Nabil H. Mustafa

k -means projective clustering

In many applications it is desirable to cluster high dimensional data along various subspaces, which we refer to as projective clustering. We propose a new objective function for projective clustering, taking into account the inherent trade-off between the dimension of a subspace and the induced clustering error. We then present an extension of the k-means clustering algorithm for projective clustering in arbitrary subspaces, and also propose techniques to avoid local minima. Unlike previous algorithms, ours can choose the dimension of each cluster independently and automatically. Furthermore, experimental results show that our algorithm is significantly more accurate than the previous approaches.

Near-Linear Time Approximation Algorithms for Curve Simplification

AbstractWe consider the problem of approximating a polygonal curve P under a given error criterion by another polygonal curve P’ whose vertices are a subset of the vertices of P. The goal is to minimize the number of vertices of P’ while ensuring that the error between P’ and P is below a certain threshold. We consider two different error measures: Hausdorff and Frechet. For both error criteria, we present near-linear time approximation algorithms that, given a parameter ε > 0, compute a simplified polygonal curve P’ whose error is less than ε and size at most the size of an optimal simplified polygonal curve with error ε/2. We consider monotone curves in ℝ2 in the case of the Hausdorff error measure under the uniform distance metric and arbitrary curves in any dimension for the Frechet error measure under Lp metrics. We present experimental results demonstrating that our algorithms are simple and fast, and produce close to optimal simplifications in practice.

PTAS for geometric hitting set problems via local search

We consider the problem of computing minimum geometric hitting sets in which, given a set of geometric objects and a set of points, the goal is to compute the smallest subset of points that hit all geometric objects. The problem is known to be strongly NP-hard even for simple geometric objects like unit disks in the plane. Therefore, unless P=NP, it is not possible to get Fully Polynomial Time Approximation Algorithms (FPTAS) for such problems. We give the first PTAS for this problem when the geometric objects are half-spaces in Re3 and when they are an r-admissible set regions in the plane (this includes pseudo-disks as they are 2-admissible). Quite surprisingly, our algorithm is a very simple local search algorithm which iterates over local improvements only.

Streaming Geometric Optimization Using Graphics Hardware

In this paper we propose algorithms for solving a variety of geometric optimization problems on a stream of points in ℝ2 or ℝ3. These problems include various extent measures (e.g. diameter, width, smallest enclosing disk), collision detection (penetration depth and distance between polytopes), and shape fitting (minimum width annulus, circle/line fitting). The main contribution of this paper is a unified approach to solving all of the above problems efficiently using modern graphics hardware. All the above problems can be approximated using a constant number of passes over the data stream. Our algorithms are easily implemented, and our empirical study demonstrates that the running times of our programs are comparable to the best implementations for the above problems. Another significant property of our results is that although the best known implementations for the above problems are quite different from each other, our algorithms all draw upon the same set of tools, making their implementation significantly easier.

Independent set of intersection graphs of convex objects in 2D

The intersection graph of a set of geometric objects is defined as a graph G = (S, E) in which there is an edge between two nodes si, sj ∈ S if si ∩ sj ≠ 0. The problem of computing a maximum independent set in the intersection graph of a set of objects is known to be NP-complete for most cases in two and higher dimensions. We present approximation algorithms for computing a maximum independent set of intersection graphs of convex objects in R2. Specifically, given (i) a set of n line segments in the plane with maximum independent set of size α, we present algorithms that find an independent set of size at least (α/(2 log(2n/α)))1/2 in time O(n3) and (α/(2log(2n/α)))1/4 in time O(n4/3 logcn), (ii) a set of n convex objects with maximum independent set of size α, we present an algorithm that finds an independent set of size at least (α/(2log(2n/α)))1/3 in time O(n3 + τ (S)), assuming that S can be preprocessed in time τ(S) to answer certain primitive operations on these convex sets, and (iii) a set of n rectangles with maximum independent set of size βn, for β≤ 1, we present an algorithm that computes an independent set of size Ω (β2n). All our algorithms use the notion of partial orders that exploit the geometric structure of the convex objects.

Dynamic simplification and visualization of large maps

In this paper, we present an algorithm that performs simplification of large geographical maps through a novel use of graphics hardware. Given a map as a collection of non‐intersecting chains and a tolerance parameter for each chain, we produce a simplified map that resembles the original map, satisfying the condition that the distance between each point on the simplified chain and the original chain is within the given tolerance parameter, and that no two chains intersect. In conjunction with this, we also present an out‐of‐core system for interactive visualization of these maps. We represent the maps hierarchically and employ different pruning strategies to accelerate the rendering. Our algorithm uses a parallel approach to do rendering as well as fetching data from the disk in a synchronous manner. We have applied our algorithm to a gigabyte sized map dataset. The memory overhead of our algorithm (the amount of main memory it requires) is output sensitive and is typically tens of megabytes, much smaller than the actual data size.

Hardware-assisted view-dependent map simplification

In this paper, we present an algorithm and a system to perform dynamic view dependent simplification of large geographical maps through a novel use of graphics hardware. Given a map as a collection of non-intersecting chains and a tolerance parameter for each chain, we produce a simplified map that resembles the original map, satisfying the condition that the distance between each point on the simplified chain and the original chain is within the given tolerance parameter, and that no two chains intersect. We also present an interactive map visualization system which uses frame-to-frame coherence to perform dynamic view-dependent simplification. Our initial results indicate that we get a 3-4 fold increase in the frame rates using our simplification algorithm on maps with 1.5-2 million vertices on an SGI Onyx workstation.

Majority Consensus and the Local Majority Rule

We study a rather generic communication/coordination/ computation problem: in a finite network of agents, each initially having one of the two possible states, can the majority initial state be computed and agreed upon by means of local computation only? We describe the architecture of networks that are always capable of reaching the consensus on the majority initial state of its agents. In particular, we show that, for any truly local network of agents, there are instances in which the network is not capable of reaching such consensus. Thus, every truly local computation approach that requires reaching consensus is not failure-free.

Fast molecular shape matching using contact maps.

In this paper, we study the problem of computing the similarity of two protein structures by measuring their contact-map overlap. Contact-map overlap abstracts the problem of computing the similarity of two polygonal chains as a graph-theoretic problem. In R3, we present the first polynomial time algorithm with any guarantee on the approximation ratio for the 3-dimensional problem. More precisely, we give an algorithm for the contact-map overlap problem with an approximation ratio of sigma where sigma = min{sigma(P1), sigma(P2)} <or= O(n(1/2)) is a decomposition parameter depending on the input polygonal chains P1 and P2. In R2, we improve the running time of the previous best known approximation algorithm from O(n(6)) to O(n(3) log n) at the cost of decreasing the approximation ratio by half. We also give hardness results for the problem in three dimensions, suggesting that approximating it better than O(n(epsilon)), for some epsilon > 0, is hard.

Conflict-Free colorings of rectangles ranges

Given the range space (