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

• Davide Bilò
• Luciano Gualà
• Guido Proietti
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6281)

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 of T w.r.t. S 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 connectivity needs to be promptly reestablished, 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 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. Such a problem has been recently solved in O(mn) time and linear space for arbitrary k, and in O(n 2 + mlogn) time and O(n 2) space for the special case k = 2. In this paper, we are interested to the prominent cases k = O(1) and k = n, which model realistic communication paradigms. For these cases, we present a linear space and $$\widetilde O(m)$$ time algorithm, and thus we improve both the above running times (but for quite dense graphs in the case k = 2, for which however it is noticeable we make use of only linear space). Moreover, we provide an accurate analysis showing that when k = n, the obtained swap tree is effective in terms of routing cost.

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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

• Davide Bilò
• 1
• Luciano Gualà
• 2
• Guido Proietti
• 1
• 3
1. 1.Dipartimento di InformaticaUniversità di L’AquilaL’AquilaItaly
2. 2.Dipartimento di MatematicaUniversità di Tor VergataRomaItaly
3. 3.Istituto di Analisi dei Sistemi ed Informatica, CNRRomaItaly

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