Constrained Centroid Clustering (CCC)

• CCC is a clustering method that enforces a maximum Euclidean distance constraint to generate geometrically bounded and compact clusters.
• It leverages a closed-form Lagrangian formulation to optimally balance cluster centroids with strict spread limits, ensuring transparent solution construction.
• Experimental evaluations show CCC reduces radial spread metrics by approximately 10% compared to conventional methods while preserving angular structure.

Constrained Centroid Clustering (CCC) generalizes conventional centroid-based clustering by enforcing explicit structural constraints—most notably on cluster spread—during the partitioning process. The methodology centers on bounding the maximum Euclidean distance between a cluster center and its extremal member, yielding clusters that are geometrically compact and suitable for domains where structure and interpretability are critical, such as sensor placement, collaborative robotics, and pattern analysis. CCC leverages a closed-form Lagrangian formulation to guarantee both optimality (subject to constraints) and clarity in solution construction.

1. Formal Lagrangian Approach to Spread-Bounded Clustering

The canonical K-means objective seeks a center $y_0$ that minimizes the sum of squared distances to $N-1$ cluster points:

$y_{0_j} = \frac{1}{N - 1} \sum_{i=1}^{N-1} y_{ij}$

CCC introduces a constraint on the squared distance to the extremal point $y_N$:

$\sum_j (y_{0_j} - y_{N_j})^2 \leq S$

where $S$ is a user-controlled spread parameter. The optimization is re-cast with a Lagrangian method:

$$\mathcal{L}(y_0, \lambda) = \sum_{i=1}^{N-1} \sum_j (y_{ij} - y_{0_j})^2 + \lambda \left(\sum_j (y_{0_j} - y_{N_j})^2 - S\right),\quad \lambda \geq 0$$

Stationarity yields the closed-form solution:

$$y_{0_j} = \frac{\sum_{i=1}^{N-1} y_{ij} + \lambda y_{N_j}}{N-1 + \lambda}$$

The multiplier $\lambda$ is chosen according to:

$$\lambda = \max\Bigl(0, \sqrt{C/S} - (N-1)\Bigr),\quad C = \sum_j [A_j - (N-1) y_{N_j}]^2,\quad A_j = \sum_{i=1}^{N-1} y_{ij}$$

This ensures that when the spread constraint is inactive ($\lambda=0$), the solution matches the centroid, and when active ($\lambda>0$), tightness is governed by $S$. Complementary slackness and feasibility are satisfied per standard Karush-Kuhn-Tucker (KKT) conditions.

2. Entropy-Based Structural Metrics

CCC is evaluated not only on geometric compactness but on its ability to preserve dataset structure. Three key entropy metrics are introduced by partitioning the data into rings (radial shells) and sectors (angular slices):

• Ring-wise Entropy ($H(P)$): Quantifies radial spread.

$$p_i = \frac{\tilde{n}_i}{\sum_r \tilde{n}_r},\quad H(P) = -\sum_i p_i \log p_i$$

• Sector-wise Entropy ($H(S)$): Quantifies angular uniformity.

$$S_k = \frac{\Gamma_k}{\sum_k \Gamma_k},\quad H(S) = -\sum_k S_k \log S_k$$

• Joint Entropy ($H(P_{i,k})$): Aggregates overall cluster structure.

Lower $H(P)$ and $H(P_{i,k})$ values indicate increased radial compactness with preservation of angular structure (indicated by unchanged $H(S)$).

3. Experimental Evaluation and Quantitative Outcomes

Experiments are performed on synthetic datasets with circular, radially symmetric distribution ($n=500$ and $n=5000$) and varied standard deviations (1.0, 1.2, 1.5).

Comparison methods include KMeans, Gaussian Mixture Models (GMM), DBSCAN, and Agglomerative Clustering. Visual analysis (see paper Figure 1) reveals CCC produces clusters tightly bounded in radius, avoiding inclusion of distant outliers.

Method	Ring Entropy	Sector Entropy	Joint Entropy
CCC	Lower	Similar	Lower
KMeans, GMM	Higher	Similar	Higher

CCC achieves ≈10% reduction in ring entropy and ≈5% in joint entropy over alternatives, while sector entropy remains largely unchanged, indicating angular structure retention.

4. Application Domains

CCC is especially suited for:

• Sensor Networks: Guarantees that all sensors remain within a controlled radius from their respective cluster centroids, supporting uniform coverage and maintenance efficiency.
• Collaborative Robotics: Ensures geometric regularity in robot task-grouping, facilitating coordination and resource allocation.
• Interpretable Pattern Analysis: The closed-form nature and entropy metrics of CCC facilitate explainable clustering decisions required for human-in-the-loop analytics.

5. Computational Properties and Implementation

• Efficiency: CCC relies on summation and one scalar multiplier computation; no iterative optimization is required, making deployment straightforward and fast.
• Interpretability: The solution $y_{0_j}$ remains a convex combination of cluster members, retaining transparency in result explanation.
• Flexibility: The approach seamlessly reduces to classical centroid computation if $S$ is large, while small $S$ enforces strict compaction.
• Constraint Enforcement: Post-processing can be required—data points lying beyond the bound are shifted inward along the radial axis to comply with the allowable spread.

6. Limitations and Parameter Sensitivity

• The method’s advantages are most pronounced in data with circular or radially-symmetric structure; for other geometries, benefit may be subdued.
• The spread threshold $S$ is pivotal: if set too small, it can lead to over-constrained clusters; too large renders the method equivalent to unconstrained centroid-based clustering.
• Adjustment of extremal points post-clustering in noisy datasets may introduce complexity.

7. Comparative Context and Impact

CCC advances centroid-based clustering for applications demanding not just compactness but explicit control over cluster geometry and interpretability. The method outperforms widely-used alternatives (KMeans, GMM) on compactness metrics without sacrificing angular or structural fidelity. Its efficient computation and reliance on interpretable geometric principles support immediate application in domains with rigid spread constraints.

By embedding constraint satisfaction into a closed-form centroid update, CCC establishes a new standard for structurally aware clustering, both for capacity-limited sensor systems and collaborative decision-making frameworks where tight spatial control is essential.