---
title: Find an Eulerian Path with the Hierholzer Algorithm
date: 2024-03-13
pseudocode: true
slug: hierholzer-algorithm
draft: true
---
An Eulerian path of a graph is a path that passes once on every edge of the graph.

The Hierholzer algorithm allows to find this path.

The algorithm proceeds as follows:
1. Choose any vertex $v_0$ of the graph $G = (V, E)$. Starting from $v_0$, build a path $c$ that passes only once per edge until it reaches $v_0$ again.
2. If $c$ is an Eulerian cycle, we have found the Eulerian path, else
    - Ignore all edges from $c$.
    - For the first vertex $u$ from $c$ whose degree is not null, build an other cycle $c'$ that passes only once per edges, and does not pass in the previously visited edges.
    - Insert $c'$ in $c$ by replacing the starting node $u$.
    - Repeat the second step until we have an Eulerian path.
  

## Pseudocode
<pre class="pseudocode">
\begin{algorithm}
\caption{Hierholzer algorithm for Eulerian path finding}
\begin{algorithmic}
\Function{EulerianPath}{$G: (V, E)$} 
\State $c \gets$ an empty Doubly Linked List
\State $E' \gets E$
\State $v_0$ becomes the first vertex from $V$
\State $u \gets v_0$
\State \Comment{Find the first cycle}
\While{$\exists (u, v) \in E'$}
    \State $c$.insert($u$)
    \State $u \gets v$
    \State $E' \gets E' \setminus \{(u, v)\}$
    \If {$u = v_0$}
        \Break
    \EndIf
\EndWhile
\State \Comment{Add the other cycles}
\While {$c$.length $< |E|$}
\State \Comment{Find the next vertex with non null degree}
\State $v_0 \gets c$.value
\While {$\nexists (v_0, v) \in E'$}
\State $c \gets c$.next
\State $v_0 \gets c$.value
\State $u \gets v_0$
\EndWhile
\State \Comment{Insert the next cycle in $c$}
\State $c$.delete() \Comment{Ensure $v_0$ is not present twice there}
\While {$\exists (u, v) \in E'$}
    \State $c$.insert($u$)
    \State $u \gets v$
    \State $E' \gets E' \setminus \{(u, v)\}$
    \If {$u = v_0$}
    \Break
    \EndIf
\EndWhile
\EndWhile
\EndFunction
\end{algorithmic}
\end{algorithm}
</pre>

## Rust implantation

```rust
use std::collections::{HashMap, LinkedList};
use std::io::Error;

/// Eulerian path
/// 
/// Warning: this assumes the graph is Eulerian
fn eulerian_path(graph: HashMap<usize, Vec<usize>>) -> Option<LinkedList<usize>>
{
    let mut n_edges: usize = 0;
    for adj_list in graph.values() {
        n_edges += adj_list.len();
    }
    let mut graph = graph.clone();
    let mut cycle: LinkedList<usize> = LinkedList::new();
    let v_0: usize = *graph.keys().next().unwrap();
    let mut visited_edges: usize = 0;
    let mut u: usize = v_0;
    loop {
        cycle.push_back(u);
        let adj_list = graph.get_mut(&u).unwrap();
        visited_edges += 1;
        if adj_list.len() > 0 {
            let v = adj_list.pop().unwrap();
            u = v;
        } else {
            return None;
        }
        if u == v_0 {
            break;
        }
    }
    while visited_edges < n_edges {
        // Find the next vertex with non null degree
        let mut v_0 = cycle.front().unwrap().clone();
        while graph.get(&v_0).unwrap().len() == 0 {
            v_0 = cycle.front().unwrap().clone();
            // Move to the next vertex without removing it from the cycle, with rotation
            let v = cycle.pop_front().unwrap();
            cycle.push_back(v);
        }
        // Insert the next cycle in c
        cycle.pop_back();
        let mut u = v_0;
        loop {
            cycle.push_back(u);
            let adj_list = graph.get_mut(&u).unwrap();
            visited_edges += 1;
            if adj_list.len() > 0 {
                let v = adj_list.pop().unwrap();
                u = v;
            } else {
                return None;
            }
            if u == v_0 {
                break;
            }
        }
    }
    Some(cycle)
}
```



## References

- _Algorithmus von Hierholzer_ (de), Wikipedia,  <https://de.wikipedia.org/wiki/Algorithmus_von_Hierholzer>.
- _Eulerkreisproblem_ (de), Wikipedia, <https://de.wikipedia.org/wiki/Eulerkreisproblem>