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Shortest Path in a Graph
Toronto
Philadelphia
Pittsburgh
Syracuse154
72220
105
217
204
288
114
127
141
135
147
159
109
289 184
198
127
Dijkstra’s Algorithm, 1959
Toronto
Philadelphia
Pittsburgh
Syracuse154
72220
105
217
204
288
114
127
141
135
147
159
109
289 184
198
127
Shortest Path in a GraphDijkstra’s Algorithm, 1959
Toronto
Philadelphia
Pittsburgh
Syracuse154
72220
105
217
204
288
114
127
141
135
147
159
109
289 184
198
127
Shortest Path in a GraphDijkstra’s Algorithm, 1959
Toronto
Philadelphia
Pittsburgh
Syracuse154
72220
105
217
204
288
114
127
141
135
147
159
109
289 184
198
127
Shortest Path in a GraphDijkstra’s Algorithm, 1959
Toronto
Philadelphia
Pittsburgh
Syracuse154
72220
105
217
204
288
114
127
141
135
147
159
109
289 184
198
127
Shortest Path in a GraphDijkstra’s Algorithm, 1959
Toronto
Philadelphia
Pittsburgh
Syracuse154
72220
105
217
204
288
114
127
141
135
147
159
109
289 184
198
127
Shortest Path in a GraphDijkstra’s Algorithm, 1959
Toronto
Philadelphia
Pittsburgh
Syracuse154
72220
105
217
204
288
114
127
141
135
147
159
109
289 184
198
127
Shortest Path in a GraphDijkstra’s Algorithm, 1959
Toronto
Philadelphia
Pittsburgh
Syracuse154
72220
105
217
204
288
114
127
141
135
147
159
109
289 184
198
127
259
322
Shortest Path in a GraphDijkstra’s Algorithm, 1959
Toronto
Philadelphia
Pittsburgh
Syracuse154
72220
105
217
204
288
114
127
141
135
147
159
109
289 184
198
127
259
322 393
394
Shortest Path in a GraphDijkstra’s Algorithm, 1959
Toronto
Philadelphia
Pittsburgh
Syracuse154
72220
105
217
204
288
114
127
141
135
147
159
109
289 184
198
127
259
322 393
394
406
Shortest Path in a GraphDijkstra’s Algorithm, 1959
Toronto
Philadelphia
Pittsburgh
Syracuse154
72220
105
217
204
288
114
127
141
135
147
159
109
289 184
198
127
259
322 393
394
406
507
Shortest Path in a GraphDijkstra’s Algorithm, 1959 -- O(m + n log n) Fredman & Tarjan, 1987
using Fibonacci heaps
S
T
Geometric Shortest Paths -- Polygon
-- O(n) Guibas, Lee & Preparata, early ‘80’sFunnel Algorithm
Geometric Shortest Paths -- Polygonal Domainhomotopic shortest path problem (shrinking an elastic band)
T
S
Geometric Shortest Paths -- Polygonal Domainhomotopic shortest path problem (shrinking an elastic band)
T
S
-- Hershberger & Snoeyink, ‘94-- Efrat & Kobourov & Lubiw, ‘02
T
S
T
S
T
S
Geometric Shortest Paths -- Polygonal Domain
- construct visibility graph
- apply Dijkstra’s graph algorithm = O(n )2O(m + n log n)}
Pocchiola &Vegter, Riviere, ‘95
reducing to a graph problem
T
S
Geometric Shortest Paths -- Polygonal - O(n log n) Mitchell, Hershberger & Suri, ‘93
shortest path map
Continuous Dijkstra
3-D Shortest Path Problem- NP-hard
- PSPACE algorithm, Canny ‘88
- approximation algorithms
- efficient algorithm for paths on polyhedral surfaces
3-D Shortest Path Problemthe general problem NP-hard -- Canny & Reif, 1987 even for the case of parallel floating triangles
A
B
there are good approximation algorithms
Shortest Path Problem on a Polyhedral Surface
the spider and the fly problem Dudeney, The Canterbury Puzzles, 1958
3-D Shortest Path Problempaths on polyhedral surfaces
-- O(n ) O’Rourke, Suri, Booth, ‘85
-- O(n ) Chen, Han, ‘96
-- O(n log n) Kapoor, ‘99
-- approximation algorithms
5
2
2
pictures from Kaneva & O’Rourke, ‘00