Results 1 
8 of
8
Finding shortest nontrivial cycles in directed graphs on surfaces
 In These Proceedings
, 2010
"... Let D be a weighted directed graph cellularly embedded in a surface of genus g, orientable or not, possibly with boundary. We describe algorithms to compute a shortest noncontractible and a shortest surface nonseparating cycle in D. This generalizes previous results that only dealt with undirected ..."
Abstract

Cited by 14 (3 self)
 Add to MetaCart
Let D be a weighted directed graph cellularly embedded in a surface of genus g, orientable or not, possibly with boundary. We describe algorithms to compute a shortest noncontractible and a shortest surface nonseparating cycle in D. This generalizes previous results that only dealt with undirected graphs. Our first algorithm computes such cycles in O(n 2 log n) time, where n is the total number of vertices and edges of D, thus matching the complexity of the best known algorithm in the undirected case. It revisits and extends Thomassen’s 3path condition; the technique applies to other families of cycles as well. We also give an algorithm with subquadratic complexity in the complexity of the input graph, if g is fixed. Specifically, we can solve the problem in O ( √ g n 3/2 log n) time, using a divideandconquer technique that simplifies the graph while preserving the topological properties of its cycles. A variant runs in O(ng log g + nlog 2 n) for graphs of bounded treewidth.
Shortest nontrivial cycles in directed surface graphs
 In Proc. 27th Ann. Symp. Comput. Geom
, 2011
"... Let G be a directed graph embedded on a surface of genus g. We describe an algorithm to compute the shortest nonseparating 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 noncontractible cy ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
(Show Context)
Let G be a directed graph embedded on a surface of genus g. We describe an algorithm to compute the shortest nonseparating 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 noncontractible cycle in G in g O(g) n log n time, matching the fastest algorithm for undirected graphs of constant genus.
Global Minimum Cuts in Surface Embedded Graphs
"... We give a deterministic algorithm to find the minimum cut in a surfaceembedded graph in nearlinear time. Given an undirected graph embedded on an orientable surface of genus g, our algorithm computes the minimum cut in g O(g) n log log n time, matching the running time of the fastest algorithm kno ..."
Abstract

Cited by 8 (5 self)
 Add to MetaCart
(Show Context)
We give a deterministic algorithm to find the minimum cut in a surfaceembedded graph in nearlinear time. Given an undirected graph embedded on an orientable surface of genus g, our algorithm computes the minimum cut in g O(g) n log log n time, matching the running time of the fastest algorithm known for planar graphs, due to Ł ˛acki and Sankowski, for any constant g. Indeed, our algorithm calls Ł ˛acki and Sankowski’s recent O(n log log n) time planar algorithm as a subroutine. Previously, the best time bounds known for this problem followed from two algorithms for general sparse graphs: a randomized algorithm of Karger that runs in O(n log³ n) time and succeeds with high probability, and a deterministic algorithm of Nagamochi and Ibaraki that runs in O(n² log n) time. We can also achieve a deterministic g O(g) n² log log n time bound by repeatedly applying the best known algorithm for minimum (s, t)cuts in surface graphs. The bulk of our work focuses on the case where the dual of the minimum cut splits the underlying surface into multiple components with positive genus.
