A Maximum Profit Coverage Algorithm with Application to Small Molecules Cluster Identification

In this article we model, and analyze the cluster identification of molecules (CIM), which is a clustering problem in a finite metric space. CIM2 has the following characteristics which separate it from other clustering models: 1. In most models outliers are a small portion of the data set, whereas in CIM they may be the vast majority of the objects. (see Figure 1) 2. The clusters identified by CIM are compact and their diameter is bounded. 3. There is a lower bound on the number of objects in a cluster. 4. Clusters may be very close to one another, as a result of the bound on the diameter. What may be considered as one cluster in other clustering models is considered as several clusters in CIM. (see Figure 2). 5. The number of clusters is not known a-priori to the clustering procedure. In this paper we present CIM and model it as a maximum profit coverage problem (MPCP). The model is a measure to be optimized, rather then a heuristic. Consider a finite set S in a metric space M with a distance function d. A ball with center t and radius r is the subset B(t, r) = {x ∈ M |d(t, x) ≤ r}. We say that the ball covers the points of S that it contains. Given a set of balls B of radius r, a coverage P = {S ′ 1, . . . , S′ l} is a set of clusters such that each of them consists of points covered by a single ball of B. Let S′ P = ∪i=1S i, and define the profit of P as ∑