We introduce the problem of finding a spanning tree along with a partition of the tree edges into the fewest number of feasible sets, where constraints on the edges define feasibility. The motivation comes from wireless networking, where we seek to model the irregularities seen in actual wireless environments. Not all node pairs may be able to communicate, even if geographically close—thus, the available pairs are specified with a link graph G= (V, E). Also, signal attenuation need not follow a nice geometric formula—hence, interference is modeled by a conflict (hyper)graph C= (E, F) on the links. The objective is to maximize the efficiency of the communication, or equivalently, to minimize the length of a schedule of the tree edges in the form of a coloring. We find that in spite of all this generality, the problem can be approximated linearly in terms of a versatile parameter, the inductive independence of the conflict graph. Specifically, we give a simple algorithm that attains a O(ρlog n) -approximation, where n is the number of nodes and ρ is the inductive independence. For an extension to Steiner trees, modeling multicasting, we obtain a O(ρlog 2n) -approximation. We also consider a natural geometric setting when only links longer than a threshold can be unavailable, and analyze the performance of a geometric minimum spanning tree.
Bibliographical noteFunding Information:
Funding was provided by Icelandic Centre for Research (Grant No. 174484-051).
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
- Network Design
- Wireless interference