Abstract
We study the problem of maintaining a (1+ε)-factor approximation of the diameter of a stream of points under the sliding window model. In one dimension, we give a simple algorithm that only needs to store \(O(\frac{1}{\epsilon}{\rm log}R)\) points at any time, where the parameter R denotes the “spread” of the point set. This bound is optimal and improves Feigenbaum, Kannan, and Zhang’s recent solution by two logarithmic factors. We then extend our one-dimensional algorithm to higher constant dimensions and, at the same time, correct an error in the previous solution. In high nonconstant dimensions, we also observe a constant-factor approximation algorithm that requires sublinear space. Related optimization problems, such as the width, are also considered in the two-dimensional case.
Work done by the first author was supported by an NSERC Research Grant and a Premiere’s Research Excellence Award. This work has appeared as part of the second author’s Master’s thesis.
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Chan, T.M., Sadjad, B.S. (2004). Geometric Optimization Problems Over Sliding Windows. In: Fleischer, R., Trippen, G. (eds) Algorithms and Computation. ISAAC 2004. Lecture Notes in Computer Science, vol 3341. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30551-4_23
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DOI: https://doi.org/10.1007/978-3-540-30551-4_23
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