Independent Tree Spanners

Fault-Tolerant Spanning Trees with Constant Distance Guarantees (Extended Abstract)
  • Dagmar Handke
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1517)


For any fixed parameter t ≥ 1, a tree t-spanner of a graph G is a spanning tree T of G such that the distance between every pair of vertices in T is at most t times their distance in G. In this paper, we incorporate a concept of 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 (resp., internally vertex) disjoint. It is shown that a pair of independent tree 2-spanners can be found in linear time, whereas the problem for arbitrary t ≥ 4 is \(\cal NP\)-complete.

As a less restrictive concept, we treat tree t-root-spanners, where the distance constraint is relaxed. Here, we show that the problem of finding an independent pair of such subgraphs is \(\cal NP\)-complete for all 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 t, whereas the vertex independent case remains \(\cal NP\)-complete.


Span Tree Parent Level Truth Assignment Span Subgraph Root Vertex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Annexstein, F.S., Berman, K.A., Swaminathan, R.: Independent spanning trees with small stretch factors. Technical Report 96-13, DIMACS (June 1996)Google Scholar
  2. 2.
    Annexstein, F.S., Berman, K.A., Swaminathan, R.: On computing nearly optimal multi-tree paths and s, t-numberings. In: Krizanc, D., Widmayer, P. (eds.) Proc. 4th Int. Colloquium on Structural Information and Communication Complexity, SIROCCO 1997, pp. 12–23. Carleton Scientific, Ottawa (1998)Google Scholar
  3. 3.
    Cai, L., Corneil, D.: Tree spanners. SIAM J. Discrete Math. 8(3), 359–387 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W H Freeman & Co. Ltd, New York (1979)zbMATHGoogle Scholar
  5. 5.
    Handke, D.: Independent tree spanners: Fault-tolerant spanning trees with distance guarantees (1998) (submitted)Google Scholar
  6. 6.
    Huck, A.: Independent trees in graphs. Graphs and Combinatorics 10, 29–45 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Itai, A., Rodeh, M.: The multi-tree approach to reliability in distributed networks. Information and Computation 79(l), 43–59 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Khuller, S., Schieber, B.: On independent spanning trees. Information Processing Letters 42, 321–323 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Levcopoulos, C., Narasimhan, G., Smid, M.: Efficient algorithms for constructing fault-tolerant geometric spanners. Preprint 16, Fakultät für Informatik, Otto-von-Guericke-Universität Magdeburg, Germany (1997)Google Scholar
  10. 10.
    Peleg, D., Ullman, J.D.: An optimal synchronizer for the hypercube. In: Proc. 6th ACM Symposium on Principles of Dist. Comp., Vancouver, pp. 77–85 (1987)Google Scholar
  11. 11.
    Santoro, N., Khatib, R.: Labelling and implicit routing in networks. The Computer Journal 28(1), 5–8 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Soares, J.: Graph spanners: a survey. Congressus Numerantium 89, 225–238 (1992)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Zehavi, A., Itai, A.: Three tree-paths. J. of Graph Theory 13, 175–188 (1989)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Dagmar Handke
    • 1
  1. 1.Fakultät für Mathematik und InformatikUniversität KonstanzKonstanzGermany

Personalised recommendations