Sandeep Sen

A simple linear time (1 + /spl epsiv/)-approximation algorithm for k-means clustering in any dimensions

We present the first linear time (1 + /spl epsiv/)-approximation algorithm for the k-means problem for fixed k and /spl epsiv/. Our algorithm runs in O(nd) time, which is linear in the size of the input. Another feature of our algorithm is its simplicity - the only technique involved is random sampling.

FSTTCS 2005: Foundations of Software Technology and Theoretical Computer Science

Invited Papers.- Semiperfect-Information Games.- Computational Complexity Since 1980.- Developments in Data Structure Research During the First 25 Years of FSTTCS.- Inference Systems for Logical Algorithms.- From Logic to Games.- Proving Lower Bounds Via Pseudo-random Generators.- Erd?s Magic.- Contributed Papers.- No Coreset, No Cry: II.- Improved Bounds on the Union Complexity of Fat Objects.- On the Bisimulation Congruence in ?-Calculus.- Extending Howes Method to Early Bisimulations for Typed Mobile Embedded Resources with Local Names.- Approximation Algorithms for Wavelength Assignment.- The Set Cover with Pairs Problem.- Non-disclosure for Distributed Mobile Code.- Quantitative Models and Implicit Complexity.- The MSO Theory of Connectedly Communicating Processes.- Reachability of Hennessy-Milner Properties for Weakly Extended PRS.- Decision Procedures for Queues with Integer Constraints.- The Directed Planar Reachability Problem.- Dimensions of Copeland-Erdos Sequences.- Refining the Undecidability Frontier of Hybrid Automata.- When Are Timed Automata Weakly Timed Bisimilar to Time Petri Nets?.- Subquadratic Algorithms for Workload-Aware Haar Wavelet Synopses.- Practical Algorithms for Tracking Database Join Sizes.- On Sampled Semantics of Timed Systems.- Eventual Timed Automata.- Causal Closure for MSC Languages.- Reachability Analysis of Multithreaded Software with Asynchronous Communication.- Probabilistic Analysis for a Multiple Depot Vehicle Routing Problem.- Computing the Expected Accumulated Reward and Gain for a Subclass of Infinite Markov Chains.- Towards a CTL* Tableau.- Bisimulation Quantified Logics: Undecidability.- Logarithmic-Time Single Deleter, Multiple Inserter Wait-Free Queues and Stacks.- Monitoring Stable Properties in Dynamic Peer-to-Peer Distributed Systems.- On the Expressiveness of TPTL and MTL.- Modal Strength Reduction in Quantified Discrete Duration Calculus.- Comparing Trees Via Crossing Minimization.- On Counting the Number of Consistent Genotype Assignments for Pedigrees.- Fixpoint Logics on Hierarchical Structures.- The Equivalence Problem for Deterministic MSO Tree Transducers Is Decidable.- Market Equilibrium for CES Exchange Economies: Existence, Multiplicity, and Computation.- Testing Concurrent Systems: An Interpretation of Intuitionistic Logic.- Proofs of Termination of Rewrite Systems for Polytime Functions.- On the Controller Synthesis for Finite-State Markov Decision Processes.- Reasoning About Quantum Knowledge.

Parallel sorting in two-dimensional VLSI models of computation

The gradual refinement of a general approach to two-dimensional sorting, the shear-sort algorithm, to more sophisticated and specialized sorting algorithms on mesh-connected computers is described. The analysis of the shear-sort algorithm gives rise to a novel perspective of two-dimensional sorting, which seems to be a very powerful tool for developing efficient algorithms. The same methods can be extended for sorting in higher dimensions, for example, in the three-dimensional mesh. The concept of clean and dirty rows can be modified to clean and dirty planes (or hyperplanes for dimensions greater than three). Although only two schemes (purely recursive and iterative) are explicitly described, the reader may construct his own algorithm using similar technique and slight modifications. Designing an O(n) algorithm for sorting on a mesh becomes much simpler using the techniques developed. >

An efficient output-sensitive hidden surface removal algorithm and its parallelization

In this paper we present an algorithm for hidden surface removal for a class of polyhedral surfaces which have a property that they can be ordered relatively quickly like the terrain maps. A distinguishing feature of this algorithm is that its running time is sensitive to the actual size of the visible image rather than the total number of intersections in the image plane which can be much larger than the visible image. The time complexity of this algorithm is &Ogr;((k +n)lognloglogn) where n and k are respectively the input and the output sizes. Thus, in a significant number of situations this will be faster than the worst case optimal algorithms which have running time &OHgr;(n2) irrespective of the output size (where as the output size k is &Ogr;(n2) only in the worst case). We also present a parallel algorithm based on a similar approach which runs in time &Ogr;(log4(n+k)) using &Ogr;((n + k)/log(n+k)) processors in a CREW PRAM model. All our bounds are obtained using ammortized analysis.

