public static void TrianglesByDegree(this PINQCollection <Pair <int> > graph, double epsilon)
{
Console.WriteLine("in TrianglesByDegree...");
// form (b, db) pairs each with weight 1/2
var bDegs = graph.GroupBy(e => e.a, e => e.b, (k, i) => new VertexData(k, i));
// form length 2 paths (a,b,c) weight 1/2db.
var path = graph.Join(graph, x => x.b, y => y.a, x => x.a, y => y.b,
(key, x, y) => new Triple(x, key, y));//.Where(x => x.a != x.c);
// form ((a,b,c), db) tuples, with weights 1/2db(1 + db).
var abc = path.Join(bDegs, x => x.b, y => y.name, x => x, y => y.edges.Length,
(key, x, y) => new Pair <Triple, int>(x, y));
// rotate to get ((c,a,b),db) or equivalently ((a,b,c),dc)
var cab = abc.Select(x => new Pair <Triple, int>(new Triple(x.a.c, x.a.a, x.a.b), x.b));
// rotate to get ((b,c,a),db) or equivalently ((a,b,c),da)
var bca = abc.Select(x => new Pair <Triple, int>(new Triple(x.a.b, x.a.c, x.a.a), x.b));
// form length ((a,b,c),da,db) tuples with weight 1/2(da(1+da) + db(1 +db))
var tuple = abc.Join(bca, x => x.a, y => y.a, x => x.b, y => y.b,
(key, x, y) => new Pair <Triple, Pair>(key, new Pair(y, x)));
// form length ((a,b,c),da,db,db) tuples with weight 1/2(da(1+da) + db(1+db) + dc(1+dc))
var tuple2 = tuple.Join(cab, x => x.a, y => y.a, x => x.b, y => y.b,
(key, x, y) => new Pair <Triple, Triple>(key, new Triple(x.a, x.b, y)));
// transform to (da,db,dc) tuples where da < db < dc
var tris = tuple2.Select(x => x.b);
tris = tris.Select(x => new Triple(Math.Min(Math.Min(x.a, x.b), x.c), median(x.a, x.b, x.c), Math.Max(x.a, Math.Max(x.b, x.c))));
// return the noisy histogram
var result = tris.Count(x => x, epsilon);
}
