找到有向图的中心

Question

I'm attempting to build a directed graph using JavaScript where there are N + 1 vertices (points) (numbered from 0 to N) and N directed edges between them. The graph network is connected; that is, ignoring the directions of edges, there is a link between each pair of vertices.

I need to find the center vertice that can be reached from all vertices on the graph, or decide that it is not there. The vertices are described by two arrays A and B of N integers each. For each integer K (0 <= K < N), there exists an edge from vertice A[K] to vertice B[K].

The function will be in this format:

function solution(A, B);

that, given two arrays A and B, returns the number of the vertice which is the center (the vertice that can be reached from all other vertices). If no such vertice exists, the function should return -1.

Given A = [1,2,3] and B = [0,0,0], the function should return 0. The center has the number 0 on the graph.

Given A = [0,1,2,4,5] and B = [2,3,3,3,2], the function should return 3. The center has the number 3 on the graph. From points 1,2 and 4, there is a direct link to point 3. From points 0 and 5, the edges to point 3 go through point 2.

Given A = [2,3,3,4] and B = [1,1,0,0], the function should return -1. There is no center on the graph.

Answer 1

The center vertice must have out-degree 0 . I feel like this sentence needs some mathematical proof, so here it goes:

Suppose the vertice c is the center. Suppose there is an edge from c to another vertice d . Given the definition of the center, there must be a path from d to c , hence there must be a cycle, and a connected graph with N + 1 vertices and N edges is the textbook definition of a tree, which contains no cycles, so such edge with origin in c cannot exist.

Obviously, all the other vertices must have out-degree bigger than 0 (since they must get to c somehow).

Knowing that, you can solve the problem in O(n) .

Just search for a single vertice with out-degree 0. If it doesn't exist or you find more than one, there is no answer, otherwise the answer is the only value you find.

A possible solution is:

function solution(A, B){
    var set = new Set();
    for(var i = 0; i <= N; i++) set.add(i);
    for(var i = 0; i < A.length; i++) set.delete(A[i]);
    if(set.size != 1) console.log("There is no answer");
    else console.log("The answer is " + set.values().next().value);
}