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

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

Problem Definition Two strings: Edit operation Taking A to B

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

Edit Graph, path and trace

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

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

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

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

Non-crossing paths

O(nmlogm) algorithm

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

References Maurice Maes (1990), “On a cyclic string-to-string correction problem”, Information Processing Letters, vol. 35, pp. 73-78.