// Determines whether a digraph has an Eulerian cycle using necessary
// and sufficient conditions (without computing the cycle itself):
// - at least one edge
// - indegree(v) = outdegree(v) for every vertex
// - the graph is connected, when viewed as an undirected graph
// (ignoring isolated vertices)
private static boolean hasEulerianCycle(Digraph G) {
// Condition 0: at least 1 edge
if (G.E() == 0) return false;
// Condition 1: indegree(v) == outdegree(v) for every vertex
for (int v = 0; v < G.V(); v++) if (G.outdegree(v) != G.indegree(v)) return false;
// Condition 2: graph is connected, ignoring isolated vertices
Graph H = new Graph(G.V());
for (int v = 0; v < G.V(); v++) for (int w : G.adj(v)) H.addEdge(v, w);
// check that all non-isolated vertices are conneted
int s = nonIsolatedVertex(G);
BreadthFirstPaths bfs = new BreadthFirstPaths(H, s);
for (int v = 0; v < G.V(); v++) if (H.degree(v) > 0 && !bfs.hasPathTo(v)) return false;
return true;
}
/**
* Computes an Eulerian cycle in the specified digraph, if one exists.
*
* @param G the digraph
*/
public DirectedEulerianCycle(Digraph G) {
// must have at least one edge
if (G.E() == 0) return;
// necessary condition: indegree(v) = outdegree(v) for each vertex v
// (without this check, DFS might return a path instead of a cycle)
for (int v = 0; v < G.V(); v++) if (G.outdegree(v) != G.indegree(v)) return;
// create local view of adjacency lists, to iterate one vertex at a time
Iterator<Integer>[] adj = (Iterator<Integer>[]) new Iterator[G.V()];
for (int v = 0; v < G.V(); v++) adj[v] = G.adj(v).iterator();
// initialize stack with any non-isolated vertex
int s = nonIsolatedVertex(G);
Stack<Integer> stack = new Stack<Integer>();
stack.push(s);
// greedily add to putative cycle, depth-first search style
cycle = new Stack<Integer>();
while (!stack.isEmpty()) {
int v = stack.pop();
while (adj[v].hasNext()) {
stack.push(v);
v = adj[v].next();
}
// add vertex with no more leaving edges to cycle
cycle.push(v);
}
// check if all edges have been used
// (in case there are two or more vertex-disjoint Eulerian cycles)
if (cycle.size() != G.E() + 1) cycle = null;
assert certifySolution(G);
}
