On Approximating Polygonal Curves in Two and Three Dimensions

Abstract Given a polygonal curve P =[ p 1 , p 2 , . . . , p n ], the polygonal approximation problem considered calls for determining a new curve P ′ = [ p ′ 1 , p ′ 2 , . . . , p ′ m ] such that (i) m is significantly smaller than n , (ii) the vertices of P ′ are an ordered subset of the vertices of P , and (iii) any line segment [ p ′ A , p ′ A + 1 of P ′ that substitutes a chain [ p B , . . . , p C ] in P is such that for all i where B ≤ i ≤ C , the approximation error of p i with respect to [ p ′ A , p ′ A + 1 ], according to some specified criterion and metric, is less than a predetermined error tolerance. Using the parallel-strip error criterion, we study the following problems for a curve P in R d , where d = 2, 3: (i) minimize m for a given error tolerance and (ii) given m , find the curve P ′ that has the minimum approximation error over all curves that have at most m vertices. These problems are called the min-# and min-ϵ problems, respectively. For R 2 and with any one of the L 1 , L 2 , or L ∞ distance metrics, we give algorithms to solve the min-# problem in O ( n 2 ) time and the min-ϵ problem in O ( n 2 log n ) time, improving the best known algorithms to date by a factor of log n . When P is a polygonal curve in R 3 that is strictly monotone with respect to one of the three axes, we show that if the L 1 and L ∞ metrics are used then the min-# problem can be solved in O ( n 2 ) time and the min-ϵ problem can be solved in O ( n 3 ) time. If distances are computed using the L 2 metric then the min-# and min-ϵ problems can be solved in O ( n 3 ) and O ( n 3 log n ) time, respectively. All of our algorithms exhibit O ( n 2 ) space complexity. Finally, we show that if it is not essential to minimize m , simple modifications of our algorithms afford a reduction by a factor of n for both time and space.