The content introduces the central spanning tree (CST) problem, a novel parameterized family of spanning trees that aims to balance data fidelity and geometric robustness. The key highlights are:
The CST problem is defined as the spanning tree that minimizes the sum of edge costs weighted by their "edge betweenness centrality", controlled by a parameter α. This subsumes previous definitions like the minimum spanning tree and minimum routing cost tree as special cases.
The authors also introduce the branched central spanning tree (BCST) problem, which allows for the addition of Steiner points to further optimize the tree structure.
Theoretical analysis shows that as α approaches infinity or the number of terminals approaches infinity with α > 1, the optimal CST/BCST converges to a star-shaped tree, which may not be desirable for modeling data structure. Conversely, as α approaches negative infinity, the optimal CST/BCST tends towards a path graph.
Empirical results demonstrate that the CST and BCST with intermediate α values exhibit greater robustness to noise compared to the minimum spanning tree and Steiner tree, while still preserving the overall data structure.
The authors propose a heuristic algorithm to efficiently approximate the optimal BCST solution by exploiting the correspondence between feasible BCST and CST topologies.
Further analysis is provided on the geometry of optimal BCST solutions, showing that degree-4 Steiner points are infeasible in the plane for α ∈ [0, 0.5] ∪ {1}.
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by Enri... lúc arxiv.org 04-10-2024
https://arxiv.org/pdf/2404.06447.pdfYêu cầu sâu hơn