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14
Succinct Representation of Labeled Graphs
 ALGORITHMICA
, 2010
"... In many applications, the properties of an object being modeled are stored as labels on vertices or edges of a graph. In this paper, we consider succinct representation of labeled graphs. Our main results are the succinct representations of labeled and multilabeled graphs (we consider planar triang ..."
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Cited by 10 (2 self)
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In many applications, the properties of an object being modeled are stored as labels on vertices or edges of a graph. In this paper, we consider succinct representation of labeled graphs. Our main results are the succinct representations of labeled and multilabeled graphs (we consider planar triangulations, planar graphs and kpage graphs) to support various label queries efficiently. The additional space cost to store the labels is essentially the informationtheoretic minimum. As far as we know, our representations are the first succinct representations of labeled graphs. We also have two preliminary results to achieve the main contribution. First, we design a succinct representation of unlabeled planar triangulations to support the rank/select of edges in ccw (counter clockwise) order in addition to the other operations supported in previous work. Second, we design a succinct representation for a kpage
Schnyder Woods for Higher Genus Triangulated Surfaces
 SCG'08
, 2008
"... Schnyder woods are a well known combinatorial structure for planar graphs, which yields a decomposition into 3 vertexspanning trees. Our goal is to extend definitions and algorithms for Schnyder woods designed for planar graphs (corresponding to combinatorial surfaces with the topology of the sphere ..."
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Cited by 4 (2 self)
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Schnyder woods are a well known combinatorial structure for planar graphs, which yields a decomposition into 3 vertexspanning trees. Our goal is to extend definitions and algorithms for Schnyder woods designed for planar graphs (corresponding to combinatorial surfaces with the topology of the sphere, i.e., of genus 0) to the more general case of graphs embedded on surfaces of arbitrary genus. First, we define a new traversal order of the vertices of a triangulated surface of genus g together with an orientation and coloration of the edges that extends the one proposed by Schnyder for the planar case. As a byproduct we show how some recent schemes for compression and compact encoding of graphs can be extended to higher genus. All the algorithms presented here have linear time complexity.
LR: compact connectivity representation for triangle meshes
 in ACM SIGGRAPH 2011 papers, SIGGRAPH ’11
, 2011
"... Figure 1: The ring (black loop) delineates two corridors of triangles. Normal T1 triangles (cream/orange) have one ring edge, deadend T2 triangles (blue) have two ring edges, and T0 triangles (green) comprising bifurcations have no ring edges. Adjacent T0 (gray/red) and T2 triangles (left) are repr ..."
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Cited by 4 (2 self)
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Figure 1: The ring (black loop) delineates two corridors of triangles. Normal T1 triangles (cream/orange) have one ring edge, deadend T2 triangles (blue) have two ring edges, and T0 triangles (green) comprising bifurcations have no ring edges. Adjacent T0 (gray/red) and T2 triangles (left) are represented internally as inexpensive T1 triangles (right), thereby significantly reducing storage. Our LR representation supports random access to connectivity, storing on average only 1.08 references or 26.2 bits per triangle. We propose LR (Laced Ring)—a simple data structure for representing the connectivity of manifold triangle meshes. LR provides the option to store on average either 1.08 references per triangle or 26.2 bits per triangle. Its construction, from an input mesh that supports constanttime adjacency queries, has linear space and time complexity, and involves ordering most vertices along a nearlyHamiltonian cycle. LR is best suited for applications that process meshes with fixed connectivity, as any changes to the connectivity require the data structure to be rebuilt. We provide an implementation of the set of standard randomaccess, constanttime operators for traversing a mesh, and show that LR often saves both space and traversal time over competing representations.
Vertices of Degree k in Random Maps ∗
"... This work is devoted to the study of the typical structure of a random map. Maps are planar graphs embedded in the plane. We investigate the degree sequences of random maps from families of a certain type, which, among others, includes fundamental map classes like those of biconnected maps, 3connec ..."
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This work is devoted to the study of the typical structure of a random map. Maps are planar graphs embedded in the plane. We investigate the degree sequences of random maps from families of a certain type, which, among others, includes fundamental map classes like those of biconnected maps, 3connected maps, and triangulations. In particular, we develop a general framework that allows us to derive relations and exact asymptotic expressions for the expected number of vertices of degree k in random maps from these classes, and also provide accompanying large deviation statements. Extending the work of Gao and Wormald (Combinatorica, 2003) on random general maps, we obtain as results of our framework precise information about the number of vertices of degree k in random biconnected, 3connected, loopless, and bridgeless maps. 1
Succinct Geometric Indexes Supporting Point Location Queries
"... We propose to design data structures called succinct geometric indexes of negligible space (more precisely, o(n) bits) that support geometric queries in optimal time, by taking advantage of the n points in the data set permuted and stored elsewhere as a sequence. Our first and main result is a succi ..."
