Is there a faster algorithm for constructing the adjacency list of this graph?

Question

My problem is to find a better algorithm to fill the adjacency list.

specs:

G = (V , E ) //the graph
V ={w} //vertex in this case each vertex is an array     
E={⟨u,v⟩| u,v ∈V ∧ u>v} // edge only if u > v
u > v only if  foreach i u̸=v ∧ u[i]≥v[i]. // (u>v and v>w => u>w)

my naive code whit complexity O((v+1)*v/2) ≈ O(n^2) is

private void riempiAdj() {
    for(int i=0;i<nodi.length;i++)
        for(int j=i+1;j<nodi.length;j++)
            if(nodi[i].eMaggiore(nodi[j]))
                nodi[i].adj.inserisci(nodi[j]);     
}

nodi is the array of vertex

adj is the adjacency list

AdjList.inserisci(Vertex t) add a vertex t into adjacency list O(1)

Vertex.eMaggiore(Vertex t) returns true in this > t O(1)

exist an algorithm whit complexity of O(v) or O(v*log()v)?

Answer 1

Since the relation is transitive, in the worst case you have $\frac{|V|(|V|-1)}{2}$ edges and so, if you want to add all edges of the graph in the list, you cannot achieve better than $O(|V|^2)$

PS: This problem is similar to an Algorithm project in my University, what a coincidence! ( ͡° ͜ʖ ͡°)