Independent tree spanners: fault-tolerant spanning trees with constant distance guarantees

For any xed rational parameter t 1, a (tree) t{spanner of a graph G is a spanning subgraph (tree) T in G such that the distance between every pair of vertices in T is at most t times their distance in G. General t{spanners and their variants have multiple applications in the eld of communication networks, distributed systems, and network design. In this paper, we combine the two concepts of simple structured, sparse t{spanners and fault-tolerance by examining independent tree t{spanners. Given a root vertex r, this is a pair of tree t{spanners, such that the two paths from any vertex to r are edge disjoint or internally vertex dijoint, respectively. It is shown that a pair of independent tree t{spanners can be found in linear time for t < 3, whereas the problem for arbitrary t 4 is NP{complete. As a less restrictive concept, we also treat tree t{root-spanners, where the distance constraint is relaxed. Here, we show that the problem of nding an independent pair of such subgraphs is NP{complete for all non-trivial, rational t. As a special case, we then consider direct tree t{root-spanners. These are tree t{root-spanners where paths from any vertex to the root have to be detour-free. In the edge independent case, a pair of these can be found in linear time for all rational t, whereas the vertex independent case remains NP{complete.