Better Bounds for Online Line Chasing

We study online competitive algorithms for the line chasing problem in Euclidean spaces ^d, where the input consists of an initial point P_0 and a sequence of lines X_1,X_2,...,X_m, revealed one at a time. At each step t, when the line X_t is revealed, the algorithm must determine a point P_t∈ X_t. An online algorithm is called c-competitive if for any input sequence the path P_0, P_1,...,P_m it computes has length at most c times the optimum path. The line chasing problem is a variant of a more general convex body chasing problem, where the sets X_t are arbitrary convex sets. To date, the best competitive ratio for the line chasing problem was 28.1, even in the plane. We significantly improve this bound, by providing a 3-competitive algorithm for any dimension d. We also improve the lower bound on the competitive ratio, from 1.412 to 1.5358.