O(1)O(1) query time algorithm for all pairs shortest distances on permutation graphs

We present an algorithm for the all pairs shortest distance problem on permutation graphs. Given a permutation model for the graph on n   vertices, after O(n)O(n) preprocessing the algorithm will deliver answers to distance queries in O(1)O(1) time. In the EREW PRAM model, preprocessing can be accomplished in O(logn)O(logn) time with O(n)O(n) work. Where the distance between query vertices is k  , a path can be delivered in O(k)O(k) time. The method is based on reduction to bipartite permutation graphs, a further reduction to unit interval graphs, and a coordinatization of unit interval graphs.