AND CHRISTOSH. PAPADIMITRIOU b'ni~'er.sit] of Cul@wu at San Dw,w, La Jollu, Ca[lfomu Abstract. The problem of determining shortest paths through a weighted planar polygonal subdivision with n vertices 1s considered. Distances are measured according to a weighted Euchdean metric: The length of a path 1sdefined to be the weighted sum of (Euchdean) lengths of the subpaths wlthm each region. An algorithm that constructs a (restricted) "shortest path map" with respect to a glt en source point N presented. The output is a partitioning of each edge of the subdlvmon mto intervals of c-ophmahty, allowing an~-optimal path to be traced from the source to any query point along any edge. The algorithm runs m worst-case time O(ES) and requmes O(E) space. where E is the number of "events" in the algorithm and S is the time it takes to run a numencal search procedure. In the worst ewe. E is bounded above by O(nq) (and we gwe an Q(ni) lower bound). but It 1s hkel} that h" will be much smaller m practice. We also show that S 1sbounded by 0(n4L ), where L 1s the preclslon of the problem instance (including the number of bits in the user-specified tolerance c). Agdm, the \alue of S should be much smaller m practice. The algorithm applies the "continuous DIJ kstra" paradigm and explolts the fact that shortest paths obey Snell's Law of RefractIon at region boundanes. a local optlrnahty property of shortest paths that is well known from the analogous optics model. The algorithm generalizes to the multi-source case to compute Voronol diagrams.