Elsevier

Discrete Optimization

Volume 7, Issues 1–2, February–May 2010, Pages 21-31
Discrete Optimization

Computing Steiner points for gradient-constrained minimum networks

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Abstract

Let Tg be a gradient-constrained minimum network, that is, a minimum length network spanning a given point set in 3-dimensional space with edges that are constrained to have gradients no more than an upper bound m. Such networks occur in underground mines where the slope of the declines (tunnels) cannot be too steep due to haulage constraints. Typically the gradient is less than 1/7. By defining a new metric, the gradient metric, the problem of finding Tg can be approached as an unconstrained problem where embedded edges can be considered as straight but measured according to their gradients. All edges in Tg are labelled by their gradients, being <m,=m or >m, in the gradient metric space. Computing Steiner points plays a central role in constructing locally minimum networks, where the topology is fixed. A degree-3 Steiner point is labelled minimal if the total length of the three adjacent edges is minimized for a given labelling. In this paper we derive the formulae for computing labelled minimal Steiner points. Then we develop an algorithm for computing locally minimal Steiner points based on information from the labellings of adjacent edges. We have tested this algorithm on uniformly distributed sets of points; our results help in finding gradient-constrained minimum networks.

Keywords

Gradient constrained
Steiner tree
Underground mining

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NICTA is funded by the Australian Government as represented by the Department of Broadband, Communications and the Digital Economy and the Australian Research Council through the ICT Centre of Excellence program.