Shortest Path ProblemMotion Planning

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

Geometric Shortest Paths

S

T

Polygon Polygonal Domain

T

S

T

S

S

T

Geometric Shortest Paths

Polygon Polygonal Domain

Geometric Shortest Paths -- Polygon

S

T

Geometric Shortest Paths -- Polygon

S

T

Geometric Shortest Paths -- Polygon

S

T

elastic band solution

Geometric Shortest Paths -- Polygon

S

T

elastic band solution

S

T

Geometric Shortest Paths -- Polygon

elastic band solution (locally shortest)

S

T

Geometric Shortest Paths -- Polygon

Funnel Algorithm

S

T

Geometric Shortest Paths -- Polygon

Funnel Algorithm

S

T

Geometric Shortest Paths -- Polygon

Funnel Algorithm

S

T

Geometric Shortest Paths -- Polygon

Funnel Algorithm

S

T

Geometric Shortest Paths -- Polygon

Funnel Algorithm

S

T

Geometric Shortest Paths -- Polygon

Funnel Algorithm

S

T

Geometric Shortest Paths -- Polygon

Funnel Algorithm

S

T

Geometric Shortest Paths -- Polygon

Funnel Algorithm

S

T

Geometric Shortest Paths -- Polygon

S

T

Funnel Algorithm

S

T

Geometric Shortest Paths -- Polygon

-- O(n) Guibas, Lee & Preparata, early ‘80’sFunnel Algorithm

Geometric Shortest Paths -- Polygonal Domain

T

S

Geometric Shortest Paths -- Polygonal Domain

multiple elastic band solutions

T

S

T

S

Geometric Shortest Paths -- Polygonal Domain

multiple elastic band solutions

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 Continuous Dijkstra

T

S

Geometric Shortest Paths -- Polygonal Continuous Dijkstra

T

S

Geometric Shortest Paths -- Polygonal Continuous Dijkstra

T

S

Geometric Shortest Paths -- Polygonal Continuous Dijkstra

T

S

Geometric Shortest Paths -- Polygonal Continuous Dijkstra

T

S

Geometric Shortest Paths -- Polygonal Continuous Dijkstra

T

S

Geometric Shortest Paths -- Polygonal Continuous Dijkstra

T

S

Geometric Shortest Paths -- Polygonal Continuous Dijkstra

T

S

Geometric Shortest Paths -- Polygonal Continuous Dijkstra

T

S

Geometric Shortest Paths -- Polygonal

shortest path map

Continuous Dijkstra

T

S

Geometric Shortest Paths -- Polygonal

shortest path map

Continuous Dijkstra

T

S

Geometric Shortest Paths -- Polygonal

shortest path map

Continuous Dijkstra

T

S

Geometric Shortest Paths -- Polygonal

shortest path map

Continuous Dijkstra

T

S

Geometric Shortest Paths -- Polygonal - O(n log n) Mitchell, Hershberger & Suri, ‘93

shortest path map

Continuous Dijkstra

3-D Shortest Path Problemthe general problem

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 Problem

the spider and the fly problem

3-D Shortest Path Problem

the spider and the fly problem

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

3-D Shortest Path Problempaths on polyhedral surfaces

pictures from Kaneva & O’Rourke, ‘00

3-D Shortest Path Problempaths on polyhedral surfaces

pictures from Kaneva & O’Rourke, ‘00

3-D Shortest Path Problempaths on polyhedral surfaces

pictures from Kaneva & O’Rourke, ‘00