Algorithms for enumeration of simple cycles in graphs.
Implementation Note: All the implementations work correctly
with loops but not with multiple duplicate edges.
Performance Notes:
The worst case time complexity of
the algorithms for finding cycles in directed graphs is:
- Tiernan - O(V.const^V)
- Tarjan - O(VEC)
- Johnson - O(((V+E)C)
- Szwarcfiter and Lauer - O(V+EC)
where V is the number of vertices, E is the number of edges and C is
the number of the simple cycles in the graph.
The worst case performance is
achieved for graphs with special structure, so on practical
workloads an algorithm with higher worst case complexity may
outperform an algorithm with lower worst case complexity. Note
also that "administrative costs" of algorithms with better worst
case performance are higher. Also higher is their memory cost
(which is in all cases O(V+E)).
The package author's workloads
contain typically several hundred nodes and from tens to several
thousand simple cycles. On these workloads the algorithms score
by performance (best to worst ) so :
- Szwarcfiter and Lauer
- Tarjan
- Johnson
- Tiernan
The worst case time complexity of the
Paton's algorithm for finding a cycle base in undirected graphs
is O(V^3)
Literature:
- J.C.Tiernan An Efficient Search Algorithm Find the
Elementary Circuits of a Graph., Communications of the ACM,
V13, 12, (1970), pp. 722 - 726.
- R.Tarjan, Depth-first search and linear graph
algorithms., SIAM J. Comput. 1 (1972), pp. 146-160.
- R. Tarjan, Enumeration of the elementary circuits of a
directed graph, SIAM J. Comput., 2 (1973), pp. 211-216.
- D.B.Johnson, Finding all the elementary circuits of a
directed graph, SIAM J. Comput., 4 (1975), pp. 77-84.
- J.L.Szwarcfiter and P.E.Lauer, Finding the elementary
cycles of a directed graph in O(n + m) per cycle, Technical
Report Series, #60, May 1974, Univ. of Newcastle upon Tyne,
Newcastle upon Tyne, England.
- P.Mateti and N.Deo, On algorithms for enumerating all
circuits of a graph., SIAM J. Comput., 5 (1978), pp. 90-99.
- L.G.Bezem and J.van Leeuwen, Enumeration in graphs.,
Technical report RUU-CS-87-7, University of Utrecht, The
Netherlands, 1987.
- K. Paton, An algorithm for finding a fundamental set of
cycles for an undirected linear graph, Comm. ACM 12 (1969),
pp. 514-518.