Some Optimal Parallel Algorithms for Shortest Path Related Problems on Interval and Circular-arc Graphs

In this paper, we consider some shortest path related problems on interval and circular-arc graphs. For the all-pair shortest path query problem on interval and circular-arc graphs, instead of using the sophisticated technique, we propose simple parallel algorithms using only the parallel prefix and suffix computations and the Euler tour technique. Our preprocessing algorithms run in O(log n) time using O(n/ log n) processors. Using the data structure constructed by our preprocessing algorithms, a query of the length of a shortest path between any two vertices can be answered in constant time by using a single processor. For the hinge vertex problem on interval graphs, we propose an O(log n) time algorithm using O(n/ log n) processors. It leads to a linear time sequential algorithm. Our algorithms are on the EREW PRAM model.