Hierholzer’s Algorithm for directed graph

Given a directed Eulerian graph, print an Euler circuit. Euler circuit is a path that traverses every edge of a graph, and the path ends on the starting vertex. Examples:

Input : Adjacency list for the below graph

Output : 0 -> 1 -> 2 -> 0 

Input : Adjacency list for the below graph

Output : 0 -> 6 -> 4 -> 5 -> 0 -> 1 
         -> 2 -> 3 -> 4 -> 2 -> 0 
Explanation:
In both the cases, we can trace the Euler circuit 
by following the edges as indicated in the output.

We have discussed the problem of finding out whether a given graph is Eulerian or not. In this post, an algorithm to print the Eulerian trail or circuit is discussed. The same problem can be solved using Fleury’s Algorithm, however, its complexity is O(E*E). Using Hierholzer’s Algorithm, we can find the circuit/path in O(E), i.e., linear time. Below is the Algorithm: ref ( wiki ). Remember that a directed graph has a Eulerian cycle if the following conditions are true (1) All vertices with nonzero degrees belong to a single strongly connected component. (2) In degree and out-degree of every vertex is the same. The algorithm assumes that the given graph has a Eulerian Circuit.

• Choose any starting vertex v, and follow a trail of edges from that vertex until returning to v. It is not possible to get stuck at any vertex other than v, because indegree and outdegree of every vertex must be same, when the trail enters another vertex w there must be an unused edge leaving w. The tour formed in this way is a closed tour, but may not cover all the vertices and edges of the initial graph.
• As long as there exists a vertex u that belongs to the current tour, but that has adjacent edges not part of the tour, start another trail from u, following unused edges until returning to u, and join the tour formed in this way to the previous tour.

Thus the idea is to keep following unused edges and removing them until we get stuck. Once we get stuck, we backtrack to the nearest vertex in our current path that has unused edges, and we repeat the process until all the edges have been used. We can use another container to maintain the final path. Let’s take an example:

Let the initial directed graph be as below


Let's start our path from 0.
Thus, curr_path = {0} and circuit = {}
Now let's use the edge 0->1 

Now, curr_path = {0,1} and circuit = {}
similarly we reach up to 2 and then to 0 again as

Now, curr_path = {0,1,2} and circuit = {}
Then we go to 0, now since 0 haven't got any unused
edge we put 0 in circuit and back track till we find
an edge

We then have curr_path = {0,1,2} and circuit = {0}
Similarly, when we backtrack to 2, we don't find any 
unused edge. Hence put 2 in the circuit and backtrack 
again.

curr_path = {0,1} and circuit = {0,2}

After reaching 1 we go to through unused edge 1->3 and 
then 3->4, 4->1 until all edges have been traversed.

The contents of the two containers look as:
curr_path = {0,1,3,4,1} and circuit = {0,2} 

now as all edges have been used, the curr_path is 
popped one by one into the circuit.
Finally, we've circuit = {0,2,1,4,3,1,0}

We print the circuit in reverse to obtain the path followed.
i.e., 0->1->3->4->1->1->2->0

Below is the implementation for the above approach: 

0 -> 1 -> 2 -> 0
0 -> 6 -> 4 -> 5 -> 0 -> 1 -> 2 -> 3 -> 4 -> 2 -> 0

Time complexity : O(V+E), where V is the number of vertices in the graph and E is the number of edges. This is because the code uses a stack to store the vertices and the stack operations push and pop have a time complexity of O(1).

Space complexity : O(V+E), as the code uses a stack to store the vertices and a vector to store the circuit, both of which take up O(V) space. Additionally, the code also uses an unordered_map to store the count of edges for each vertex, which takes up O(E) space.

Alternate Implementation: Below are the improvements made from the above code 

The above code kept a count of the number of edges for every vertex. This is unnecessary since we are already maintaining the adjacency list. We simply deleted the creation of edge_count array. In the algorithm we replaced `if edge_count[current_v]` with `if adj[current_v]` 

The above code pushes the initial node twice to the stack. Though the way he coded the result is correct, this approach is confusing and inefficient. We eliminated this by appending the next vertex to the stack, instead of the current one. 

In the main part, where the author tests the algorithm, the initialization of the adjacency lists `adj1` and `adj2`were a little weird. That potion is also improved. 

0 -> 1 -> 2 -> 0
0 -> 6 -> 4 -> 5 -> 0 -> 1 -> 2 -> 3 -> 4 -> 2 -> 0

Time Complexity :  O(V + E), where V is the number of vertices and E is the number of edges in the graph. The reason for this is because the algorithm performs a depth-first search (DFS) and visits each vertex and each edge exactly once. So, for each vertex, it takes O(1) time to visit it and for each edge, it takes O(1) time to traverse it.

Space complexity  : O(V + E), as the algorithm uses a stack to store the current path and a list to store the final circuit. The maximum size of the stack can be V + E at worst, so the space complexity is O(V + E).

The article contains also inputs from Nitish Kumar.