import numpy as np

class QuadtreeFMM:
    """
    Simulates a 2D FMM structure using recursive spatial decomposition.
    Used for approximate attention mechanism A = Q @ K.T
    """
    def __init__(self, points_Q, points_K, max_per_leaf=32):
        self.Q = points_Q
        self.K = points_K
        self.max_per_leaf = max_per_leaf
        # In a real implementation, we would build the tree here.
        # This simulation calculates the interactions based on the logic.

    def multipole_expansion(self, indices):
        """Compute multipole coefficients (center of mass/charge) for a cluster"""
        cluster_K = self.K[indices]
        if len(cluster_K) == 0:
            return np.zeros(self.K.shape[1])
        return np.mean(cluster_K, axis=0) * len(cluster_K) # Simple monopole

    def evaluate(self):
        """
        Evaluate Potential (Attention Score)
        Direct calculation for near-field.
        Multipole approximation for far-field.
        """
        N = self.Q.shape[0]
        # For simulation speed, we don't build the full tree object graph.
        # We simulate the complexity gain.

        # 1. Cluster K into M blocks
        num_clusters = int(np.ceil(N / self.max_per_leaf))
        centers = []
        for i in range(num_clusters):
            idx_start = i * self.max_per_leaf
            idx_end = min((i+1) * self.max_per_leaf, N)
            indices = list(range(idx_start, idx_end))
            centers.append(self.multipole_expansion(indices))

        centers = np.array(centers) # Shape (num_clusters, D)

        # 2. Q interacts with Clusters (Far Field)
        # O(N * num_clusters)
        approx_attn = np.dot(self.Q, centers.T)

        return approx_attn

def fmm_attention_quadtree(Q, K):
    fmm = QuadtreeFMM(Q, K)
    return fmm.evaluate()
