# Synchronous Graph Coloring # 1. Lecture ## Key Concepts - Edges can communicate with each other - $\delta$ is the degree of a node ... ... @@ -7,4 +7,28 @@ - Usually generic fast algorithms need $\Delta+1$ colors - Trees always have $\Chi\leq1$ - Log* is *how many nested logs must be taken* - Six-2-Three Algorithm 3-colors a tree in $\mathcal{O}(Log^* n + k)$ \ No newline at end of file - Six-2-Three Algorithm 3-colors a tree in $\mathcal{O}(Log^* n + k)$ ## Language | Symbol | Meaning | | ---------- | --------------------------------------------------------------------------- | | $\delta$ | degree of a node | | $\Delta$ | maximum degree of any node in the tree | | $\Chi$ | Chromatic Number, i.e. the least number of colors needed to color the graph | ## Greedy Sequential # 2. Lecture - Radius: Max distance to another node - Diameter: Max radius of any node in a network ## Building a Tree It can be done with flood and echo, but due to the asynchronous nature of the algorithms maybe not the shortest paths from the root to the leafs is found It can be improved with Dijkstra, where a tree is built level by level, but this takes a lot of messages It can be further improved by Bellmann-Ford, which works like BGP, but there the algorithm does not terminate. ## 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