Dispersion on Trees
Abstract
In the $k$-dispersion problem, we need to select $k$ nodes of a given graph so as to maximize the minimum distance between any two chosen nodes. This can be seen as a generalization of the independent set problem, where the goal is to select nodes so that the minimum distance is larger than 1. We design an optimal $O(n)$ time algorithm for the dispersion problem on trees consisting of $n$ nodes, thus improving the previous $O(n\log n)$ time solution from 1997. We also consider the weighted case, where the goal is to choose a set of nodes of total weight at least $W$. We present an $O(n\log^2n)$ algorithm improving the previous $O(n\log^4 n)$ solution. Our solution builds on the search version (where we know the minimum distance $\lambda$ between the chosen nodes) for which we present tight $\Theta(n\log n)$ upper and lower bounds.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2017
- DOI:
- 10.48550/arXiv.1706.09185
- arXiv:
- arXiv:1706.09185
- Bibcode:
- 2017arXiv170609185G
- Keywords:
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- Computer Science - Data Structures and Algorithms