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27
OutputSensitive Algorithm for the EdgeWidth of an Embedded Graph
, 2010
"... Let G be an unweighted graph of complexity n cellularly embedded in a surface (orientable or not) of genus g. We describe improved algorithms to compute (the length of) a shortest noncontractible and a shortest nonseparating cycle of G. If k is an integer, we can compute such a nontrivial cycle w ..."
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Cited by 6 (1 self)
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Let G be an unweighted graph of complexity n cellularly embedded in a surface (orientable or not) of genus g. We describe improved algorithms to compute (the length of) a shortest noncontractible and a shortest nonseparating cycle of G. If k is an integer, we can compute such a nontrivial cycle with length at most k in O(gnk) time, or correctly report that no such cycle exists. In particular, on a fixed surface, we can test in linear time whether the edgewidth or facewidth of a graph is bounded from above by a constant. This also implies an outputsensitive algorithm to compute a shortest nontrivial cycle that runs in O(gnk) time, where k is the length of the 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 ..."
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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 tok plus the genus of the surface.
Maximum Flows and Parametric Shortest Paths in Planar Graphs
"... We observe that the classical maximum flow problem in any directed planar graph G can be reformulated as a parametric shortest path problem in the oriented dual graph G ∗. This reformulation immediately suggests an algorithm to compute maximum flows, which runs in O(n log n) time. As we continuously ..."
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We observe that the classical maximum flow problem in any directed planar graph G can be reformulated as a parametric shortest path problem in the oriented dual graph G ∗. This reformulation immediately suggests an algorithm to compute maximum flows, which runs in O(n log n) time. As we continuously increase the parameter, each change in the shortest path tree can be effected in O(log n) time using standard dynamic tree data structures, and the special structure of the parametrization implies that each directed edge enters the evolving shortest path tree at most once. The resulting maximumflow algorithm is identical to the recent algorithm of Borradaile and Klein [J. ACM 2009], but our new formulation allows a simpler presentation and analysis. On the other hand, we demonstrate that for a similarly structured parametric shortest path problem on the torus, the shortest path tree can change Ω(n²) times in the worst case, suggesting that a different method may be required to efficiently compute maximum flows in highergenus graphs.
Finding shortest contractible and shortest separating cycles in embedded graphs
 ACM Transactions on Algorithms
"... embedded graphs ∗ ..."
LinearSpace Approximate Distance Oracles for Planar, BoundedGenus, and MinorFree Graphs
"... Abstract. A (1 + ɛ)approximate distance oracle for a graph is a data structure that supports approximate pointtopoint shortestpathdistance queries. The relevant measures for a distanceoracle construction are: space, query time, and preprocessing time. There are strong distanceoracle construct ..."
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Cited by 5 (2 self)
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Abstract. A (1 + ɛ)approximate distance oracle for a graph is a data structure that supports approximate pointtopoint shortestpathdistance queries. The relevant measures for a distanceoracle construction are: space, query time, and preprocessing time. There are strong distanceoracle constructions known for planar graphs (Thorup) and, subsequently, minorexcluded graphs (Abraham and Gavoille). However, these require Ω(ɛ −1 n lg n) space for nnode graphs. We argue that a very low space requirement is essential. Since modern computer architectures involve hierarchical memory (caches, primary memory, secondary memory), a high memory requirement in effect may greatly increase the actual running time. Moreover, we would like data structures that can be deployed on small mobile devices, such as handhelds, which have relatively small primary memory. In this paper, for planar graphs, boundedgenus graphs, and minorexcluded graphs we give distanceoracle constructions that require only
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 ..."
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Cited by 4 (2 self)
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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.
Testing contractibility in planar Rips complexes
 In Proc. Symp. on Comp. Geom. (SoCG) 2008
"... The (Vietoris)Rips complex of a discrete pointset P is an abstract simplicial complex in which a subset of P defines a simplex if and only if the diameter of that subset is at most 1. We describe an efficient algorithm to determine whether a given cycle in a planar Rips complex is contractible. Ou ..."
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Cited by 3 (1 self)
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The (Vietoris)Rips complex of a discrete pointset P is an abstract simplicial complex in which a subset of P defines a simplex if and only if the diameter of that subset is at most 1. We describe an efficient algorithm to determine whether a given cycle in a planar Rips complex is contractible. Our algorithm requires O(m log n) time to preprocess a set of n points in the plane in which m pairs have distance at most 1; after preprocessing, deciding whether a cycle of k Rips edges is contractible requires O(k) time. We also describe an algorithm to compute the shortest noncontractible cycle in a planar Rips complex in O(n 2 log n + mn) time.
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 ..."
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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 log 3 n) time and succeeds with high probability, and a deterministic algorithm of Nagamochi and Ibaraki that runs in O(n 2 log n) time. We can also achieve a deterministic g O(g) n 2 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. 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. ..."
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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.
unknown title
"... Computational topology is an exciting new area which is emerging at the intersection of theoretical computer science and mathematics. Algorithmic techniques in topology are far from new, but the increasing use of computational geometry in many disciplines has led to an explosion of new results in bo ..."
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Computational topology is an exciting new area which is emerging at the intersection of theoretical computer science and mathematics. Algorithmic techniques in topology are far from new, but the increasing use of computational geometry in many disciplines has led to an explosion of new results in both computer science and mathematics. There is currently great demand for algorithms which can make topological guarantees, as well as great opportunity to apply algorithmic proofs in topology. Problems in computational topology are motivated by a wide variety of areas; applications abound in areas such as shape modeling in graphics, pathfinding in motion planning, geometric and topological modeling for protein docking prediction, encoding low dimensional spans for high dimensional data in statistical analysis, and modeling networks using cell complexes and meshes. Many topological questions are provably hard or even unsolvable in the most general settings, making it necessary to examine specific instances. My research focuses on several instances in this larger setting of computational topology. In each instance, the goal is to find an interesting topological feature, usually a cycle or path with certain desired properties. Combinatorial Surfaces