NC Coloring Algorithms for Permutation Graphs

We show that the problem of coloring a permutation graph of size n can be solved in O(logn logk) time using O(kn2 / logk log2n) processors on the CREW PRAM model of computation, where 1 < k < n. We estimate the parameter k on random permutation graphs and show that the coloring problem can be solved in O(logn loglogn) time in the average-case on the CREW PRAM model of computation with O(n2) processors. Our computational strategy goes as follows: Given a permutation  or its corresponding permutation graph G[], we first construct a directed acyclic graph G*[] using certain combinatorial properties of , and then compute longest paths in the directed acyclic graph using divideand-conquer techniques. We show that the problem of coloring a permutation graph G[] is equivalent to finding longest paths in its acyclic digraph G*[]. The best-known parallel algorithms for the same problem run in O(log2n) time using O(n3 / logn) processors on the CREW PRAM model of computation. CR Classification: F.2.2, G.2.2