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要旨:
In this paper we consider the problem of computing minimum-cost spanning trees
with depth restrictions. Specifically, we are given an $n$-node complete graph
$G$, a metric cost-function $c$ on its edges, and an integer $k\geq 1$. The
goal in the minimum-cost $k$-hop spanning tree problem is to compute a spanning
tree $T$ in $G$ of minimum total cost such that the longest root-leaf-path in
the tree has at most $k$ edges.
Our main result is an algorithm that computes a tree of depth at most $k$ and
total expected cost $O(\log n)$ times that of a minimum-cost $k$-hop
spanning-tree. The result is based upon earlier work on metric space
approximation due to Fakcharoenphol et al.[FRT03] and Bartal [Bart96,Bart98].
In particular we show that the problem can be solved exactly in polynomial time
when the cost metric $c$ is induced by a so-called hierarchically
well-separated tree.
