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Author Mitchell, Joseph S B ♦ Papadimitriou, Christos H.
Source ACM Digital Library
Content type Text
Publisher Association for Computing Machinery (ACM)
File Format PDF
Copyright Year ©1991
Language English
Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Subject Keyword Dijkstra's algorithm ♦ Voronoi diagrams ♦ Shortest paths ♦ Terrain navigation ♦ Weighted distance functions
Abstract The problem of determining shortest paths through a weighted planar polygonal subdivision with $\textit{n}$ vertices is considered. Distances are measured according to a weighted Euclidean metric: The length of a path is defined to be the weighted sum of (Euclidean) lengths of the subpaths within each region. An algorithm that constructs a (restricted) “shortest path map” with respect to a given source point is presented. The output is a partitioning of each edge of the subdivion into intervals of ε-optimality, allowing an ε-optimal path to be traced from the source to any query point along any edge. The algorithm runs in worst-case time $\textit{O}(\textit{ES})$ and requires $\textit{O}(\textit{E})$ space, where $\textit{E}$ is the number of “events” in our algorithm and $\textit{S}$ is the time it takes to run a numerical search procedure. In the worst case, $\textit{E}$ is bounded above by $\textit{O}(\textit{n}4)$ (and we give an &OHgr;(n</>4) lower bound), but it is likeky that $\textit{E}$ will be much smaller in practice. We also show that $\textit{S}$ is bounded by $\textit{O}(\textit{n}4\textit{L}),$ where $\textit{L}$ is the precision of the problem instance (including the number of bits in the user-specified tolerance ε). Again, the value of $\textit{S}$ should be smaller in practice. The algorithm applies the “continuous Dijkstra” paradigm and exploits the fact that shortest paths obey Snell's Law of Refraction at region boundaries, a local optimaly property of shortest paths that is well known from the analogous optics model. The algorithm generalizes to the multi-source case to compute Voronoi diagrams.
ISSN 00045411
Age Range 18 to 22 years ♦ above 22 year
Educational Use Research
Education Level UG and PG
Learning Resource Type Article
Publisher Date 1991-01-03
Publisher Place New York
e-ISSN 1557735X
Journal Journal of the ACM (JACM)
Volume Number 38
Issue Number 1
Page Count 56
Starting Page 18
Ending Page 73


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Source: ACM Digital Library