Results 1  10
of
23
Multiple source shortest paths in a genus g graph
 Proc. 18th Ann. ACMSIAM Symp. Discrete Algorithms
"... We give an O(g2n log n) algorithm to represent the shortest path tree from all the vertices on a single specified face f in a genus g graph. From this representation, any query distance from a vertex in f can be obtained in O(log n) time. The algorithm uses a kinetic data structure, where the source ..."
Abstract

Cited by 34 (13 self)
 Add to MetaCart
We give an O(g2n log n) algorithm to represent the shortest path tree from all the vertices on a single specified face f in a genus g graph. From this representation, any query distance from a vertex in f can be obtained in O(log n) time. The algorithm uses a kinetic data structure, where the source of the tree iteratively movesacrossedgesinf. In addition, we give applications using these shortest path trees in order to compute the shortest noncontractible cycle and the shortest nonseparating cycle embedded on an orientable 2manifold in O(g3n log n) time. 1
Splitting (complicated) surfaces is hard
 COMPUT. GEOM. THEORY APPL
, 2006
"... Let M be an orientable surface without boundary. A cycle on M is splitting if it has no selfintersections and it partitions M into two components, neither homeomorphic to a disk. In other words, splitting cycles are simple, separating, and noncontractible. We prove that finding the shortest splitt ..."
Abstract

Cited by 24 (10 self)
 Add to MetaCart
Let M be an orientable surface without boundary. A cycle on M is splitting if it has no selfintersections and it partitions M into two components, neither homeomorphic to a disk. In other words, splitting cycles are simple, separating, and noncontractible. We prove that finding the shortest splitting cycle on a combinatorial surface is NPhard but fixedparameter tractable with respect to the surface genus. Specifically, we describe an algorithm to compute the shortest splitting cycle in g^O(g) n log n time.
Minimum Cuts and Shortest Homologous Cycles
 SYMPOSIUM ON COMPUTATIONAL GEOMETRY
, 2009
"... We describe the first algorithms to compute minimum cuts in surfaceembedded graphs in nearlinear 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 spec ..."
Abstract

Cited by 20 (7 self)
 Add to MetaCart
We describe the first algorithms to compute minimum cuts in surfaceembedded graphs in nearlinear 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 minimumcut algorithm computes a minimumcost subgraph in every Z2homology class. We also prove that finding a minimumcost subgraph homologous to a single input cycle is NPhard.
Localized homology
 In Shape Modeling International
, 2007
"... In this paper, we provide the theoretical foundation and an effective algorithm for localizing topological attributes such as tunnels and voids. Unlike previous work that focused on 2manifolds with restricted geometry, our theory is general and localizes arbitrarydimensional attributes in arbitrar ..."
Abstract

Cited by 15 (3 self)
 Add to MetaCart
In this paper, we provide the theoretical foundation and an effective algorithm for localizing topological attributes such as tunnels and voids. Unlike previous work that focused on 2manifolds with restricted geometry, our theory is general and localizes arbitrarydimensional attributes in arbitrary spaces. We implement our algorithm to validate our approach in practice. 1
Computing the shortest essential cycle
, 2008
"... An essential cycle on a surface is a simple cycle that cannot be continuously deformed to a point or a single boundary. We describe algorithms to compute the shortest essential cycle in an orientable combinatorial surface in O(n 2 log n) time, or in O(n log n) time when both the genus and number of ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
An essential cycle on a surface is a simple cycle that cannot be continuously deformed to a point or a single boundary. We describe algorithms to compute the shortest essential cycle in an orientable combinatorial surface in O(n 2 log n) time, or in O(n log n) time when both the genus and number of boundaries are fixed. Our result corrects an error in a paper of Erickson and HarPeled.
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 ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
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.
Cuttings for Disks and AxisAligned Rectangles in ThreeSpace ∗
"... We present new asymptotically tight bounds on cuttings, a fundamental data structure in computational geometry. For n objects in space and a parameter r ∈ N, an 1 rcutting is a covering of the space with simplices such that the interior of each simplex intersects at most n/rcutting of objects. For ..."
Abstract
 Add to MetaCart
We present new asymptotically tight bounds on cuttings, a fundamental data structure in computational geometry. For n objects in space and a parameter r ∈ N, an 1 rcutting is a covering of the space with simplices such that the interior of each simplex intersects at most n/rcutting of objects. For n pairwise disjoint disks in R3 and a parameter r ∈ N, we construct a 1 r size O(r2). For n axisaligned rectangles in R3, we construct a 1 rcutting of size O(r3/2). As an application related to multipoint location in threespace, we present tight bounds on the cost of spanning trees across barriers. Given n points and a finite set of disjoint disk barriers in R 3, the points can be connected with a straight line spanning tree such that every disk is stabbed by at most O ( √ n) edges of the tree. If the barriers are axisaligned rectangles, then there is a straight line spanning tree such that every rectangle is stabbed by O(n 1/3) edges. Both bounds are best possible. 1
Mesh Enhanced Persistent Homology ∗
, 2009
"... We apply ideas from mesh generation to improve the time and space complexity of computing the persistent homology of a point set in R d. The traditional approach to persistence starts with the αcomplex of the point set and thus incurs the O(n ⌊d/2 ⌋) size of the Delaunay triangulation. The common a ..."
Abstract
 Add to MetaCart
We apply ideas from mesh generation to improve the time and space complexity of computing the persistent homology of a point set in R d. The traditional approach to persistence starts with the αcomplex of the point set and thus incurs the O(n ⌊d/2 ⌋) size of the Delaunay triangulation. The common alternative is to use a Rips complex and then to truncate the filtration when the size of the complex becomes prohibitive, possibly before discovering relevant topological features. Given a point set P of n points in R d, we construct a new filtration, the αmesh, of size O(n) in time O(n 2) with persistent homology approximately the same as that of the αshape filtration. This makes it possible to compute the complete persistence barcode in O(n 3) time, where n is the number of points. Previously, this bound was only achievable (with exponentially worse constants) for computing partial barcodes from uniform samples from manifolds. The constants in this paper are all singly exponential in d, making them suitable for medium dimensions. 1
Recursive Geometry of the Flow Complex and Topology of the Flow Complex Filtration ⋆
"... The flow complex is a geometric structure, similar to the Delaunay tessellation, to organize a set of (weighted) points in R k. Flow shapes are topological spaces corresponding to substructures of the flow complex. The flow complex and flow shapes have found applications in surface reconstruction, s ..."
Abstract
 Add to MetaCart
The flow complex is a geometric structure, similar to the Delaunay tessellation, to organize a set of (weighted) points in R k. Flow shapes are topological spaces corresponding to substructures of the flow complex. The flow complex and flow shapes have found applications in surface reconstruction, shape matching, and molecular modeling. In this article we give an algorithm for computing the flow complex of weighted points in any dimension. The algorithm reflects the recursive structure of the flow complex. On the basis of the algorithm we establish a topological similarity between flow shapes and the nerve of a corresponding ball set, namely homotopy equivalence.