Largest Common Point Set (LCP) problem

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Largest Common Point Set (LCP) problem
Given egt0 and two point sets A and B find a
transformation T and equally sized subsets A (a
subset of A) and B (a subset of B) of maximal
cardinality such that dist(A,T(B)) e.
Bottleneck metric optimal solution in O(n32.5)
C. Ambuhl et al. 2000
RMSD metric open problem
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alignment technique

• 1. Generate 3D transformations. For example
  for each pair of triplets, one from each molecule
  which define e-congruent triangles compute the
  rigid transformation that superimposes them.
• 2. For each transformation compute the maximal
  matching (for the bottleneck metric apply the
  bipartite matching).

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Approximation Algorithm
Assume that L is the size of the LCP (bottleneck)
with error e. Definition A ß-approximation
algorithm for the LCP problem guarantees to find
the LCP of size at least L with error (eß).
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R2 Construct transformations between each
point pair (a,b) and (p,q) in the following way



p
q
a
b
Let T(p)p , T(q)q. Let Topt be the
optimal transformation and Topt(p)p ,
Topt(q)q.
Select the grid size to be ß/3 Select ? v2
ß/3 -gt d(p,p) SQRT((ß/3)2 (ß/3)2)/2
ß/(3v2) -gt minT d(q,T(q)) SQRT( (ß/(3v2))2
(?/2)2) ß/3
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Estimate the error, i.e. d(T(v), Topt(v)) lt
? d(T(v), Topt(v)) d(R?I?T(v), T(v)) where R
and I are defined as follows
I(T(p))p R(I(T(q))q -gt R?I?T Topt
q
p
p
q
a
b
d(R?I?T(v), T(v)) d(R?I?T(v), I?T(v))
d(I?T(v), T(v)) d(R?I?T(v),
I?T(v)) ß/3 d(R?I?T(v), I?T(v)) d(R?I?T(q),
I?T(q)) why? d(R?I?T(q), I?T(q))
d(q,q) d(q, I?T(q)) ß/3 ß/3 d(R?I?T(v),
T(v)) ß
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Time Complexity for R2 O(n2 n2 (2e/(ß/3))2
(2e/(v2 ß/3)) ) O(n4 (2e/(ß/3))2
(2e/(v2 ß/3)) ) O(n4 (e/ß)3) with large constant
153. For example if e3 and we want ß0.5 -gt
3.2 104 n4 On the other hand, the optimal
algorithm for R2 is O(n8.5) For n 300
-gt 1.51011n4