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Homology flows, cohomology cuts
 ACM SYMPOSIUM ON THEORY OF COMPUTING
, 2009
"... We describe the first algorithms to compute maximum flows in surfaceembedded graphs in nearlinear time. Specifically, given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, we can compute a maximum (s, t)flow in O(g 7 n log 2 n log 2 C) time fo ..."
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Cited by 30 (10 self)
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We describe the first algorithms to compute maximum flows in surfaceembedded graphs in nearlinear time. Specifically, given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, we can compute a maximum (s, t)flow in O(g 7 n log 2 n log 2 C) time for integer capacities that sum to C, or in (g log n) O(g) n time for real capacities. Except for the special case of planar graphs, for which an O(n log n)time algorithm has been known for 20 years, the best previous time bounds for maximum flows in surfaceembedded graphs follow from algorithms for general sparse graphs. Our key insight is to optimize the relative homology class of the flow, rather than directly optimizing the flow itself. A dual formulation of our algorithm computes the minimumcost cycle or circulation in a given (real or integer) homology class.
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 8 (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.
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 ..."
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Cited by 5 (3 self)
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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
Annotating simplices with a homology basis and its applications
 In Proc. 13th Scandinavian Symp. and Workshop on Algorithm Theory
, 2012
"... Let K be a simplicial complex and g the rank of its pth homology group Hp(K) defined with Z2 coefficients. We show that we can compute a basis H of Hp(K) and annotate each psimplex of K with a binary vector of length g with the following property: the annotations, summed over all psimplices in an ..."
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Cited by 4 (3 self)
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Let K be a simplicial complex and g the rank of its pth homology group Hp(K) defined with Z2 coefficients. We show that we can compute a basis H of Hp(K) and annotate each psimplex of K with a binary vector of length g with the following property: the annotations, summed over all psimplices in any pcycle z, provide the coordinate vector of the homology class [z] in the basis H. The basis and the annotations for all simplices can be computed in O(n ω) time, where n is the size of K and ω < 2.376 is a quantity so that two n × n matrices can be multiplied in O(n ω) time. The precomputation of annotations permits answering queries about the independence or the triviality of pcycles efficiently. Using annotations of edges in 2complexes, we derive better algorithms for computing optimal basis and optimal homologous cycles in 1dimensional homology. Specifically, for computing an optimal basis of H1(K), we improve the time complexity known for the problem from O(n 4) to O(n ω + n 2 g ω−1). Here n denotes the size of the 2skeleton of K and g the rank of H1(K). Computing an optimal cycle homologous to a given 1cycle is NPhard even for surfaces and an algorithm taking 2 O(g) n log n time is known for surfaces. We extend this algorithm to work with arbitrary 2complexes in O(n ω) + 2 O(g) n 2 log n time using annotations. 1
An Efficient Computation of Handle and Tunnel Loops via Reeb Graphs
"... A special family of nontrivial loops on a surface called handle and tunnel loops associates closely to geometric features of “handles” and “tunnels” respectively in a 3D model. The identification of these handle and tunnel loops can benefit a broad range of applications from topology simplificatio ..."
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Cited by 2 (0 self)
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A special family of nontrivial loops on a surface called handle and tunnel loops associates closely to geometric features of “handles” and “tunnels” respectively in a 3D model. The identification of these handle and tunnel loops can benefit a broad range of applications from topology simplification / repair, and surface parameterization, to feature and shape recognition. Many of the existing efficient algorithms for computing nontrivial loops cannot be used to compute these special type of loops. The two algorithms known for computing handle and tunnel loops provably have a serious drawback that they both require a tessellation of the interior and exterior spaces bounded by the surface. Computing such a tessellation of three dimensional space around the surface is a nontrivial task and can be quite expensive. Furthermore, such a tessellation may need to refine the surface mesh, thus causing the undesirable sideeffect of outputting the loops on an altered surface mesh. In this paper, we present an efficient algorithm to compute a basis for handle and tunnel loops without requiring any 3D tessellation. This saves time considerably for large meshes making the algorithm scalable while computing the loops on the original input mesh and not on some refined version of it. We use the concept of the Reeb graph which together with several key theoretical insights on linking number provide an initial set of loops that provably constitute a handle and a tunnel basis. We further develop a novel strategy to tighten these handle and tunnel basis loops to make them geometrically relevant. We demonstrate the efficiency and effectiveness of our algorithm as well as show its robustness against noise, and other anomalies in the input.
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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Cited by 2 (0 self)
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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.
COMBINATORIAL OPTIMIZATION ON EMBEDDED CURVES
, 2012
"... We describe several algorithms for classifying, comparing and optimizing curves on surfaces. We give algorithms to compute the minimum member of a given homology class, particularly computing the maximum flow and minimum cuts, in surface embedded graphs. We describe approximation algorithms to compu ..."
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We describe several algorithms for classifying, comparing and optimizing curves on surfaces. We give algorithms to compute the minimum member of a given homology class, particularly computing the maximum flow and minimum cuts, in surface embedded graphs. We describe approximation algorithms to compute certain similarity measures for embedded curves on a surface. Finally, we present algorithms to solve computational problems for compactly presented curves. We describe the first algorithms to compute the shortest representative of a Z2homology class. Given a directed graph embedded on a surface of genus g with b boundary cycles, we can compute the shortest single cycle Z2homologous to a given even subgraph in 2O(g+b)n log n time. As a consequence we obtain an algorithm to compute the shortest directed nonseparating cycle in 2O(g)n log n time, which improves the previous best algorithm by a factor of O( p n) if the genus is
FAST ALGORITHMS FOR SURFACE EMBEDDED GRAPHS VIA HOMOLOGY
, 2013
"... We describe several results on combinatorial optimization problems for graphs where the input comes with an embedding on an orientable surface of small genus. While the specific techniques used differ between problems, all the algorithms we describe share one common feature in that they rely on the ..."
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We describe several results on combinatorial optimization problems for graphs where the input comes with an embedding on an orientable surface of small genus. While the specific techniques used differ between problems, all the algorithms we describe share one common feature in that they rely on the algebraic topology construct of homology. We describe algorithms to compute global minimum cuts and count minimum s, tcuts. We describe new algorithms to compute short cycles that are topologically nontrivial. Finally, we describe ongoing work in designing a new algorithm for computing maximum s, tflows in surface embedded graphs. We begin by describing an algorithm to compute global minimum cuts in edge weighted genus g graphs in gO(g)n log log n time. When the genus is a constant, our algorithm’s running time matches the best time bound known for planar graphs due to La̧cki and Sankowski. In our algorithm, we reduce to the problem of finding a minimum weight separating subgraph in the dual