The.Algorithm.Design.Manual.Springer-Verlag.1998

Eulerian Cycle / Chinese Postman
are of even degree. These two vertices will be the start and end points of the path.
● A directed graph contains an Eulerian cycle iff (1) it is connected and (2) each vertex has the
same in-degree as out-degree.
● Finally, a directed graph contains an Eulerian path iff (1) it is connected and (2) all but two
vertices have the same in-degree as out-degree, and these two vertices have their in-degree and
out-degree differ by one.
Given this characterization of Eulerian graphs, it is easy to test in linear time whether such a cycle exists:
test whether the graph is connected using DFS or BFS, and then count the number of odd-degree
vertices. Actually constructing such a cycle also takes linear time. Use DFS to find a cycle in the graph.
Delete this cycle and repeat until the entire set of edges has been partitioned into a set of edge-disjoint
cycles. Since deleting a cycle reduces each vertex degree by an even number, the remaining graph will
continue to satisfy the same Eulerian degree-bound conditions. For any connected graph, these cycles
will have common vertices, and so by splicing these cycles in a ``figure eight'' at a shared vertex, we can
construct a single circuit containing all of the edges.
An Eulerian cycle, if one exists, solves the motivating snowplow problem, since any tour that visits each
edge only once must have minimum length. However, it is unlikely that any real road network would
happen to satisfy the degree conditions that make it Eulerian. We need to solve the more general
Chinese postman problem, which minimizes the length of a cycle that traverses every edge at least once.
In fact, it can be shown that this minimum cycle never visits any edge more than twice, so good tours
exist for any road network.
The optimal postman tour can be constructed by adding the appropriate edges to the graph G so as to
make it Eulerian. Specifically, we find the shortest path between each pair of odd-degree vertices in G.
Adding a path between two odd-degree vertices in G turns both of them to even-degree, thus moving us
closer to an Eulerian graph. Finding the best set of shortest paths to add to G reduces to identifying a
minimum-weight perfect matching in a graph on the odd-degree vertices, where the weight of edge (i,j) is
the length of the shortest path from i to j. For directed graphs, this can be solved using bipartite matching,
where the vertices are partitioned depending on whether they have more ingoing or outgoing edges. Once
the graph is Eulerian, the actual cycle can be extracted in linear time using the procedure described
above.
Implementations: Unfortunately, we have not able to identify a suitable Chinese postman
implementation. However, it should not be difficult for you to roll your own by using a matching code
from Section and all-pairs shortest path from Section . Matching is by far the hardest part of the
algorithm.
Combinatorica [Ski90] provides Mathematica implementations of Eulerian cycles and de Bruijn
sequences. See Section .
file:///E|/BOOK/BOOK4/NODE165.HTM (2 of 4) [19/1/2003 1:31:06]