Figure 6:

Computation of the bid for a point in a small cluster by the neighboring clusters in η-th iteration. (left) x_k is a point represented by a small red disc, in a small cluster c_k enclosed by solid red line. The dashed red line enclose U_k. Assume that the cluster c_k is small so that $\left|{\mathcal{C}}_{c_k}\right|\in \left(0,\eta \right)$. Clusters m₁, m₂, m₃ and m₄ are enclosed by solid colored lines too. Note that m₁, m₂ and m₃ lie in $c_{U_k}$ (the nonempty overlap between these clusters and U_k indicate that), while $m_4\notin c_{U_k}$. Thus, the bid by m₄ for x_k is zero. Since the size of cluster m₃ is less than the size of cluster c_k i.e.$\left|{\mathcal{C}}_{m_3}\right|<\left|{\mathcal{C}}_{c_k}\right|$, the bid by m₃ for x_k is also zero. Since clusters m₁ and m₂ satisfy all the conditions, the bids by m₁ and m₂ for x_k are to be computed. (right) The bid $b_{m_1\leftarrow x_k}$, is given by the inverse of the distortion of ${\tilde{\mathrm{\Phi }}}_{m_1}$ on $U_k\cup {\tilde{U}}_{m_1}$, where the dashed blue line enclose ${\tilde{U}}_{m_1}$. If the bid $b_{m_1\leftarrow x_k}$ is greater (less) than the bid $b_{m_2\leftarrow x_k}$, then the clustering procedure would favor relabelling of x_k to m₁ (m₂).