Shortest path problems

1. Shortest path problems Lecture 2 © Alexander & Michael Bronstein tosca.cs.technion.ac.il/book Numerical geometry of non-rigid shapes Stanford University, Winter 2009

2. How to compute the intrinsic metric? • So far, we represented itself. • Our model of non-rigid shapes as metric spaces involves the intrinsic metric • Sampling procedure requires as well. • We need a tool to compute geodesic distances on .

3. Shortest path problem Brussels 943 566 Prague 183 Paris 504 1542 194 902 407 285 Vienna 346 271 Munich 1146 Bern

4. Shapes as graphs • Sample the shape at vertices . • Represent shape as an undirected graph • set of edges representing adjacent vertices. • Define length function measuring local distances as Euclidean ones,

5. Shapes as graphs • Path between is an ordered set of connected edges where and . • Path length = sum of edge lengths

6. Geodesic distance • Shortest path between • Length metric in graph • Approximates the geodesic distance on the shape. • Shortest path problem: compute and between any . • Alternatively:given a source point , compute the distance map .

7. Bellman’s principle of optimality • Let be shortest path between and a point on the path. • Then, and are shortest sub-paths between , and . • Suppose there exists a shorter path . • Contradiction to being shortest path. Richard Bellman (1920-1984)

8. Dynamic programming • How to compute the shortest path between source and on ? • Bellman principle: there exists such that • has to minimize path length • Recursive dynamic programming equation.

9. Edsger Wybe Dijkstra(1930–2002) [‘ɛtsxər ‘wibə ‘dɛɪkstra]

10. Dijkstra’s algorithm • Initialize and for the rest of the graph; Initialize queue of unprocessed vertices . • While • Find vertex with smallest value of , • For each unprocessed adjacent vertex , • Remove from . • Return distance map .

11. 183 183 183   749 749 749 749 0 183 0 346 0 617 679  617 679 617 617 904 904 904 904  904 346 346 346  346 Dijkstra’s algorithm Brussels 566 Prague 183 Paris 504 407 194 285 346 271 Munich Vienna Bern

12. Dijkstra’s algorithm – complexity • While there are still unprocessed vertices • Find and remove minimum • For each unprocessed adjacent vertex • Perform update • Every vertex is processed exactly once: outer iterations. • Minimum extraction straightforward complexity: • Can be reduced to using binary or Fibonacci heap. • Updating adjacent vertices is in general . • In our case, graph is sparsely connected, update in . • Total complexity: .

13. Troubles with the metric • Grid with 4-neighbor connectivity. • True Euclidean distance • Shortest path in graph (not unique) • Increasing sampling density does not help.

14. Metrication error 4-neighbor topology Manhattan distance 8-neighbor topology Continuous Euclidean distance • Graph representation induces an inconsistent metric. • Increasing sampling size does not make it consistent. • Neither does increasing connectivity.

15. Metrication error • How to approximate the metric consistently? Solution 1 • Stick to graph representation. • Change connectivity and sampling. • Under certain conditions consistency is guaranteed. Solution 2 • Stick to given sampling (and connectivity). • Compute distance map on a surface in some representation (e.g., mesh). • Requires a new algorithm.