... ... @@ -31,4 +31,17 @@ It can be further improved by Bellmann-Ford, which works like BGP, but there the ## MST The GHS algorithm works as follows: Nodes are split into fragments with roots. The root then asks for the cheapest outgoing edge from the fragment (called blue edge). This edge then is added to the fragment and the fragments are merged. The node who found the cheapest edge becomes the new root and repeats the process until there are no other fragments left. \ No newline at end of file Nodes are split into fragments with roots. The root then asks for the cheapest outgoing edge from the fragment (called blue edge). This edge then is added to the fragment and the fragments are merged. The node who found the cheapest edge becomes the new root and repeats the process until there are no other fragments left. # 3. Lecture ## Cover Free Families A set of sets. If one set is chosen then it contains always an element that is not contained in the union of any other set. ## Livindals Algorithm Assign a set from the cover free sets to each of the nodes. Then if every node learns which set his neighbors have, it can then choose a color that is not in the union of the sets of his neighbors. This has complexity $\mathcal{O}(1)$, but is not very optimal yet. ## Color Reduction To further reduce the colors needed to $\Delta+1$, the colors are grouped into groups with size $2\Delta+2$. Then in every round the node with the highest color in the group chooses a color from the lower half of the group. Different groups do this in parallel, since they cannot choose the same color. If one iteration is complete, the remaining colors are regrouped again in the same way, but now there are only half as many groups. This is done until only one group is left.