The weighted region problem

Abstract

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.

Optimal path finding in direction, location, and time dependent environments

“…Asymmetric direction-dependence is occasionally considered in literature [12,61], however the introduced anisotropy makes a strong assumption of the distance function convexity which we relax in our analysis. The path-finding problems in a locationdependent environment examine a presence of polygonal obstacles [3,31,37,42,44] and uniform-weighted regions [11,47,67]. On the other hand, all the problems studied in the field of computational geometry are predominantly static, and timedependence is not considered in these settings.…”

“…However, the majority of the to date work concentrates on determining the Euclidean shortest paths (see Mitchell's extensive survey [45]). Even though a number of extensions to optimal path planning have been considered (e.g., traversing through polygonal constantly-weighted regions [47,67]), most work is restricted to isotropic metrics, where the cost function is assumed to be independent of the traveling direction.…”

Algorithms for Approximate Shortest Path Queries on Weighted Polyhedral Surfaces

“…Cost of a path on the weighted surface is computed as the weighted sums of the Euclidean lengths of their portions inside the faces of the surface. The weighted model was introduced in [36] for planar subdivisions. Solutions of the exact problems in the weighted case seem to be infeasible, since computing of exact distances even on very simple weighted surfaces with just a few faces requires a solution of high degree algebraic equations.…”

“…So, for the case of weighted planar subdivision of size n, the algorithm presented in [36] constructs -a so called -restricted shortest path map from a fixed source point to the edges of the weighted planar subdivision of size O(n 4 ) and in roughly O(n 8 log n ε ) time. The map then can be used to answer approximate distance queries from the source to points on the edges of the surface in roughly O(n 7 log n ε ) time.…”

“…This work is inspired by our work and that of Reif and Sun [38], and the authors were able to remove the dependence on the geometric parameters at the expense of increasing the dependence on n as well as the ratio of the weights (max weight to min weight). Their analysis requires the bound on the maximum number of links possible in a weighted shortest path; which is Ω(n 2 ) as shown in [36].…”