Algorithm 2 Selection based on Cluster-level Constraints

Input:     Let $T=\{C_2,...,C_k\}$ be the set of cluster candidates of the condensed cluster tree (excluding the root), $L\subset T$ the set of leaf clusters and $X\subset T$ the set of constraint clusters. Let $A_i=\left\{j\right|C_j$ is ancestor of $C_i\}$ denote the set of ancestor indices for a cluster $C_i$. Let ${\delta }_i\in \{0,1\}$ be Boolean indicator whether a cluster is selected or not.

Initialize ${\delta }_i=0$ for $i=2,...,k$. $\forall i:C_i\in X$, go up the tree and set $X=X\bigcup \left\{C_j\right\}$ for every unprocessed $j\in A_i$ $\forall i:C_i\in L$, go up the tree and set ${\delta }_i=1$ if $C_{PARENT\left(C_i\right)}\in X$ or root