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Algorithmica

, Volume 40, Issue 2, pp 119–132 | Cite as

Nearly Linear Time Minimum Spanning Tree Maintenance for Transient Node Failures

  • Enrico NardelliEmail author
  • Guido ProiettiEmail author
  • Peter WidmayerEmail author
Article

Abstract

Given a 2-node connected, real weighted, and undirected graph $G=(V,E)$, with $n$ nodes and $m$ edges, and given a minimum spanning tree (MST) $T=(V,E_T)$ of $G$, we study the problem of finding, for every node $v \in V$, a set of replacement edges which can be used for constructing an MST of $G-v$ (i.e., the graph $G$ deprived of $v$ and all its incident edges). We show that this problem can be solved on a pointer machine in ${\cal O}(m \cdot \alpha(m,n))$ time and ${\cal O}(m)$ space, where $\alpha$ is the functional inverse of Ackermann’s function. Our solution improves over the previously best known ${\cal O}(\min\{m \cdot \alpha(n,n), m + n \log n\})$ time bound, and allows us to close the gap existing with the fastest solution for the edge-removal version of the problem (i.e., that of finding, for every edge $e \in E_T$, a replacement edge which can be used for constructing an MST of $G-e=(V,E \backslash \{e\})$). Our algorithm finds immediate application in maintaining MST-based communication networks undergoing temporary node failures. Moreover, in a distributed environment in which nodes are managed by selfish agents, it can be used to design an efficient, truthful mechanism for building an MST.

Graph algorithms Minimum spanning tree Transient node failures Fault tolerance Algorithmic mechanism design 

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Copyright information

© Springer-Verlag 2004

Authors and Affiliations

  1. 1.Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy and Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti”, CNR, Viale Manzoni 30, 00185 RomaItaly
  2. 2.Dipartimento di Informatica, Università di L’Aquila, Via Vetoio, 67010 L’Aquila, Italy and Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti”, CNR, Viale Manzoni 30, 00185 RomaItaly
  3. 3.Institut für Theoretische Informatik, ETH Zentrum, CLWC 2, Clausiusstrasse 49, 8092 ZürichSwitzerland

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