Approximate distance oracles for unweighted graphs in expected O ( n 2 ) time

Let <i>G</i> = (<i>V</i>, <i>E</i>) be an undirected graph on <i>n</i> vertices, and let Δ(<i>u</i>, <i>v</i>) denote the distance in <i>G</i> between two vertices <i>u</i> and <i>v</i>. Thorup and Zwick showed that for any positive integer <i>t</i>, the graph <i>G</i> can be preprocessed to build a data structure that can efficiently report <i>t</i>-approximate distance between any pair of vertices. That is, for any <i>u</i>, <i>v</i> ∈ <i>V</i>, the distance reported is at least Δ(<i>u</i>, <i>v</i>) and at most <i>t</i>Δ(<i>u</i>, <i>v</i>). The remarkable feature of this data structure is that, for <i>t</i>≥3, it occupies subquadratic space, that is, it does not store all-pairs distances explicitly, and still it can answer any <i>t</i>-approximate distance query in constant time. They named the data structure “approximate distance oracle” because of this feature. Furthermore, the trade-off between the stretch <i>t</i> and the size of the data structure is essentially optimal.In this article, we show that we can actually construct approximate distance oracles in expected <i>O</i>(<i>n</i>²) time if the graph is unweighted. One of the new ideas used in the improved algorithm also leads to the first expected linear-time algorithm for computing an optimal size (2, 1)-spanner of an unweighted graph. A (2, 1) spanner of an undirected unweighted graph <i>G</i> = (<i>V</i>, <i>E</i>) is a subgraph (<i>V</i>, Ê), Ê ⊆ <i>E</i>, such that for any two vertices <i>u</i> and <i>v</i> in the graph, their distance in the subgraph is at most 2Δ(<i>u</i>, <i>v</i>) + 1.

A simple linear time algorithm for computing a (2k - 1)-spanner of o(n 1+1/k ) size in weighted graphs

Let G(V, E) be an undirected weighted graph with |V| = n, and |E| = m. A t-spanner of the graph G(V, E) is a sub-graph G(V, ES) such that the distance between any pair of vertices in the spanner is at most t times the distance between the two in the given graph. A 1963 girth conjecture of Erdos implies that Ω(n1+1/k) edges are required in the worst case for any (2k - 1)-spanner, which has been proved for k = 1, 2, 3, 5. There exist polynomial time algorithms that can construct spanners with the size that matches this conjectured lower bound, and the best known algorithm takes O(mn1/k) expected running time. In this paper, we present an extremely simple linear time randomized algorithm that constructs (2k - 1)-spanner of size matching the conjectured lower bound. Our algorithm requires local information for computing a spanner, and thus can be adapted suitably to obtain efficient distributed and parallel algorithms.

Linear-time approximation schemes for clustering problems in any dimensions

We present a general approach for designing approximation algorithms for a fundamental class of geometric clustering problems in arbitrary dimensions. More specifically, our approach leads to simple randomized algorithms for the <i>k</i>-means, <i>k</i>-median and discrete <i>k</i>-means problems that yield (1+ϵ) approximations with probability ≥ 1/2 and running times of <i>O</i>(2^{(<i>k</i>/ϵ)^<i>O</i>(1)} <i>dn</i>). These are the first algorithms for these problems whose running times are linear in the size of the input (<i>nd</i> for <i>n</i> points in <i>d</i> dimensions) assuming <i>k</i> and ϵ are fixed. Our method is general enough to be applicable to clustering problems satisfying certain simple properties and is likely to have further applications.

The distance bound for sorting on mesh-connected processor arrays is tight

In this paper, We consider the problem of sorting n2 numbers, initially distributed randomly in an n × n mesh-connected processor array with one element per processor. We show a lower bound, based on distance arguments, of 4n routing steps on mesh-connected processors operating in an SIMD mode with no wraparounds in rows or columns, We present an algorithm using a novel approach, which is optimal upto the conslant of the leading term, and hence, succeed in proving the tightness of the lower bound based on distance. Keeping in mind the practical difficulties in implementation of this algorithm, we also present an extremely practical O(n) algorithm amenable for VLSI implementation and for existing mesh- connected computers. All the results in this paper were derived by using a new method of analysis inspired by the discovery of shear-sort or row-column sort.

Optimal Randomized Parallel Algorithms for Computational Geometry I

We present parallel algorithms for some fundamental problems in computational geometry which have a running time ofO(logn) usingn processors, with very high probability (approaching 1 asn → ∞). These include planar-point location, triangulation, and trapezoidal decomposition. We also present optimal algorithms for three-dimensional maxima and two-set dominance counting by an application of integer sorting. Most of these algorithms run on a CREW PRAM model and have optimal processor-time product which improve on the previously best-known algorithms of Atallah and Goodrich [5] for these problems. The crux of these algorithms is a useful data structure which emulates the plane-sweeping paradigm used for sequential algorithms. We extend some of the techniques used by Reischuk [26] and Reif and Valiant [25] for flashsort algorithm to perform divide and conquer in a plane very efficiently leading to the improved performance by our approach.

Polling: a new randomized sampling technique for computational geometry

We introduce a new randomized sampling technique, called Polling which has applications to deriving efficient parallel algorithms. As an example of its use in computational geometry, we present an optimal parallel randomized algorithm for intersection of half-spaces in three dimensions. Because of well-known reductions, our methods also yield equally efficient algorithms for fundamental problems like the convex hull in three dimensions, Voronoi diagram of point sites on a plane and Euclidean minimal spanning tree. Our algorithms run in time T = O(logn) for worst-case inputs and uses P = O(n) processors in a CREW PRAM model where n is the input size. They are randomized in the sense that they use a total of only O(log<supscrpt>2</supscrpt> <italic>n</italic>) random bits and terminate in the claimed time bound with probability 1 - <italic>n</italic><supscrpt>-α</supscrpt> for any α > 0. They are also optimal in <italic>P</italic> . <italic>T</italic> product since the sequential time bound for all these problems is &OHgr;(<italic>nlogn</italic>). The best known determistic parallel algorithms for 2-D Voronoi-diagram and 3-D Convex hull run in O(log<supscrpt>2</supscrpt> <italic>n</italic>) and O(log<supscrpt>2</supscrpt> <italic>nlog</italic> * <italic>n</italic>) time respectively while using O(n) processors.