Spanner for the \(0/1/\infty\) weighted region problem

We consider the problem of computing an approximate weighted shortest path in a weighted subdivision, with weights assigned from the set \(\{0, 1, \infty\}\). We present a data structure \(B\), which stores a set of convex, non-overlapping regions. These include zero-cost regions (0-regions) with a weight of \(0\) and obstacles with a weight of \(\infty\), all embedded in a plane with a weight of \(1\). The data structure \(B\) can be constructed in expected time \(O(N + (n/\varepsilon^3)(\log(n/\varepsilon) + \log N))\), where \(n\) is the total number of regions, \(N\) represents the total complexity of the regions, and \(1 + \varepsilon\) is the approximation factor, for any \(0 < \varepsilon < 1\). Using \(B\), one can compute an approximate weighted shortest path from any point \(s\) to any point \(t\) in \(O(N + n/\varepsilon^3 + (n/\varepsilon^2) \log(n/\varepsilon) + (\log N)/\varepsilon)\) time. In the special case where the 0-regions and obstacles are polygons (not necessarily convex), \(B\) contains a \((1 + \varepsilon)\)-spanner of the input vertices.