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An Improved Algorithm for Computing All the Best Swap Edges of a Tree Spanner

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Abstract

A tree \(\sigma \)-spanner of a positively real-weighted n-vertex and m-edge undirected graph G is a spanning tree T of G which approximately preserves (i.e., up to a multiplicative stretch factor \(\sigma \)) distances in G. Tree spanners with provably good stretch factors find applications in communication networks, distributed systems, and network design. However, finding an optimal or even a good tree spanner is a very hard computational task. Thus, if one has to face a transient edge failure in T, the overall effort that has to be afforded to rebuild a new tree spanner (i.e., computational costs, set-up of new links, updating of the routing tables, etc.) can be rather prohibitive. To circumvent this drawback, an effective alternative is that of associating with each tree edge a best possible (in terms of resulting stretch) swap edge—a well-established approach in the literature for several other tree topologies. Correspondingly, the problem of computing all the best swap edges of a tree spanner is a challenging algorithmic problem, since solving it efficiently means to exploit the structure of shortest paths not only in G, but also in all the scenarios in which an edge of T has failed. For this problem we provide a very efficient solution, running in \(O(n^2 \log ^4 n)\) time, which drastically improves (almost by a quadratic factor in n in dense graphs) on the previous known best result.

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Notes

  1. While it is natural to think that T optimizes \(\phi \), we actually allow T to be any spanning tree of G.

  2. A formal definition of critical edge will be given in Sect. 3.

  3. More precisely, the above definition allows us to relate critical edges to the upper envelopes of suitable sets of linear functions, as we will discuss in Sect. 4.

  4. The authors of [8] design a dynamic data structure to maintain the convex hull of a set of points in the plane. Then, they explain how point-line duality can be used to convert such a structure into one maintaining the upper envelope of a set of linear functions.

  5. Notice that each key \((\varPsi , \varLambda )\) can be uniquely represented by a pair of integers. This follows from the fact that the subtree \(\varPsi \) (resp. \(\varLambda \)) of T is one of the O(n) vertices of \(\mathcal {T}\).

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Authors and Affiliations

  1. Università di Sassari, Sassari, Italy

    Davide Bilò

  2. Gran Sasso Science Institute, L’Aquila, Italy

    Feliciano Colella

  3. Università di Roma “Tor Vergata”, Rome, Italy

    Luciano Gualà

  4. ETH Zürich, Zürich, Switzerland

    Stefano Leucci

  5. Università degli Studi dell’Aquila, L’Aquila, Italy

    Guido Proietti

  6. Istituto di Analisi dei Sistemi ed Informatica, CNR, Rome, Italy

    Guido Proietti

Authors
  1. Davide Bilò
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  2. Feliciano Colella
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  3. Luciano Gualà
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  4. Stefano Leucci
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  5. Guido Proietti
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Correspondence to Guido Proietti.

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Davide Bilò: This work has been partially supported by the 2016 Research Grant funded by “Fondazione di Sardegna”

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Bilò, D., Colella, F., Gualà, L. et al. An Improved Algorithm for Computing All the Best Swap Edges of a Tree Spanner. Algorithmica 82, 279–299 (2020). https://doi.org/10.1007/s00453-019-00549-w

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