Results 1  10
of
17
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.
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 ..."
Abstract

Cited by 10 (4 self)
 Add to MetaCart
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.
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.
Multiplesource shortest paths in embedded graphs
, 2012
"... Let G be a directed graph with n vertices and nonnegative weights in its directed edges, embedded on a surface of genus g, and let f be an arbitrary face of G. We describe an algorithm to preprocess the graph in O(gn log n) time, so that the shortestpath distance from any vertex on the boundary of ..."
Abstract

Cited by 7 (5 self)
 Add to MetaCart
Let G be a directed graph with n vertices and nonnegative weights in its directed edges, embedded on a surface of genus g, and let f be an arbitrary face of G. We describe an algorithm to preprocess the graph in O(gn log n) time, so that the shortestpath distance from any vertex on the boundary of f to any other vertex in G can be retrieved in O(log n) time. Our result directly generalizes the O(n log n)time algorithm of Klein [Multiplesource shortest paths in planar graphs. In Proc. 16th Ann. ACMSIAM Symp. Discrete Algorithms, 2005] for multiplesource shortest paths in planar graphs. Intuitively, our preprocessing algorithm maintains a shortestpath tree as its source point moves continuously around the boundary of f. As an application of our algorithm, we describe algorithms to compute a shortest noncontractible or nonseparating cycle in embedded, undirected graphs in O(g² n log n) time.
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 ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
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
Finding shortest contractible and shortest separating cycles in embedded graphs
 ACM Transactions on Algorithms
"... embedded graphs ∗ ..."
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 5 (3 self)
 Add to MetaCart
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.
Exact Distance Oracles for Planar Graphs
, 2010
"... We provide the first linearspace data structure with provable sublinear query time for exact pointtopoint shortest path queries in planar graphs. We prove that for any planar graph G with nonnegative arc lengths and for any ɛ> 0 there is a data structure that supports exact shortest path and dist ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
We provide the first linearspace data structure with provable sublinear query time for exact pointtopoint shortest path queries in planar graphs. We prove that for any planar graph G with nonnegative arc lengths and for any ɛ> 0 there is a data structure that supports exact shortest path and distance queries in G with the following properties: the data structure can be created in time O(n lg(n) lg(1/ɛ)), the space required is O(n lg(1/ɛ)), and the query time is O(n 1/2+ɛ). Previous data structures by Fakcharoenphol and Rao (JCSS’06), Klein, Mozes, and Weimann (TransAlg’10), and Mozes and WulffNilsen (ESA’10) with query time O(n 1/2 lg 2 n) use space at least Ω(n lg n / lg lg n). We also give a construction with a more general tradeoff. We prove that for any integer S ∈ [n lg n, n 2], we can construct in time Õ(S) a data structure of size O(S) that answers distance queries in O(nS −1/2 lg 2.5 n) time per query. Cabello (SODA’06) gave a comparable construction for the smaller range S ∈ [n 4/3 lg 1/3 n, n 2]. For the range S ∈ (n lg n, n 4/3 lg 1/3 n), only data structures of size O(S) with query time O(n 2 /S) had been known (Djidjev, WG’96). Combined, our results give the best query times for any shortestpath data structure for planar graphs with space S = o(n 4/3 lg 1/3 n). As a consequence, we also obtain an algorithm that computes k–many distances in planar graphs in time O((kn) 2/3 (lg n) 2 (lg lg n) −1/3 + n(lg n) 2 / lg lg n). 1
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.