Computing the Shortest Essential Cycle

Erickson, Jeff; Worah, Pratik

doi:10.1007/s00454-010-9241-8

Cited by 17 publications

(11 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This is the fastest algorithm known for arbitrary surface-embedded graphs; however, several faster algorithms are known when the genus g of the underlying surface is small [12,6,46,7,8]. For related results, see [10,11,13,32].…”

Section: Related Resultsmentioning

confidence: 99%

Minimum Cuts and Shortest Non-Separating Cycles via Homology Covers

Erickson¹,

Nayyeri²

2011

Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms

Self Cite

View full text Add to dashboard Cite

Let G be a directed graph with weighted edges, embedded on a surface of genus g. We describe an algorithm to compute a shortest directed cycle in G in any given 2 -homology class in 2 O(g) n log n time; this problem is NP-hard even for undirected graphs. We also present two applications of our algorithm. The first is an algorithm to compute a shortest nonseparating directed cycle in G in 2 O(g) n log n time, improving the recent algorithm of Cabello et al. [SOCG 2010] for all g = o(log n). The second is a combinatorial algorithm to compute minimum (s, t)-cuts in undirected surface graphs in 2 O(g) n log n time, improving on previous combinatorial algorithms, and in particular the recent of Chambers et al. [SOCG 2009], for all g = o(log n). Unlike earlier algorithms for surface graphs that construct and search finite portions of the universal cover, our algorithms use another canonical covering space, called the 2 -homology cover.

show abstract

Section: Related Resultsmentioning

confidence: 99%

Minimum Cuts and Shortest Non-Separating Cycles via Homology Covers

Erickson¹,

Nayyeri²

2011

Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms

Self Cite

View full text Add to dashboard Cite

show abstract

“…Cellular graph embeddings are equivalent to the combinatorial surfaces introduced by Colin de Verdière [17] and used by several other authors to formulate optimization problems for surface-embedded graphs [9,10,11,13,18,19,20,21,27,28,29,52]. A combinatorial surface S = (Σ, G) consists of an abstract surface Σ together with a cellularly embedded graph G with positively weighted edges.…”

Section: Graph Embeddingsmentioning

confidence: 99%

Minimum cuts and shortest homologous cycles

Chambers

Erickson

Nayyeri

2009

Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry

Self Cite

121

View full text Add to dashboard Cite

We describe the first algorithms to compute minimum cuts in surface-embedded graphs in near-linear time. Given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, our algorithm computes a minimum (s, t)-cut in g O(g) n log n time. Except for the special case of planar graphs, for which O(n log n)-time algorithms have been known for more than 20 years, the best previous time bounds for finding minimum cuts in embedded graphs follow from algorithms for general sparse graphs. A slight generalization of our minimum-cut algorithm computes a minimum-cost subgraph in every 2 -homology class. We also prove that finding a minimum-cost subgraph homologous to a single input cycle is NP-hard.

show abstract

“…All of these faster algorithms exploit the observation by Cabello and Mohar [12] that the shortest non-trivial cycle crosses any shortest path at most once. For related results and extensions, see [5,9,10,11,13,15,24]. Both Thomassen's 3-path condition [44] and Cabello and Mohar's crossing condition [12] are consequences of the following easy observation: For any four vertices s, t, u, v in an undirected surface graph, there is a shortest path from s to t and a shortest path from u to v that cross at most once.…”

Section: Introductionmentioning

confidence: 99%

Shortest non-trivial cycles in directed surface graphs

Erickson

2011

Proceedings of the Twenty-Seventh Annual Symposium on Computational Geometry

Self Cite

View full text Add to dashboard Cite

Let G be a directed graph embedded on a surface of genus g. We describe an algorithm to compute the shortest non-separating cycle in G in O(g 2 n log n) time, exactly matching the fastest algorithm known for undirected graphs. We also describe an algorithm to compute the shortest non-contractible cycle in G in g O(g) n log n time, matching the fastest algorithm for undirected graphs of constant genus.

show abstract

Computing the Shortest Essential Cycle

Cited by 17 publications

References 31 publications

Minimum Cuts and Shortest Non-Separating Cycles via Homology Covers

Minimum Cuts and Shortest Non-Separating Cycles via Homology Covers

Minimum cuts and shortest homologous cycles

Shortest non-trivial cycles in directed surface graphs

Contact Info

Product

Resources

About