Optimal LCP for translations in R 2 C. Ambühl et al. ’00 Input: A= {a i }, B= {b j }, ε ≥ 0 Time complexity: |A|=n |B|=m The number of cells n 2 m 2 Bipartite matching O(nm sqrt(n+m)) O(n 3 m 3 sqrt(n+m)) Link to the paper Define B ij = {v : |a i – (v+b j )| ≤ ε} {B ij } defines an arrangement A({B ij }) of balls of radius ε Observation 1: Enough to consider translations v only from U B ij (given some translation v we create a bipartite graph E={|a i – (v+b j )| ≤ ε} and apply bipartite matching) Observation 2: Enough to consider only one v for each cell of A({B ij }) (any two vectors from the same cell define exactly the same bipartite graph)

Optimal LCP for rotations in R 2 S.Chakraborty et al. ’99 Rotations around the point p. Dome D ij is the circular figure traced out by the point p’ on the surface of S p by rotations which cause a i to lie within the ε-ball around b j. Time complexity: |A|=n |B|=m The number of domes nm -> nm intersections on the circle (there are 2nm segment end points) Bipartite matching O(nm sqrt(n+m)) O(n 2 m 2 sqrt(n+m)) Link to the paper A({D ij }) defines an arrangement of circular segments on S p Observation: Enough to consider only one rotation for each cell (segment) of A({D ij })

Optimal LCP for rotations+translations in R 2 C. Ambühl et al. ’00 Input: A= {a i }, B= {b j }, ε ≥ 0 Time complexity: |A|=n |B|=m O(n 4 m 4 sqrt(n+m)) Link to the paper B ij = {v : |a i – (v+b j )| ≤ ε} B ij becomes a function of rotation angle r. Arrangement A({B ij }) becomes a function of r as well. Observation: The number of combinatorial changes of the arrangement A({B ij }) is (nm) 3 There are two cases: III -> there are (nm) 2 + (nm) 3 relevant points on the interval [0,2π)