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Finding Best Swap Edges Minimizing the Routing Cost of a Spanning Tree

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Abstract

Given an n-node, undirected and 2-edge-connected graph G=(V,E) with positive real weights on its m edges, given a set of k source nodes S⊆V, and given a spanning tree T of G, the routing cost from S of T is the sum of the distances in T from every source s∈S to all the other nodes of G. If an edge e of T undergoes a transient failure, and one needs to promptly reestablish the connectivity, then to reduce set-up and rerouting costs it makes sense to temporarily replace e by means of a swap edge, i.e., an edge in G reconnecting the two subtrees of T induced by the removal of e. Then, a best swap edge for e is a swap edge which minimizes the routing cost from S of the tree obtained after the swapping. As a natural extension, the all-best swap edges problem is that of finding a best swap edge for every edge of T, and this has been recently solved in O(mn) time and linear space. In this paper, we focus our attention on the relevant cases in which k=O(1) and k=n, which model realistic communication paradigms. For these cases, we improve the above result by presenting an \(\widetilde{O}(m)\) time and linear space algorithm. Moreover, for the case k=n, we also provide an accurate analysis showing that the obtained swap tree is effective in terms of routing cost. Indeed, if the input tree T has a routing cost from V which is a constant-factor away from that of a minimum routing-cost spanning tree (whose computation is a problem known to be in APX), and if in addition nodes in T enjoys a suitable distance stretching property from a tree centroid (which can be constructively induced, as we show), then the tree obtained after the swapping has a routing cost from V which is still a constant-ratio approximation of that of a new (i.e., in the graph deprived of the failed edge) minimum routing-cost spanning tree.

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Algorithm 1

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Notes

  1. In the same paper, the authors define also several other swap criteria.

  2. Notice that, to be more precise, our improvement holds for \(k 2^{O (\alpha(2m)^{k}\log \alpha(m) )}=o(n/\log^{2} n)\), which includes k=O(1).

  3. Recall that for ℓ,t∈ℕ+, a sequence U=(u 1,…,u p ) of integers u i ∈[1,ℓ] is a Davenport-Schinzel sequence of order t with ℓ integers, if \(u_{i} \not= u_{i+1}, \forall i < p\), and there exists no subsequence \((u_{i_{1}},\ldots, u_{i_{t+2}})\) in U having two alternating values. If DS(ℓ,t) denotes the set of all Davenport-Schinzel sequences of order t with ℓ integers, then the complexity of DS(ℓ,t) is defined as the length of a longest sequence in DS(ℓ,t).

  4. Notice that, if the same M is chosen for all functions, then the number of inversion points between pairs of functions remains unchanged.

  5. For the case S=V, the second kind of potential inversion points are related to the two zeros of the function \(\Delta^{V}_{\chi(f),\chi(g)}\) seen as a quadratic function on the variable |T Y | with domain ℝ.

  6. Observe that for k constant, the time complexity is \(\widetilde{O}(m)\).

  7. Recall that a tree has at most 2 centroids.

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Acknowledgements

We wish to thank the anonymous reviewers for their helpful suggestions.

This work was partially supported by the PRIN 2008 research project COGENT (COmputational and GamE-theoretic aspects of uncoordinated NeTworks), funded by the Italian Ministry of Education, University, and Research.

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Authors and Affiliations

  1. Dipartimento di Informatica, Università di L’Aquila, 67010, L’Aquila, Italy

    Davide Bilò & Guido Proietti

  2. Dipartimento di Matematica, Università di Tor Vergata, 00133, Roma, Italy

    Luciano Gualà

  3. Istituto di Analisi dei Sistemi ed Informatica, CNR, 00185, Roma, Italy

    Guido Proietti

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  1. Davide Bilò
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Correspondence to Guido Proietti.

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Bilò, D., Gualà, L. & Proietti, G. Finding Best Swap Edges Minimizing the Routing Cost of a Spanning Tree. Algorithmica 68, 337–357 (2014). https://doi.org/10.1007/s00453-012-9674-y

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