Geometric Approximation Algorithms in the Online and Data Stream Models

The online and data stream models of computation have recently attracted considerable research attention due to many real-world applications in various areas such as data mining, machine learning, distributed computing, and robotics. In both these models, input items arrive one at a time, and the algorithms must decide based on the partial data received so far, without any secure information about the data that will arrive in the future. In this thesis, we investigate efficient algorithms for a number of fundamental geometric optimization problems in the online and data stream models. The problems studied in this thesis can be divided into two major categories: geometric clustering and computing various extent measures of a set of points. In the online setting, we show that the basic unit clustering problem admits non-trivial algorithms even in the simplest one-dimensional case: we show that the naive upper bounds on the competitive ratio of algorithms for this problem can be beaten using randomization. In the data stream model, we propose a new streaming algorithm for maintaining "core-sets" of a set of points in fixed dimensions, and also, introduce a new simple framework for transforming a class of offline algorithms to their equivalents in the data stream model. These results together lead to improved streaming approximation algorithms for a wide variety of geometric optimization problems in fixed dimensions, including diameter, width, k-center, smallest enclosing ball, minimum-volume bounding box, minimum enclosing cylinder, minimum-width enclosing spherical shell/annulus, etc. In high-dimensional data streams, where the dimension is not a constant, we propose a simple streaming algorithm for the minimum enclosing ball (the 1-center) problem with an improved approximation factor

Online unit clustering in higher dimensions

We revisit the online Unit Clustering and Unit Covering problems in higher dimensions: Given a set of $n$ points in a metric space, that arrive one by one, Unit Clustering asks to partition the points into the minimum number of clusters (subsets) of diameter at most one; while Unit Covering asks to cover all points by the minimum number of balls of unit radius. In this paper, we work in $\mathbb{R}^d$ using the $L_\infty$ norm. We show that the competitive ratio of any online algorithm (deterministic or randomized) for Unit Clustering must depend on the dimension $d$. We also give a randomized online algorithm with competitive ratio $O(d^2)$ for Unit Clustering}of integer points (i.e., points in $\mathbb{Z}^d$, $d\in \mathbb{N}$, under $L_{\infty}$ norm). We show that the competitive ratio of any deterministic online algorithm for Unit Covering is at least $2^d$. This ratio is the best possible, as it can be attained by a simple deterministic algorithm that assigns points to a predefined set of unit cubes. We complement these results with some additional lower bounds for related problems in higher dimensions.Comment: 15 pages, 4 figures. A preliminary version appeared in the Proceedings of the 15th Workshop on Approximation and Online Algorithms (WAOA 2017

Preface to the Special Issue “Sufism in the Modern World”

“Sufism is the major sacrifice offered by Islam on the altar of its modernization”, declares a contemporary scholar while explaining the modern challenges faced by Sufism (Weismann 2015, p [...

Online Coloring Co-interval Graphs

We study the problem of online coloring co-interval graphs. In this problem, a set of intervals on the real line is presented to the online algorithm in some arbitrary order, and the algorithm must assign each interval a color that is different from the colors of all previously presented intervals not intersecting the current interval. It is known that the competitive ratio of the simple First-Fit algorithm on the class of co-interval graphs is at most 2. We show that for the class of unit co-interval graphs, where all intervals have equal length, the 2-bound on the competitive ratio of First-Fit is tight. On the other hand, we show that no deterministic online algorithm for coloring unit co-interval graphs can be better than 3/2-competitive. We then study the effect of randomization in our problem, and show a lower bound of 4/3 on the competitive ratio of any randomized algorithm for the unit co-interval coloring problem. We also prove that for the class of general co-interval graphs no randomized algorithm has competitive ratio better than 3/2