Bilò, Davide and Gualà, Luciano and Proietti, Guido (2013) *A Faster computation of all the best swap edges of a shortest paths tree.* In: *21. Annual European Symposium on Algorithms: ESA 2013; proceedings*, September 2-4, 2013, Sophia Antipolis, France. Heidelberg [etc.], Springer. p. 157-168. (Lecture notes in computer science, 8125/2013). ISSN 0302-9743. eISBN 978-3-642-40450-4. Conference or Workshop Item.

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DOI: 10.1007/978-3-642-40450-4_14

## Abstract

We consider a 2-edge connected, non-negatively weighted graph *G*, with *n* nodes and *m* edges, and a *single-source shortest paths tree* (SPT) of G rooted at an arbitrary node. If an edge of the SPT is temporarily removed, a widely recognized approach to reconnect the nodes 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 we instead rebuild 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 nodes. For the former criteria, we present an *O*(*m* logα(*m,n*)) time algorithm to find a best swap edge for *every* edge of the SPT, thus improving onto the previous *O*(*m* log*n*) time algorithm (B. Gfeller, *ESA’08*). 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 favorably with the *O*(*m*+*n*α(*n,n*)log^{2} *n*) 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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