Connected Components

include/spatial/connected_components.hpp provides num::connected_components, a template BFS labelling function with a pre-allocated flat queue (no heap allocation per call, no recursion).

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Problem

Connected-component labeling partitions included vertices of a finite graph into maximal connected subsets. For Ising nucleation, included vertices are the spin-down sites and the largest component is the nucleus observable.

connected_components uses caller-supplied predicates for site inclusion and neighbor traversal.

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Routine Reference

struct num::ClusterResult {
    std::vector<int> id;        // id[i]: -2 = excluded, >=0 = cluster index
    std::vector<int> sizes;     // sizes[c] = number of sites in cluster c
    int largest_id   = -1;
    int largest_size = 0;
};
 
template<typename InCluster, typename Neighbors>
num::ClusterResult num::connected_components(
    int n_sites,
    InCluster&&  in_cluster,   // bool(int i)        -- include site i?
    Neighbors&&  neighbors);   // void(int i, F&& f) -- call f(nb) per neighbor

Labels

id[i]	Meaning
-2	Excluded – in_cluster(i) returned false
>=0	Cluster index

Sites with id[i] == ClusterResult::largest_id belong to the largest cluster.

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Algorithm

Iterative BFS with a pre-allocated flat queue of size n_sites:

for each unvisited included site s:
    assign cluster id, push to queue
    while queue not empty:
        pop i, increment size
        for each neighbor nb of i:
            if unvisited: label nb, push
    update largest_id / largest_size

No heap allocations after the initial ClusterResult construction.

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Ising Nucleation Observable

num::PBCLattice2D nbr(N);
num::ClusterResult det;
 
// In the Metropolis order-parameter measurement:
det = num::connected_components(N*N,
    [&](int i) { return spins[i] < 0.0; },      // spin-down sites only
    [&](int i, auto&& visit) {
        visit(nbr.up[i]); visit(nbr.dn[i]);
        visit(nbr.lt[i]); visit(nbr.rt[i]);
    });
int nucleus_size = det.largest_size;
 
// Rendering: highlight the largest cluster red
if (det.id[i] == det.largest_id) { /* draw red */ }

Generalizations

The callable interface accepts any graph topology:

// 3D cubic lattice with PBC
det = num::connected_components(nx*ny*nz,
    [&](int i) { return active[i]; },
    [&](int i, auto&& visit) {
        for (int nb : six_neighbors(i, nx, ny, nz))
            visit(nb);
    });
 
// Irregular graph from adjacency list
det = num::connected_components(n_nodes,
    [&](int i) { return !excluded[i]; },
    [&](int i, auto&& visit) {
        for (int nb : adj[i]) visit(nb);
    });