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Distributed Algorithms for Graph Coloring: Key Results and Insights

This paper discusses notable results in distributed algorithms for graph coloring. It establishes that any tree can be efficiently colored using only two colors, while trees with a maximum node degree of Δ can be colored with (Δ + 1) colors. The paper also touches on planar graphs, which can be colored with four colors, though no efficient distributed algorithm exists. Highlighted is the O(log* n) tree coloring algorithm by Cole and Vishkin, demonstrating its efficiency based on logarithmic operations, showcasing the slow growth of the log* function for large n.

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Distributed Algorithms for Graph Coloring: Key Results and Insights

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  1. Distributed Algorithms forGraph coloring

  2. A few known results • Any tree can be colored using two colors only • Any graph whose maximum node degree is ∆ can be colored using (∆+1) colors • Any planar graph can be colored using four colors, but no distributed algorithm is known and the centralized algorithm is also extremely cumbersome. • Any tree of size n can be colored using three colors in log*(n) rounds (Cole and Vishkin)

  3. O(log*n) coloring of a treeCole & Vishkin’s algorithm Log*(n) is the smallest number of log-operations needed to bring n down to ≤2. For example, let n = one trillion. Now, log (one trillion) = 40, log (log (one trillion)) = 5.322, and log (log (log (log (one trillion)))) < 2. This means that log* (one trillion) = 4. This also illustrates that the function grows very slowly with the value of the argument.

  4. O(log*n) coloring of a tree Consider a rooted tree. The algorithm assumes that initially the color of each node is its id. Each non-root node v is aware of its parent p(v) . Interpret each color c as a little-endian bit string ck-1ck-2 ck-3 …c0, and let |c| denote the size of the bit string.

  5. O(log*n) coloring of a tree

  6. O(log*n) coloring of a tree

  7. O(log*n) coloring of a tree

  8. O(log*n) coloring of a tree

  9. O(log*n) coloring of a tree

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