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A Faster Computation of All the Best Swap Edges of a Shortest Paths Tree

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Abstract

We consider a two-edge connected, non-negatively real-weighted graph G with n vertices and m edges, and a single-source shortest paths tree (SPT) of G rooted at an arbitrary vertex. If an edge of the SPT is temporarily removed, a widely recognized approach to reconnect the vertices disconnected from the root consists of joining the two resulting subtrees by means of a single non-tree edge, called a swap edge. This allows to reduce consistently the set-up and computational costs which are incurred if one instead rebuilds a new optimal SPT from scratch. In the past, several optimality criteria have been considered to select a best possible swap edge, and here we restrict our attention to arguably the two most significant measures: the minimization of either the maximum or the average distance between the root and the disconnected vertices. For the former criteria, we present an \(O(m \log \alpha (m,n))\) time algorithm—where \(\alpha \) is the inverse of the Ackermann function—to find a best swap edge for every edge of the SPT, thus improving onto the previous \(O(m \log n)\) time algorithm. Concerning the latter criteria, we provide an \(O(m+n \log n)\) time algorithm for the special but important case where G is unweighted, which compares favourably with the \(O\left( m+n \, \alpha (n,n)\log ^2n\right) \) time bound that one would get by using the fastest algorithm known for the weighted case—once this is suitably adapted to the unweighted case.

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Notes

  1. Notice that for the MST, edge weights can also be negative, and the swap tree is actually an MST of the graph deprived of the failed edge.

  2. Actually, in [6] the authors claim an \(O(m \log ^2 n)\) time bound, but this must be augmented by an \(O(\alpha (m,m))\) factor, as pointed out in [2].

  3. Indeed, we can perform a sensitivity analysis of an MST as follows: (i) we first make the graph \(G\) non-negatively weighted, by summing up to all the edge weights the absolute value of the lightest edge, then (ii) we root the MST at any arbitrary vertex, hence (iii) we set the weight of every edge of the MST to 0, and finally (iv) we solve the ABSE problem w.r.t. the maximum distance from the root. Clearly, for every edge \(e\) of the MST, the best swap edge computed by the algorithm is an edge of minimum weight cycling with \(e\).

  4. Indeed, we can arbitrarily number the tree vertices with different numbers, and then choose, among all paths of equal length starting at \(v'\), the one minimizing the number associated with its endpoint different from \(v'\).

  5. Observe that, if \(z=c_x\), then \(\mathcal{{G}}(x,z)=V(T_{c_x})\); if \(z\ne c_x\), then \(\mathcal{{G}}(x,z)=V(T_{z})\setminus V(T_{v})\), where \(v\) is the (unique) child of \(z\) in \(P(x,c_x)\).

  6. Notice that, from a topological point of view, the \(\mathtt{split}({\mathcal{{S}}},v)\) operation can be viewed as the removal of edge \((r',v)\) from \(T'\).

  7. Observe that a decreasekey operation on \({\mathcal{{S}}_1}\) modifies one group while a split operation on \({\mathcal{{S}}_1}\) splits one group into two groups, i.e., it creates two new groups.

  8. More precisely, each creation, insertion, merge, and findmin operation requires only constant time.

  9. Notice that all these Fibonacci heaps can be inherited in \(O(1)\) time by simply changing their reference from \(q_1\) to \(x\).

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Acknowledgments

This work was partially supported by the Research Grant PRIN 2010 “ARS TechnoMedia” (Algorithms for Techno-Mediated Social Networks), funded by the Italian Ministry of Education, University, and Research.

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Authors and Affiliations

  1. Dipartimento di Scienze Umanistiche e Sociali, University of Sassari, Sassari, Italy

    Davide Bilò

  2. Dipartimento di Ingegneria dell’Impresa, University of Rome “Tor Vergata”, Rome, Italy

    Luciano Gualà

  3. Dip. di Ingegneria e Scienze dell’Informazione e Matematica, Univ. of L’Aquila, L’Aquila, Italy

    Guido Proietti

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

    Guido Proietti

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  1. Davide Bilò
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  2. Luciano Gualà
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  3. Guido Proietti
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Correspondence to Guido Proietti.

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Bilò, D., Gualà, L. & Proietti, G. A Faster Computation of All the Best Swap Edges of a Shortest Paths Tree. Algorithmica 73, 547–570 (2015). https://doi.org/10.1007/s00453-014-9912-6

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  • DOI: https://doi.org/10.1007/s00453-014-9912-6

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