A truthful mechanism for the non-utilitarian minimum radius spanning tree problem
Proceedings of the seventeenth annual ACM symposium on Parallelism in algorithms and architectures
A truthful (2 - 2/k)-approximation mechanism for the steiner tree problem with k terminals*
COCOON'05 Proceedings of the 11th annual international conference on Computing and Combinatorics
Efficient truthful mechanisms for the single-source shortest paths tree problem
Euro-Par'05 Proceedings of the 11th international Euro-Par conference on Parallel Processing
Distributed computation of all node replacements of a minimum spanning tree
Euro-Par'07 Proceedings of the 13th international Euro-Par conference on Parallel Processing
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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.