Optimal LCP for translations in R2 C. Ambühl et al. ’00

Input: A= {ai}, B= {bj}, ε ≥ 0

Time complexity: |A|=n |B|=m

The number of cells n2m2

Bipartite matching O(nm sqrt(n+m))

O(n3m3 sqrt(n+m))

Link to the paper

Define Bij = {v : |ai – (v+bj)| ≤ ε}

{Bij} defines an arrangement A({Bij}) of balls of radius ε

Observation 1: Enough to consider translations v only from U Bij

(given some translation v we create a bipartite graph E={|ai – (v+bj)| ≤ ε} and apply bipartite matching)

Observation 2: Enough to consider only one v for each cell of A({Bij})

(any two vectors from the same cell define exactly the same bipartite graph)

12

3

7

4

89

1110

56 12

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Optimal LCP for rotations in R2 S.Chakraborty et al. ’99

Rotations around the point p.

Dome Dij is the circular figure traced out by the point p’ on the surface of Sp by rotations which cause ai to lie within the ε-ball around bj.

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(n2m2 sqrt(n+m))

Link to the paper

A({Dij}) defines an arrangement of circular segments on Sp

Observation: Enough to consider only one rotation for each cell (segment) of A({Dij})

Optimal LCP for rotations+translations in R2 C. Ambühl et al. ’00

Input: A= {ai}, B= {bj}, ε ≥ 0

Time complexity: |A|=n |B|=m

O(n4m4 sqrt(n+m))

Link to the paper

Bij = {v : |ai – (v+bj)| ≤ ε}

Bij becomes a function of rotation angle r. Arrangement A({Bij}) becomes a function of r as well.

Observation: The number of combinatorial changes of the arrangement A({Bij}) is (nm)3

There are two cases:

I II

-> there are (nm)2 + (nm)3 relevant points on the interval [0,2π)