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Minimum Cuts and Shortest NonSeparating Cycles via Homology Covers
 SYMPOSIUM ON DISCRETE ALGORITHMS
, 2011
"... Let G be a directed graph with weighted edges, embedded on a surface of genus g with b boundaries. We describe an algorithm to compute the shortest directed cycle in G in any given � 2homology class in 2 O(g+b) n log n time; this problem is NPhard even for undirected graphs. We also present two ap ..."
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Cited by 10 (4 self)
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Let G be a directed graph with weighted edges, embedded on a surface of genus g with b boundaries. We describe an algorithm to compute the shortest directed cycle in G in any given � 2homology class in 2 O(g+b) n log n time; this problem is NPhard even for undirected graphs. We also present two applications of our algorithm. The first is an algorithm to compute the 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 an algorithm of Chambers et al. [SOCG 2009] for all positive g. 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 Z 2homology cover.
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.
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 3 (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
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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Cited by 2 (2 self)
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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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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.
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, ..."
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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.
Homology Annotations via Matrix Reduction
"... In this work, we propose an alternative algorithm for annotating simplices of a simplicial complex K with subbases of a basis B of its pdimensional homology group H1(K). Such annotations, summed over psimplices in any pcycle z, provide an expression of z in B. This allows us to answer queries of ..."
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In this work, we propose an alternative algorithm for annotating simplices of a simplicial complex K with subbases of a basis B of its pdimensional homology group H1(K). Such annotations, summed over psimplices in any pcycle z, provide an expression of z in B. This allows us to answer queries of null homology and independence of pcycles efficiently and improve the running time of the greedy algorithm to computea shortest basis of H1(K). The best known algorithm for the shortest basis problem that does not use annotations has a time complexity of O(n 4), where n is the size of the 2skeleton of K. We improve it to O(n 3 +n 2 g 2), where g is the rank of H1(K). Annotating simplices with a homology basis has been considered before [1]. The existing approachcan preprocess the simplicial complex and assign annotations in subcubictime. However, this involves computing the LSPdecomposition of the boundary matrix, which can be computationally cumbersome. We present a simple and implementationfriendly O(n 3) approach that fits nicely to the family of matrix reduction algorithms such as the persistence algorithm and the classic Smith normal form reduction. Our analysis also reveals interesting connections to the persistence algorithm. Namely, our matrix reduction method computes pairing between simplices under homomorphisms between homology groups that are not necessarily induced by inclusions between the subcomplexes of the filtration of K. 1.
The Complexity of Separating Points in the Plane
, 2013
"... We study the following separation problem: Given n connected curves and two points s and t in the plane, compute the minimum number of curves one needs to retain so that any path connecting s to t intersects some of the retained curves. We give the first polynomial (O(n 3)) time algorithm for the pr ..."
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We study the following separation problem: Given n connected curves and two points s and t in the plane, compute the minimum number of curves one needs to retain so that any path connecting s to t intersects some of the retained curves. We give the first polynomial (O(n 3)) time algorithm for the problem, assuming that the curves have reasonable computational properties. The algorithm is based on considering the intersection graph of the curves, defining an appropriate family of closed walks in the intersection graph that satisfies the 3pathcondition, and arguing that a shortest cycle in the family gives an optimal solution. The 3pathcondition has been used mainly in topological graph theory, and thus its use here makes the connection to topology clear. We also show that the generalized version, where several input points are to be separated, is NPhard for natural families of curves, like segments in two directions or unit circles. 1
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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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
MINIMUM CELL CONNECTION AND SEPARATION IN LINE SEGMENT ARRANGEMENTS Helmut Alt
"... Minimum cell connection and separation in line segment arrangements ∗ ..."