Shortest Nontrivial Cycles in Directed and Undirected Surface Graphs
"... Let G be a graph embedded on a surface of genus g with b boundary cycles. We describe algorithms to compute multiple types of nontrivial cycles in G, using different techniques depending on whether or not G is an undirected graph. If G is undirected, then we give an algorithm to compute a shortest ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
Let G be a graph embedded on a surface of genus g with b boundary cycles. We describe algorithms to compute multiple types of nontrivial cycles in G, using different techniques depending on whether or not G is an undirected graph. If G is undirected, then we give an algorithm to compute a shortest nonseparating cycle in G in 2O(g) n log log n time. Similar algorithms are given to compute a shortest noncontractible or nonnullhomologous cycle in 2O(g+b) n log log n time. Our algorithms for undirected G combine an algorithm of Kutz with known techniques for efficiently enumerating homotopy classes of curves that may be shortest nontrivial cycles. Our main technical contributions in this work arise from assuming G is a directed graph with possibly asymmetric edge weights. For this case, we give an algorithm to compute a shortest noncontractible cycle in G in O((g 3 + g b)n log n) time. In order to achieve this time bound, we use a restriction of the infinite cyclic cover that may be useful in other contexts. We also describe an algorithm to compute a shortest nonnullhomologous cycle in G in O((g 2 + g b)n log n) time, extending a known algorithm of Erickson to compute a shortest nonseparating cycle. In both the undirected and directed cases, our algorithms improve the best time bounds known for many values of g and b. 1
On the homotopy test on surfaces
 in Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS
"... Let G be a graph cellularly embedded in a surface S. Given two closed walks c and d in G, we take advantage of the RAM model to describe linear time algorithms to decide if c and d are homotopic in S, either freely or with fixed basepoint. We restrict S to be orientable for the free homotopy test, b ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
Let G be a graph cellularly embedded in a surface S. Given two closed walks c and d in G, we take advantage of the RAM model to describe linear time algorithms to decide if c and d are homotopic in S, either freely or with fixed basepoint. We restrict S to be orientable for the free homotopy test, but allow nonorientable surfaces when the basepoint is fixed. After O(G) time preprocessing independent of c and d, our algorithms answer the homotopy test in O(c  + d) time, where G, c  and d  are the respective numbers of edges of G, c and d. As a byproduct we obtain linear time algorithms for the word problem and the conjugacy problem in surface groups. These results were previously announced by Dey and Guha (1999). Their approach was based on small cancellation theory from combinatorial group theory. However, several flaws in their algorithms make their approach fails, leaving the complexity of the homotopy test problem still open. We present a geometric approach, based on previous works by Colin de Verdière and Erickson, that provides optimal homotopy tests. 1
Faster shortest noncontractible cycles in directed surface graphs
 CoRR
"... Let G be a directed graph embedded on a surface of genus g with b boundary cycles. We describe an algorithm to compute the shortest noncontractible cycle in G in O((g 3 + g b)n log n) time. Our algorithm improves the previous best known time bound of (g + b) O(g+b) n log n for all positive g and b. ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Let G be a directed graph embedded on a surface of genus g with b boundary cycles. We describe an algorithm to compute the shortest noncontractible cycle in G in O((g 3 + g b)n log n) time. Our algorithm improves the previous best known time bound of (g + b) O(g+b) n log n for all positive g and b. We also describe an algorithm to compute the shortest nonnullhomologous cycle in G in O((g 2 + g b)n log n) time, generalizing a known algorithm to compute the shortest nonseparating cycle.
Finding cycles with topological properties in embedded graphs∗
, 2010
"... Let G be a graph cellularly embedded on a surface. We consider the problem of determining whether G contains a cycle (i.e. a closed walk without repeated vertices) of a certain topological type. We show that the problem can be answered in linear time when the topological type is one of the following ..."
Abstract
 Add to MetaCart
Let G be a graph cellularly embedded on a surface. We consider the problem of determining whether G contains a cycle (i.e. a closed walk without repeated vertices) of a certain topological type. We show that the problem can be answered in linear time when the topological type is one of the following: contractible, noncontractible, or nonseparating. In either case we obtain the same time complexity if we require the cycle to contain a given vertex. On the other hand, we prove that the problem is NPcomplete when considering separating or splitting cycles. We also show that deciding the existence of a separating or a splitting cycle of length at most k is fixedparameter tractable with respect to k plus the genus of the surface. 1
Shortest Cut Graph of a Surface with Prescribed Vertex Set
, 2010
"... We describe a simple greedy algorithm whose input is a set P of vertices on a combinatorial surface S without boundary and that computes a shortest cut graph of S with vertex set P. (A cut graph is an embedded graph whose removal leaves a single topological disk.) If S has genus g and complexity n, ..."
Abstract
 Add to MetaCart
We describe a simple greedy algorithm whose input is a set P of vertices on a combinatorial surface S without boundary and that computes a shortest cut graph of S with vertex set P. (A cut graph is an embedded graph whose removal leaves a single topological disk.) If S has genus g and complexity n, the runningtime is O(nlog n+(g + P )n). This is an extension of an algorithm by Erickson and Whittlesey [Proc. ACMSIAM Symp. on Discrete Algorithms, 1038–1046 (2005)], which computes a shortest cut graph with a single given vertex. Moreover, our proof is simpler and also reveals that the algorithm actually computes a minimumweight basis of some matroid.