# Algorithms for updating minimum spanning trees datingworkshop org

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.

In another proof of the theorem stated differently you can also find the meaning of a spanning tree being "a local minimum", although it is not needed to understand that in order to understand this answer.Alternatively, suppose that it did use \$e'\$, then there is a path (without loss of generality) from \$u\$ to one endpoint of \$e'\$ and from the other endpoint of \$e'\$ to \$v\$.There is also a path from one endpoint of \$e'\$ to the other endpoint via \$e''\$ (around the cycle), all within \$T'\$.Armed with the above theorem, the readers are encouraged to construct a proof by themselves, which is probably easier than reading my rigorous proof below.Let us reuse all notations in OP's definition of the algorithm.

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