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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Why do I consider my answer is better than other answers?