Add sparse EMD solver with unit tests

  - Implement sparse bipartite graph EMD solver in C++
  - Add Python bindings for sparse solver (emd_wrap.pyx, _network_simplex.py)
  - Add unit tests to verify sparse and dense solvers produce identical results
  - Tests use augmented k-NN approach to ensure fair comparison
  - Update setup.py to include sparse solver compilation

  Both test_emd_sparse_vs_dense() and test_emd2_sparse_vs_dense() verify:
    * Identical costs between sparse and dense solvers
    * Marginal constraint satisfaction for both solvers

Author: nathanneike

ot/lp/EMD.h

@@ -32,6 +32,24 @@ enum ProblemType {
 int EMD_wrap(int n1,int n2, double *X, double *Y,double *D, double *G, double* alpha, double* beta, double *cost, uint64_t maxIter);
 int EMD_wrap_omp(int n1,int n2, double *X, double *Y,double *D, double *G, double* alpha, double* beta, double *cost, uint64_t maxIter, int numThreads);
 
+int EMD_wrap_sparse(
+    int n1,                      
+    int n2,                       
+    double *X,                   
+    double *Y,                   
+    uint64_t n_edges,            // Number of edges in sparse graph
+    int64_t *edge_sources,       // Source indices for each edge (n_edges)
+    int64_t *edge_targets,       // Target indices for each edge (n_edges)
+    double *edge_costs,          // Cost for each edge (n_edges)
+    int64_t *flow_sources_out,   // Output: source indices of non-zero flows
+    int64_t *flow_targets_out,   // Output: target indices of non-zero flows
+    double *flow_values_out,     // Output: flow values
+    uint64_t *n_flows_out,       
+    double *alpha,               // Output: dual variables for sources (n1)
+    double *beta,                // Output: dual variables for targets (n2)
+    double *cost,                // Output: total transportation cost
+    uint64_t maxIter             // Maximum iterations for solver
+);
 
 
 #endif

ot/lp/EMD_wrapper.cpp

@@ -15,8 +15,10 @@
 
 #include "network_simplex_simple.h"
 #include "network_simplex_simple_omp.h"
+#include "sparse_bipartitegraph.h"
 #include "EMD.h"
 #include <cstdint>
+#include <unordered_map>
 
 
 int EMD_wrap(int n1, int n2, double *X, double *Y, double *D, double *G,
@@ -216,3 +218,156 @@ int EMD_wrap_omp(int n1, int n2, double *X, double *Y, double *D, double *G,
 
