On approximating polygonal curves in two and three dimensions

Given a polygonal curve P = [P₁, P₂ …, 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 [PB, …, PC] in P is such that for all i where B ≤ i ≤ C, the approximation error of Pi 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 R² and with any one of the L₁, L₂, or L_∞ distance metrics, we give algorithms to solve the min-# problem in O (n²) time and the min-ε problem in O (n² log n) time, improving the best known algorithms to date by a factor of log n. When P is a polygonal curve in R³ that is strictly monotone with respect to one of the three axes, we show that if the L₁ and L_∞ metrics are used then the min-# problem can be solved in O(n²) time and the min-ε problem can be solved in O(n³) time. If distances are computed using the L₂ metric then the min-# and min-ε problems can be solved in O(n³) and O (n³ log n) time, respectively. All of our algorithms exhibit O(n²) 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.