gather in MapReduce which runs in O(... 

We focus on two metric clusterings namely r-gather and (r, ?)-gather. The objective of r-gather is to minimize the radius of clustering, such that each cluster has at least r points. (r, ?)-gather is a version of r-gather with the extra condition that at most n? points can be left unclustered (outliers). MapReduce is a model used for processing big data. In each round, it distributes data to multiple servers, then simultaneously processes each server's data. We prove a lower bound 2 on the approximation factor of metric r-gather in the MapReduce model, even if an optimal algorithm for r-gather exists. Then, we give a (4+ δ)-approximation algorithm for r-gather in MapReduce which runs in O(... 

Given a set of n segments and a query shape Q, the windowing length query asks for finding the sum of the lengths of the parts of the segments that lie inside Q. The popular places problem of a set of curves asks for the subset of the plane where each query shape centered at a point of that region intersects with at least f distinct curves. For square queries, an optimal O(n2) time algorithm and a matching lower bound exist. We solve the length query problem for convex polygons and disks as query shapes, with O(log n+ k) query time and polynomial preprocessing time that depends on the complexity of the query shape. We define a new version of the problem of finding popular places in a set of... 

Given a set of n segments and a query shape Q, the windowing length query asks for finding the sum of the lengths of the parts of the segments that lie inside Q. The popular places problem of a set of curves asks for the subset of the plane where each query shape centered at a point of that region intersects with at least f distinct curves. For square queries, an optimal O(n2) time algorithm and a matching lower bound exist. We solve the length query problem for convex polygons and disks as query shapes, with O(log n+ k) query time and polynomial preprocessing time that depends on the complexity of the query shape. We define a new version of the problem of finding popular places in a set of... 

Given a set of n segments and a query shape Q, the windowing length query asks for finding the sum of the lengths of the parts of the segments that lie inside Q. The popular places problem of a set of curves asks for the subset of the plane where each query shape centered at a point of that region intersects with at least f distinct curves. For square queries, an optimal O(n2) time algorithm and a matching lower bound exist. We solve the length query problem for convex polygons and disks as query shapes, with O(log n+ k) query time and polynomial preprocessing time that depends on the complexity of the query shape. We define a new version of the problem of finding popular places in a set of... 

Packedness is a measure defined for curves as the ratio of maximum curve length inside any disk divided by its radius. Sparsification allows us to reduce the number of candidate disks for maximum packedness to a polynomial amount in terms of the number of vertices of the polygonal curve. This gives an exact algorithm for computing packedness. We prove that using a fat shape, such as a square, instead of a disk gives a constant factor approximation for packedness. Further sparsification using well-separated pair decomposition improves the time complexity at the cost of losing some accuracy. By adjusting the ratio of the separation factor and the size of the query, we improve the approximation...