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### Computing Minimum Spanning Trees with Uncertainty

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### Abstract

We consider the minimum spanning tree problem in a setting where
information about the edge weights of the given graph is uncertain.
Initially, for each edge \$e\$ of the graph only a set \$A_e\$, called
an uncertainty area, that contains the actual edge weight
\$w_e\$ is known. The algorithm can `update' \$e\$ to obtain the edge
weight \$w_e in A_e\$. The task is to output the edge set of a
minimum spanning tree after a minimum number of updates. An
algorithm is \$k\$-update competitive if it makes at most \$k\$ times
as many updates as the optimum. We present a \$2\$-update
competitive algorithm if all areas \$A_e\$ are open or trivial, which
is the best possible among deterministic algorithms. The condition
on the areas \$A_e\$ is to exclude degenerate inputs for which no
constant update competitive algorithm can exist.

Next, we consider a setting where the vertices of the graph
correspond to points in Euclidean space and the weight of an edge
is equal to the distance of its endpoints. The location of each
point is initially given as an uncertainty area, and an update
reveals the exact location of the point. We give a general
relation between the edge uncertainty and the vertex uncertainty
versions of a problem and use it to derive a \$4\$-update competitive
algorithm for the minimum spanning tree problem in the vertex
uncertainty model. Again, we show that this is best possible among
deterministic algorithms.

### BibTeX - Entry

```@InProceedings{hoffmann_et_al:LIPIcs:2008:1358,
author =	{Michael Hoffmann and Thomas Erlebach and Danny Krizanc and Mat{\'u}s Mihal'{\'a}k and Rajeev Raman},
title =	{{Computing Minimum Spanning Trees with Uncertainty}},
booktitle =	{25th International Symposium on Theoretical Aspects of Computer Science},
pages =	{277--288},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-06-4},
ISSN =	{1868-8969},
year =	{2008},
volume =	{1},
editor =	{Susanne Albers and Pascal Weil},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
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