Abstract
Consider a line segment R consisting of n facilities. Each facility is a point on R and it needs to be assigned exactly one of the colors from a given palette of c colors. At an instant of time only the facilities of one particular color are ‘active’ and all other facilities are ‘dormant’. For the set of facilities of a particular color, we compute the one dimensional Voronoi diagram, and find the cell, i.e, a segment of maximum length. The users are assumed to be uniformly distributed over R and they travel to the nearest among the facilities of that particular color that is active. Our objective is to assign colors to the facilities in such a way that the length of the longest cell is minimized. We solve this optimization problem for various values of n and c. We propose an optimal coloring scheme for the number of facilities n being a multiple of c as well as for the general case where n is not a multiple of c. When n is a multiple of c, we compute an optimal scheme in Θ(n) time. For the general case, we propose a coloring scheme that returns the optimal in O(n 2logn) time.
This research is partially supported by NSERC and the Commonwealth Scholarship Program of DFAIT, Canada.
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© 2014 Springer International Publishing Switzerland
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Das, S., Maheshwari, A., Nandy, A., Smid, M. (2014). A Facility Coloring Problem in 1-D. In: Gu, Q., Hell, P., Yang, B. (eds) Algorithmic Aspects in Information and Management. AAIM 2014. Lecture Notes in Computer Science, vol 8546. Springer, Cham. https://doi.org/10.1007/978-3-319-07956-1_9
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DOI: https://doi.org/10.1007/978-3-319-07956-1_9
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-07955-4
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