1. Cyclic string-to-string correction
Vida Movahedi
Elderlab, October 2009

2. Contents
Problem Definition
Linear string-to-string correction
Dynamic Programming
Cyclic strings
A faster approach
Application: curve similarity

3. Problem Definition
Two strings:
Edit operation
Taking A to B

4. Linear string-to-string correction
Cost of edit
Example: edit ‘high’ to ‘low’
Edit sequence: delete ‘h’, change ‘i’ to ‘l’, delete ‘g’, change ‘h’ to ‘o’, insert ‘w’
Goal: find edit sequence with minimum cost

5. Edit Graph, path and trace

6. Dynamic Programming Why is dynamic programming an option?
Complexity: O(nm)

7. Cyclic strings Cyclic shifts Edit cost if cyclic shifts
m possible shifts, m runs of dynamic programming: O(nm2)

8. A faster approach
All edit graphs are included in edit graph of A and BB (let’s call it graph H)

9. Non-crossing Paths Consider shifts j, k, l where
Traces corresponding to the optimal edit sequences are non-crossing on graph H: P(j), P(k), P(l)
Reducing necessary calculations

10. Non-crossing paths

11. O(nmlogm) algorithm

12. An Application: Curve Similarity
Two curves as two strings A and B
Edit cost: Euclidean distance
Minimum edit cost corresponds to optimal matching
Symmetric cost for each edit operation  Symmetric distance
Contour Mapping Distance=7.73

13. References
Maurice Maes (1990), “On a cyclic string-to-string correction problem”, Information Processing Letters, vol. 35, pp