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In order to design a drug curing a given pathology, an approach
which is commonly used consists in first selecting those that work
best among molecules known to treat similar symptoms. In view of the
number of known molecules, an exhaustive approach is often
impossible. Instead, it is a common strategy to pick at random a
number of molecules in a large database; and then concentrate on those
that are chemically close to the ones that performed well. It is
therefore important to be able to find those molecules in such a
database. The aim of this work is to present a probabilistic algorithm
for this task.
1 Molecules and similarity
The database is represented as an array of n~10,000 molecules,
each molecule being characterized by the presence or absence
of d~1,500 molecular fragments. Molecules are close when they
differ by few molecular fragments. More precisely, one defines the sizes(m) of a molecule m as the number of its fragments and the
similarity sim(m,M) between two molecules as the number of
common fragments. Finally, two molecules m and M are
called (a,b)-similar for aÎ[0,1] and b³1
when
Note that this is not an equivalence relation. Other measures of
similarity might also be of interest in practice.
2 Algorithm
The aim of this work is
to find efficiently as many (a,b)-similar pairs in the
database as possible. Obviously, an exhaustive search in n(n-1)/2
operations is possible, but expensive. The idea instead is to use a
divide-and-conquer partitioning process. A fragment is selected at
random and the database is partitioned into two subsets according to
the presence or absence of this fragment. When such a subset has less
than a fixed number K of elements an exhaustive search is
performed; otherwise, the same process is applied recursively.
This technique finds a proportion t N of the number N
of (a,b)-similar pairs. Heuristically, repeating the same
process from scratch yields t(1-t)^{i-1}N new pairs at the ith
iteration. Thus a few iterations of this idea yield a very large
proportion of N.
3 Implementation
The parameter K plays an important part in the efficiency of the
algorithm. When K is small the search is faster but finds less
pairs, so that the number of times it has to be repeated to obtain the
same number of pairs can be larger than for higher values of K. The
optimal value of K also depends on the efficiency of the different
stages of the algorithm. Any improvement on the partitioning and
exhaustive search part shift the optimum to larger values of K,
while a good data-structure for checking whether a pair has already been
found shifts it in the other direction. Here are implementation ideas
that lead to an efficient program:
The database is stored as an array of bits;
the entries in the database are accessed by chunks (bytes or
words), a constant array making it fast to count the number of bits equal
to 1 in a chunk;
computing the similarity between to molecules is then performed
by bitwise and;
the sizes of the molecules are computed once at the beginning;
the partitioning is done like in Quicksort, on an array of
pointers to the molecules;
the set of pairs is stored as an array of binary search trees (a
hash table would also do).
In practice, with this implementation and K around 150, then 4 or 5 runs
of the partitioning yield more than 90% of the pairs in a matter of
minutes. The exhaustive search would take several hours.
Conclusion
It would be nice to find the optimal value of K by a complexity
analysis. However, the Bernoulli distribution for the bits in the database
does not give a good model. It is necessary to take into account the
fact that the database was arrived at by a historical process where
many of the molecules are variants of each other.
Cazals (Frédéric). --
Effective nearest neighbours searching on the hyper-cube, with
applications to molecular clustering. In ACM Symposium on Computational
Geometry. pp. 222--230. --
ACM Press, 1998. Also available at the url http://www.inria.fr/prisme/personnel/cazals/xfc_research.html.
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