2,469 research outputs found

    Approximation algorithms for multi-facility location

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    This thesis deals with the development and implementation of efficient algorithms to obtain acceptable solutions for the location of several facilities to serve customer sites. The general version of facility location problem is known to be NP-hard; For locating multiple facilities we use Voronoi diagram of initial facility locations to partition the customer sites into k clusters. On each Voronoi region, solutions for single facility problem is obtained by using both Weizfield\u27s algorithm and Center of Gravity. The customer space is again partitioned by using the newly computed locations. This iteration is continued to obtain a better solution for multi-facility location problem. We call the resulting algorithm: Voronoi driven k-median algorithm ; We report experimental results on several test data that include randomly distributed customers and distinctly clustered customers. The observed results show that the proposed approximation algorithm produces good results

    On Variants of k-means Clustering

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    \textit{Clustering problems} often arise in the fields like data mining, machine learning etc. to group a collection of objects into similar groups with respect to a similarity (or dissimilarity) measure. Among the clustering problems, specifically \textit{kk-means} clustering has got much attention from the researchers. Despite the fact that kk-means is a very well studied problem its status in the plane is still an open problem. In particular, it is unknown whether it admits a PTAS in the plane. The best known approximation bound in polynomial time is 9+\eps. In this paper, we consider the following variant of kk-means. Given a set CC of points in Rd\mathcal{R}^d and a real f>0f > 0, find a finite set FF of points in Rd\mathcal{R}^d that minimizes the quantity fF+pCminqFpq2f*|F|+\sum_{p\in C} \min_{q \in F} {||p-q||}^2. For any fixed dimension dd, we design a local search PTAS for this problem. We also give a "bi-criterion" local search algorithm for kk-means which uses (1+\eps)k centers and yields a solution whose cost is at most (1+\eps) times the cost of an optimal kk-means solution. The algorithm runs in polynomial time for any fixed dimension. The contribution of this paper is two fold. On the one hand, we are being able to handle the square of distances in an elegant manner, which yields near optimal approximation bound. This leads us towards a better understanding of the kk-means problem. On the other hand, our analysis of local search might also be useful for other geometric problems. This is important considering that very little is known about the local search method for geometric approximation.Comment: 15 page

    Lorentzian Iterative Hard Thresholding: Robust Compressed Sensing with Prior Information

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    Commonly employed reconstruction algorithms in compressed sensing (CS) use the L2L_2 norm as the metric for the residual error. However, it is well-known that least squares (LS) based estimators are highly sensitive to outliers present in the measurement vector leading to a poor performance when the noise no longer follows the Gaussian assumption but, instead, is better characterized by heavier-than-Gaussian tailed distributions. In this paper, we propose a robust iterative hard Thresholding (IHT) algorithm for reconstructing sparse signals in the presence of impulsive noise. To address this problem, we use a Lorentzian cost function instead of the L2L_2 cost function employed by the traditional IHT algorithm. We also modify the algorithm to incorporate prior signal information in the recovery process. Specifically, we study the case of CS with partially known support. The proposed algorithm is a fast method with computational load comparable to the LS based IHT, whilst having the advantage of robustness against heavy-tailed impulsive noise. Sufficient conditions for stability are studied and a reconstruction error bound is derived. We also derive sufficient conditions for stable sparse signal recovery with partially known support. Theoretical analysis shows that including prior support information relaxes the conditions for successful reconstruction. Simulation results demonstrate that the Lorentzian-based IHT algorithm significantly outperform commonly employed sparse reconstruction techniques in impulsive environments, while providing comparable performance in less demanding, light-tailed environments. Numerical results also demonstrate that the partially known support inclusion improves the performance of the proposed algorithm, thereby requiring fewer samples to yield an approximate reconstruction.Comment: 28 pages, 9 figures, accepted in IEEE Transactions on Signal Processin

    Real-time human ambulation, activity, and physiological monitoring:taxonomy of issues, techniques, applications, challenges and limitations

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    Automated methods of real-time, unobtrusive, human ambulation, activity, and wellness monitoring and data analysis using various algorithmic techniques have been subjects of intense research. The general aim is to devise effective means of addressing the demands of assisted living, rehabilitation, and clinical observation and assessment through sensor-based monitoring. The research studies have resulted in a large amount of literature. This paper presents a holistic articulation of the research studies and offers comprehensive insights along four main axes: distribution of existing studies; monitoring device framework and sensor types; data collection, processing and analysis; and applications, limitations and challenges. The aim is to present a systematic and most complete study of literature in the area in order to identify research gaps and prioritize future research directions

