Computing simple paths from given points inside a polygon

Abstract

Abstract Given a set X of points inside a polygonal region P , two distinguished points s , t ∈ X , we study the problem of finding a simple polygonal path that uses only the points of X as vertices and lies in P , from s to t . We present an O ( n 2 + n log n log ( m n ) + m log m ) time and O ( n 2 + m ) space algorithm for computing a simple path from s to t or reporting no simple paths exist, where n is the number of points of X and m is the number of vertices of P . This gives a significant improvement upon the previously known O ( m 2 n 2 ) time and space algorithm, and O ( n 3 log m + m n ) time and O ( n 3 + m ) space algorithm. Our result is mainly obtained by showing that the visibility graph of X inside P , together with an implicit representation of the shortest paths from s to all other points of X , is sufficient to give a solution to the considered problem.