Approximating the Minimum Spanning Tree of Set of Points in the Hausdorff Metric

We study the problem of approximating MST(P ), the Euclidean minimum spanning tree of a set P of n points in [0, 1], by a spanning tree of some subset Q ⊂ P . We show that if the weight of MST(P ) is to be approximated, then in general Q must be large. If the shape of MST(P ) is to be approximated, then this is always possible with a small Q. More specifically, for any 0 < ε < 1 we prove: (i) There are sets P ⊂ [0, 1] of arbitrarily large size n with the property that any subset Q′ ⊂ P that admits a spanning tree T ′ with ∣ ∣|T ′| − |MST(P )| ∣ ∣ < ε · |MST(P )| must have size at least Ω(n1−1/d). (Here |T | denotes the weight, i.e. the sum of the edge lengths of tree T .) (ii) For any P ⊂ [0, 1] of size n there exists a subset Q ⊆ P of size O(1/ε) that admits a spanning tree T that is ε-close to MST(P ) in terms of Hausdorff distance (which measures shape dissimilarity). (iii) This setQ and this spanning tree T can be computed in time O(τd(n) + 1/ε d log(1/ε)) for any fixed dimension d. Here τd(n) denotes the time necessary to compute the minimum spanning tree of n points in R, which is known to be O(n log n) for d = 2, O((n log n)) for d = 3, and O(n2−2/(⌈d/2⌉+1)+φ), with φ > 0 arbitrarily small, for d > 3 (see [1]). All the results hold not only for the Euclidean metric L2 but also for any Lp metric with 1 ≤ p ≤ ∞ as underlying metric.