    Learning visual docking for non-holonomic autonomous vehicles

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    This paper presents a new method of learning visual docking skills for non-holonomic vehicles by direct interaction with the environment. The method is based on a reinforcement algorithm, which speeds up Q-learning by applying memorybased sweeping and enforcing the “adjoining property”, a filtering mechanism to only allow transitions between states that satisfy a fixed distance. The method overcomes some limitations of reinforcement learning techniques when they are employed in applications with continuous non-linear systems, such as car-like vehicles. In particular, a good approximation to the optimal behaviour is obtained by a small look-up table. The algorithm is tested within an image-based visual servoing framework on a docking task. The training time was less than 1 hour on the real vehicle. In experiments, we show the satisfactory performance of the algorithm

    Deterministic Clustering in High Dimensional Spaces: Sketches and Approximation

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    In all state-of-the-art sketching and coreset techniques for clustering, as well as in the best known fixed-parameter tractable approximation algorithms, randomness plays a key role. For the classic kk-median and kk-means problems, there are no known deterministic dimensionality reduction procedure or coreset construction that avoid an exponential dependency on the input dimension dd, the precision parameter ε1\varepsilon^{-1} or kk. Furthermore, there is no coreset construction that succeeds with probability 11/n1-1/n and whose size does not depend on the number of input points, nn. This has led researchers in the area to ask what is the power of randomness for clustering sketches [Feldman, WIREs Data Mining Knowl. Discov'20]. Similarly, the best approximation ratio achievable deterministically without a complexity exponential in the dimension are Ω(1)\Omega(1) for both kk-median and kk-means, even when allowing a complexity FPT in the number of clusters kk. This stands in sharp contrast with the (1+ε)(1+\varepsilon)-approximation achievable in that case, when allowing randomization. In this paper, we provide deterministic sketches constructions for clustering, whose size bounds are close to the best-known randomized ones. We also construct a deterministic algorithm for computing (1+ε)(1+\varepsilon)-approximation to kk-median and kk-means in high dimensional Euclidean spaces in time 2k2/εO(1)poly(nd)2^{k^2/\varepsilon^{O(1)}} poly(nd), close to the best randomized complexity. Furthermore, our new insights on sketches also yield a randomized coreset construction that uses uniform sampling, that immediately improves over the recent results of [Braverman et al. FOCS '22] by a factor kk.Comment: FOCS 2023. Abstract reduced for arxiv requirement

    Improved Bounds for Metric Capacitated Covering Problems

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    In the Metric Capacitated Covering (MCC) problem, given a set of balls ? in a metric space P with metric d and a capacity parameter U, the goal is to find a minimum sized subset ?\u27 ? ? and an assignment of the points in P to the balls in ?\u27 such that each point is assigned to a ball that contains it and each ball is assigned with at most U points. MCC achieves an O(log |P|)-approximation using a greedy algorithm. On the other hand, it is hard to approximate within a factor of o(log |P|) even with ? < 3 factor expansion of the balls. Bandyapadhyay et al. [SoCG 2018, DCG 2019] showed that one can obtain an O(1)-approximation for the problem with 6.47 factor expansion of the balls. An open question left by their work is to reduce the gap between the lower bound 3 and the upper bound 6.47. In this current work, we show that it is possible to obtain an O(1)-approximation with only 4.24 factor expansion of the balls. We also show a similar upper bound of 5 for a more generalized version of MCC for which the best previously known bound was 9

    Custom Integrated Circuits

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    Contains table of contents for Part III, table of contents for Section 1 and reports on eleven research projects.IBM CorporationMIT School of EngineeringNational Science Foundation Grant MIP 94-23221Defense Advanced Research Projects Agency/U.S. Army Intelligence Center Contract DABT63-94-C-0053Mitsubishi CorporationNational Science Foundation Young Investigator Award Fellowship MIP 92-58376Joint Industry Program on Offshore Structure AnalysisAnalog DevicesDefense Advanced Research Projects AgencyCadence Design SystemsMAFET ConsortiumConsortium for Superconducting ElectronicsNational Defense Science and Engineering Graduate FellowshipDigital Equipment CorporationMIT Lincoln LaboratorySemiconductor Research CorporationMultiuniversity Research IntiativeNational Science Foundatio
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