Directed Shortest Paths via Approximate Cost Balancing

Approximate Shortest Descending Paths

We present an approximation algorithm for the shortest descending path problem. Given a source s and a destination t on a terrain, a shortest descending path from s to t is a path of minimum Euclidean length on the terrain subject to the constraint that the height decreases monotonically as we traverse that path from s to t. Given any ε ∈ (0, 1), our algorithm returns in O(n log(n/ε)) time a de...

Maximum st-Flow in Directed Planar Graphs via Shortest Paths

Minimum cuts have been closely related to shortest paths in planar graphs via planar duality – so long as the graphs are undirected. Even maximum flows are closely related to shortest paths for the same reason – so long as the source and the sink are on a common face. In this paper, we give a correspondence between maximum flows and shortest paths via duality in directed planar graphs with no c...

Approximate Shortest Homotopic Paths in Weighted Regions

Let P be a path between two points s and t in a polygonal subdivision T with obstacles and weighted regions. Given a relative error tolerance ε ∈ (0, 1), we present the first algorithm to compute a path between s and t that can be deformed to P without passing over any obstacle and the path cost is within a factor 1+ ε of the optimum. The running time is O( 3 ε2 knpolylog(k, n, 1 ε )), where k ...

Computing Approximate Shortest Paths on Convex Polytopes1

The algorithms for computing a shortest path on a polyhedral surface are slow, complicated, and numerically unstable. We have developed and implemented a robust and efficient algorithm for computing approximate shortest paths on a convex polyhedral surface. Given a convex polyhedral surface P in R3, two points s, t ∈ P , and a parameter ε > 0, it computes a path between s and t on P whose lengt...

Euclidean Shortest Paths - Exact or Approximate Algorithms