Approximating shortest path for the skew lines problem in time doubly logarithmic in 1/epsilon

• Authors: D. Burago;D. Grigoriev;A. Slissenko
• Venue: Theoretical Computer Science - Algebraic and numerical algorithm
• Year: 2004

We consider two three-dimensional situations when a polytime algorithm for approximating a shortest path can be constructed. The main part of the paper treats a well-known problem of constructing a shortest path touching lines in R3: given a list of straight lines L = (L1,..., Ln) in R3 and two points s and t, find a shortest path that, starting from s, touches the lines Li in the given order and ends at t. We remark that such a shortest path is unique. We show that it can be length--position ε-approximated (i.e. both its length and its position can be found approximately) in time (Rn/dα)16 + O(n2 log log 1/ε), where d is the minimal distance between consecutive lines of L, α˜ is the minimum of sines of angles between consecutive lines, and R is the radius of a ball where the initial approximation can be placed (such a radius can be easily computed from the initial data).As computational model we take real RAM extended by square and cubic roots extraction. This problem of constructing a shortest path touching lines is known for quite some time to be a challenging problem. The existing methods for approximating shortest paths based on adding Steiner points which form a grid and subsequently applying Dijkstra's algorithm for finding a shortest path in the grid, provide a complexity bound which depends polynomially on 1/ε, while our algorithm for the problem under consideration has complexity linear in log log 1/ε. Our algorithm is motivated by the observation that the shortest path in question is a geodesic in a certain length space of non-positive curvature (in the sense of A.D. Alexandrov), and it relies on the (elementary) theory of CAT(0)-spaces.In the second part of the paper we analyze very simple grid approximations. We assume that a parameter a 0 describing separability of obstacles is given and the part of a grid with mesh size a outside the obstacles is built (for semi-algebraic obstacles all these precalculations are polytime). We show that there is an algorithm of time complexity O((1/a)6) which, given a-separated obstacles in a unit cube, finds a path (between given vertices s and t of the grid) whose length is bounded from above by (84π* + 96a), where π* is the length of a shortest path. On the other hand, as we show by an example, one cannot approximate the length of a shortest path better than 7π* if one uses only grid polygons (constructed only from grid edges). For semi-algebraic obstacles our computational model is bitwise. For a general type of obstacles the model is bitwise modulo constructing the part of the grid admissible for our paths. Observe that the existing methods for approximating shortest paths are not directly applicable for semi-algebraic obstacles since they usually place the Steiner points forming a grid on the edges of polyhedral obstacles.