deletions | additions       diff --git a/subsection_Shape_Indexing_and_Similarity__.tex b/subsection_Shape_Indexing_and_Similarity__.tex  index 1e38d93..2e03a6b 100644  --- a/subsection_Shape_Indexing_and_Similarity__.tex  +++ b/subsection_Shape_Indexing_and_Similarity__.tex 
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As with VAMS\cite{VAMS}, we use a matching and packing\cite{Koes_2014} bulk-loading algorithm to initialize an efficient data structure for volumetric shape constraint searches.  Briefly, shapes are stored in a GSS-tree\cite{keim1999} where each leaf of the tree is a single molecular shape and each internal node includes a maximum included volume (MIV) and minimum surrounding volume (MSV). The MSV is the union of all the molecular shapes beneath the node in the tree while the MIV is the intersection. By appropriately applying minimum and maximum shape constraints to the MIV and MSV, it can be determined if any of the shapes lower in the tree have the potential to match the constraints. If not, the entire subtree of molecular shapes can be eliminated from consideration, resulting in a sub-linear running time.  %Shape Shape  constraints combined with shape indexing provide a rapid way to filter a virtual library. %Alternatively, Alternatively,  instead of serving as hard constraints, they can also be used to rank molecular shapes by similarity to the shape constraint query. We use the shape Tanimoto\cite{RushIII2005} to compute the similarity of two shapes: %$$\delta(A,B) $$\delta(A,B)  = \frac{A \cap B}{A \cup B}$$ %where where  a larger score indicates a greater degree of similarity. %The We consider both similarities with a single query ligand and similarities with shape constraints. The  similarity of a shape, $A$, with the minimum, $MIN$, and maximum, $MAX$, shape constraints is computed by combining adding  their shape Tanimotos: %$$\delta(A,MIN,MAX) $$\delta(A,MIN,MAX)  = \delta(A,MIN) + \delta(A,MAX) = \frac{A \cap MIN}{A \cup MIN} + \frac{A \cap MAX}{A \cup MAX }$$