Sharif Digital Repository

Given a 2.5D terrain and a query point p on or above it, we want to find the triangles of terrain that are visible from p. We present an approximation algorithm to solve this problem. We implement the algorithm and test it on real data sets. The experimental results show that our approximate solution is very close to the exact solution and compared to the other similar works, the computational cost of our algorithm is lower. We analyze the computational complexity of the algorithm. We consider the visibility testing problem where the goal is to test whether a given triangle of the terrain is visible or not with respect to p. We present an algorithm for this problem and show that the average... 

Given a set P of n points in the plane, the all nearest neighbors problem asks for finding the closest point in P for each point in the set. The following folklore algorithm is used for the problem in practice: Pick a line in a random direction, project all points onto the line, and then search for the nearest neighbor of each point in a small vicinity of that point on the line. It is widely believed that the expected number of points needed to be checked by the algorithm in the vicinity of each point is O(pn) on average. We confirm this conjecture in affirmative by providing a careful analysis on the expected number of comparisons made by the algorithm. We also present a matching lower... 

Given a set P of n points in the plane, the all nearest neighbors problem asks for finding the closest point in P for each point in the set. The following folklore algorithm is used for the problem in practice: Pick a line in a random direction, project all points onto the line, and then search for the nearest neighbor of each point in a small vicinity of that point on the line. It is widely believed that the expected number of points needed to be checked by the algorithm in the vicinity of each point is O(pn) on average. We confirm this conjecture in affirmative by providing a careful analysis on the expected number of comparisons made by the algorithm. We also present a matching lower... 

Given a set S of n disjoint line segments in ℝ2, the visibility counting problem (VCP) is to preprocess S such that the number of segments in S visible from any query point p can be computed quickly. This problem can trivially be solved in logarithmic query time using O(n4) preprocessing time and space. Gudmundsson and Morin proposed a 2-approximation algorithm for this problem with a tradeoff between the space and the query time. They answer any query in Oε(n1−α) with Oε(n2+2α) of preprocessing time and space, where α is a constant 0 ≤ α ≤ 1, ε > 0 is another constant that can be made arbitrarily small, and Oε(f(n)) = O(f(n)nε). In this paper, we propose a randomized approximation algorithm... 

We consider load balancing in a network of caching servers delivering contents to end users. Randomized load balancing via the so-called power of two choices is a well-known approach in parallel and distributed systems that reduces network imbalance. In this paper, we propose a randomized load balancing scheme which simultaneously considers cache size limitation and proximity in the server redirection process. Since the memory limitation and the proximity constraint cause correlation in the server selection process, we may not benefit from the power of two choices in general. However, we prove that in certain regimes, in terms of memory limitation and proximity constraint, our scheme results... 

Given a set S of n disjoint line segments in R2, the visibility counting problem (VCP) is to preprocess S such that the number of segments in S visible from any query point p can be computed quickly. This problem can be solved trivially in O(log⁡n) query time using O(n4log⁡n) preprocessing time and O(n4) space. Gudmundsson and Morin (2010) [10] proposed a 2-approximation algorithm for this problem with a tradeoff between the space and the query time. For any constant 0≤α≤1, their algorithm answers any query in Oϵ(m(1−α)/2) time with Oϵ(m1+α) of preprocessing time and space, where ϵ>0 is a constant that can be made arbitrarily small and Oϵ(f(n))=O(f(n)nϵ) and m=O(n2) is a number that depends... 

We study two well-known planar visibility problems, namely visibility testing and visibility counting, in a model where there is uncertainty about the input data. The standard versions of these problems are defined as follows: we are given a set S of n segments in R 2 , and we would like to preprocess S so that we can quickly answer queries of the form: is the given query segment s∈S visible from the given query point q∈R 2 (for visibility testing) and how many segments in S are visible from the given query point q∈R 2 (for visibility counting). In our model of uncertainty, each segment may or may not exist, and if it does, it is located in one of finitely many possible locations, given by a... 

This paper addresses the problem of finding the nearest neighbor (or one of the R-nearest neighbors) of a query object q in a database of n objects, when we can only use a comparison oracle. The comparison oracle, given two reference objects and a query object, returns the reference object most similar to the query object. The main problem we study is how to search the database for the nearest neighbor (NN) of a query, while minimizing the questions. The difficulty of this problem depends on properties of the underlying database. We show the importance of a characterization: combinatorial disorder D which defines approximate triangle inequalities on ranks. We present a lower bound of Ω(Dlog... 

For a set of n disjoint line segments S in R2, the visibility testing problem (VTP) is to test whether the query point p sees a query segment s∈S. For this configuration, the visibility counting problem (VCP) is to preprocess S such that the number of visible segments in S from any query point p can be computed quickly. In this paper, we solve VTP in expected logarithmic query time using quadratic preprocessing time and space. Moreover, we propose a (1+δ)-approximation algorithm for VCP using at most quadratic preprocessing time and space. The query time of this method is Oε (1/δ 2√n) where Oε (f(n))=O(f(n)nε) and ε>0 is an arbitrary constant number  

Motivated by the bus escape routing problem in printed circuit boards, we study the following rectangle escape problem: given a set S of n axis-aligned rectangles inside an axis-aligned rectangular region R, extend each rectangle in S toward one of the four borders of R so that the maximum density over the region R is minimized. The density of each point p∈R is defined as the number of extended rectangles containing p. We show that the problem is hard to approximate to within a factor better than 3/2 in general. When the optimal density is sufficiently large, we provide a randomized algorithm that achieves an approximation factor of 1+ε with high probability improving over the current best... 

We study a weighted balanced version of the k-center problem, where each center has a fixed capacity, and each element has an arbitrary demand. The objective is to assign demands of the elements to the centers, so as the total demand assigned to each center does not exceed its capacity, while the maximum distance between centers and their assigned elements is minimized. We present a deterministic O(1)-approximation algorithm for this generalized version of the k-center problem in the distributed setting, where data is partitioned among a number of machines. Our algorithm substantially improves the approximation factor of the current best randomized algorithm available for the problem. We... 

We revisit the classical edge coloring problem for general graphs in the streaming model. In this model, the input graph is presented as a stream of edges, and the algorithm must report colors assigned to the edges in a streaming fashion, using a memory of size O(n polylog n) on graphs of up to O(n2) edges. In ESA 2019 and SOSA 2021, two elegant randomized algorithms were presented for this problem in the adversarial edge arrival model, where the latest one colors any input graph using O(Δ2/s) colors with high probability in O (ns) space. In this short note, we propose two extremely simple streaming algorithms that achieve the same color and space bounds deterministically. Besides being...