     return ret;
 }
+
+// ============================================================================
+// SPARSE VERSION: Accepts edge list instead of dense cost matrix
+// ============================================================================
+int EMD_wrap_sparse(
+    int n1,
+    int n2,
+    double *X,
+    double *Y,
+    uint64_t n_edges,
+    int64_t *edge_sources,
+    int64_t *edge_targets,
+    double *edge_costs,
+    int64_t *flow_sources_out,
+    int64_t *flow_targets_out,
+    double *flow_values_out,
+    uint64_t *n_flows_out,
+    double *alpha,
+    double *beta,
+    double *cost,
+    uint64_t maxIter
+) {
+    using namespace lemon;
+    
+    uint64_t n = 0;  
+    for (int i = 0; i < n1; i++) {
+        double val = *(X + i);
+        if (val > 0) {
+            n++;
+        } else if (val < 0) {
+            return INFEASIBLE; 
+        }
+    }
+    
+    uint64_t m = 0;
+    for (int i = 0; i < n2; i++) {
+        double val = *(Y + i);
+        if (val > 0) {
+            m++;
+        } else if (val < 0) {
+            return INFEASIBLE; 
+        }
+    }
+
+    std::vector<uint64_t> indI(n);  // indI[graph_idx] = original_source_idx
+    std::vector<uint64_t> indJ(m);  // indJ[graph_idx] = original_target_idx
+    std::vector<double> weights1(n);  // Source masses (positive only)
+    std::vector<double> weights2(m);  // Target masses (negative for demand)
+    
+    // Create reverse mapping: original_idx → graph_idx
+    std::vector<int64_t> source_to_graph(n1, -1);  
+    std::vector<int64_t> target_to_graph(n2, -1);
+    
+    uint64_t cur = 0;
+    for (int i = 0; i < n1; i++) {
+        double val = *(X + i);
+        if (val > 0) {
+            weights1[cur] = val;           // Store the mass
+            indI[cur] = i;                 // Forward map: graph → original
+            source_to_graph[i] = cur;      // Reverse map: original → graph
+            cur++;
+        }
+    }
+    
+    cur = 0;
+    for (int i = 0; i < n2; i++) {
+        double val = *(Y + i);
+        if (val > 0) {
+            weights2[cur] = -val;         
+            indJ[cur] = i;                 // Forward map: graph → original
+            target_to_graph[i] = cur;      // Reverse map: original → graph
+            cur++;
+        }
+    }
+    
+    typedef SparseBipartiteDigraph Digraph;
+    DIGRAPH_TYPEDEFS(Digraph);
+
+    Digraph di(n, m);  
+
+    std::vector<std::pair<int, int>> edges;  // (source, target) pairs
+    std::vector<uint64_t> edge_to_arc;       // edge_to_arc[k] = arc ID for edge k
+    std::vector<double> arc_costs;            // arc_costs[arc_id] = cost (for O(1) lookup)
+    edges.reserve(n_edges);
+    edge_to_arc.reserve(n_edges);
+
+    uint64_t valid_edge_count = 0;
+    for (uint64_t k = 0; k < n_edges; k++) {
+        int64_t src_orig = edge_sources[k];
+        int64_t tgt_orig = edge_targets[k];
+        int64_t src = source_to_graph[src_orig];
+        int64_t tgt = target_to_graph[tgt_orig];
+
+        if (src >= 0 && tgt >= 0) {
+            edges.emplace_back(src, tgt + n);
+            edge_to_arc.push_back(valid_edge_count);
+            arc_costs.push_back(edge_costs[k]);  // Store cost indexed by arc ID
+            valid_edge_count++;
+        } else {
+            edge_to_arc.push_back(UINT64_MAX);  
+        }
+    }
+
+
+    di.buildFromEdges(edges);
+
+    NetworkSimplexSimple<Digraph, double, double, node_id_type> net(
+        di, true, (int)(n + m), di.arcNum(), maxIter
+    );
+
+    net.supplyMap(&weights1[0], (int)n, &weights2[0], (int)m);
+
+    for (uint64_t k = 0; k < n_edges; k++) {
+        if (edge_to_arc[k] != UINT64_MAX) {
+            net.setCost(edge_to_arc[k], edge_costs[k]);
+        }
+    }
+    
+    int ret = net.run();
+
+    if (ret == (int)net.OPTIMAL || ret == (int)net.MAX_ITER_REACHED) {
+        *cost = 0;
+        *n_flows_out = 0; 
+        
+        Arc a;
+        di.first(a);
+        for (; a != INVALID; di.next(a)) {
+            uint64_t i = di.source(a); 
+            uint64_t j = di.target(a);  
+            double flow = net.flow(a);
+            
+            uint64_t orig_i = indI[i];
+            uint64_t orig_j = indJ[j - n];  
+
+
+            double arc_cost = arc_costs[a]; 
+
+            *cost += flow * arc_cost;
+            
+
+            *(alpha + orig_i) = -net.potential(i);
+            *(beta + orig_j) = net.potential(j);
+            
+            if (flow > 1e-15) {  
+                flow_sources_out[*n_flows_out] = orig_i;
+                flow_targets_out[*n_flows_out] = orig_j;
+                flow_values_out[*n_flows_out] = flow;
+                (*n_flows_out)++;
+            }
+        }
+    }
+    return ret;
+}