Tree Spanners

A tree $t$-spanner $T$ of a graph $G$ is a spanning tree in which the distance between every pair of vertices is at most $t$ times their distance in $G$. This notion is motivated by applications in communication networks, distributed systems, and network design. This paper studies graph-theoretic, algorithmic, and complexity issues about tree spanners. It is shown that a tree 1-spanner, if it exists, in a weighted graph with $m$ edges and $n$ vertices is a minimum spanning tree and can be found in $O(m \log \beta(m, n))$ time, where $\beta(m, n) = \min\{i\mid\log^{(i)}n \leq m/n\}$. On the other hand, for any fixed $t > 1$, the problem of determining the existence of a tree $t$-spanner in a weighted graph is proven to be NP-complete. For unweighted graphs, it is shown that constructing a tree 2-spanner takes linear time, whereas determining the existence of a tree $t$-spanner is NP-complete for any fixed $t \geq 4$. A theorem that captures the structure of tree 2-spanners is presented for unweighted graphs. For digraphs, an $O((m + n)\alpha(m, n))$ algorithm is provided for finding a tree $t$-spanner with $t$ as small as possible, where $\alpha(m, n)$ is a functional inverse of Ackerman's function. The results for tree spanners on undirected graphs are extended to "quasi-tree spanners" on digraphs. Furthermore, linear-time algorithms are derived for verifying tree spanners and quasi-tree spanners.