Fleury’s Algorithm: Finding Eulerian tours in a graph

• Algorithm
• Complexity
• Pseudocode
• Problems
• Applications
• Discussions

Reading time: 10 minutes | Coding time: 12 minutes

Fleury's algorithm is a simple algorithm for finding Eulerian paths or tours. It proceeds by repeatedly removing edges from the graph in such way, that the graph remains Eulerian.

The steps of Fleury's algorithm is as follows:

• Start with any vertex of non-zero degree.
• Choose any edge leaving this vertex, which is not a bridge (cut edges).
• If there is no such edge, stop.
• Otherwise, append the edge to the Euler tour, remove it from the graph, and repeat the process starting with the other endpoint of this edge.

Complexity

• Worst case time complexity: Θ((V+E)^2)
• Average case time complexity: Θ((V+E)^2)
• Best case time complexity: Θ((V+E)^2)
• Space complexity: Θ(V^2)

Pseudocode

vector E
dfs (v):
        color[v] = gray
        for u in adj[v]:
                erase the edge v-u and dfs(u)
        color[v] = black
        push v at the end of e

Problems

• CodeForces: Strongly Connected City
• CodeForces: New Year Santa Network

Applications

• Find Eulerian path in a graph