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We propose to design data structures called succinct geometric indexes of negligible space (more precisely, o(n) bits) that support geometric queries in optimal time, by taking advantage of the n points in the data set permuted and stored elsewhere as a sequence. Our first and main result is a succinct geometric index that can answer point location queries, a fundamental problem in computational geometry, on planar triangulations in O(lg n) time1. We also design three variants of this index. The first supports point location using lg n +2 √ lg n + O(lg 1/4 n) pointline comparisons. The second supports point location in o(lg n) time when the coordinates are integers bounded by U. The last variant can answer point location queries in O(H +1) expected time, where H is the entropy of the query distribution. These results match the query efficiency of previous point location structures that occupy O(n) words or O(n lg n) bits, while saving drastic amounts of space. We generalize our succinct geometric index to planar subdivisions, and design indexes for other types of queries. Finally, we apply our techniques to design the first implicit data structures that support point location in O(lg 2 n) time. 1
InPlace 2d Nearest Neighbor Search
, 2007
"... Abstract We revisit a classic problem in computational geometry: preprocessing a planar npoint set to answer nearest neighbor queries. In SoCG 2004, Br"onnimann, Chan, and Chen showed that it is possible to design an efficient data structure that takes no extra space at all other than the inpu ..."
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Abstract We revisit a classic problem in computational geometry: preprocessing a planar npoint set to answer nearest neighbor queries. In SoCG 2004, Br"onnimann, Chan, and Chen showed that it is possible to design an efficient data structure that takes no extra space at all other than the input array holding a permutation of the points. The best query time known for such "inplace data structures " is O(log 2 n). In this paper, we break the O(log 2 n) barrier by providing a method that answers nearest neighbor queries in time O((log n) log3=2 2 log log n) = O(log
CATALOGBASED REPRESENTATION OF 2D TRIANGULATIONS
 INTERNATIONAL JOURNAL OF COMPUTATIONAL GEOMETRY & APPLICATIONS
, 2011
"... Several Representations and Coding schemes have been proposed to represent efficiently 2D triangulations. In this paper we propose a new practical approach to reduce the main memory space needed to represent an arbitrary triangulation, while maintaining constant time for some basic queries. This wor ..."
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Several Representations and Coding schemes have been proposed to represent efficiently 2D triangulations. In this paper we propose a new practical approach to reduce the main memory space needed to represent an arbitrary triangulation, while maintaining constant time for some basic queries. This work focuses on the connectivity information of the triangulation, rather than the geometric information (vertex coordinates), since the combinatorial data represents the main part of the storage. The main idea is to gather triangles into patches, to reduce the number of pointers by eliminating the internal pointers in the patches and reducing the multiple references to vertices. To accomplish this, we define and use stable catalogs of patches that are closed under basic standard update operations such as insertion and deletion of vertices, and edge flips. We present some bounds and results concerning special catalogs, and some experimental results that exhibits the practical gain of such methods.
ESQ: Editable SQuad representation for triangle meshes
"... Abstract—We consider the problem of designing space efficient solutions for representing the connectivity information of manifold triangle meshes. Most mesh data structures are quite redundant, storing a large amount of information in order to efficiently support mesh traversal operators. Several co ..."
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Abstract—We consider the problem of designing space efficient solutions for representing the connectivity information of manifold triangle meshes. Most mesh data structures are quite redundant, storing a large amount of information in order to efficiently support mesh traversal operators. Several compact data structures have been proposed to reduce storage cost while supporting constanttime mesh traversal. Some recent solutions are based on a global reordering approach, which allows to implicitly encode a map between vertices and faces. Unfortunately, these compact representations do not support efficient updates, because local connectivity changes (such as edgecontractions, edgeflips or vertex insertions) require reordering the entire mesh. Our main contribution is to propose a new way of designing compact data structures which can be dynamically maintained. In our solution, we push further the limits of the reordering approaches: the main novelty is to allow to reorder vertex data (such as vertex coordinates), and to exploit this vertex permutation to easily maintain the connectivity under local changes. We describe a new class of data structures, called Editable SQuad (ESQ), offering the same navigational and storage performance as previous works, while supporting local editing in amortized constant time. As far as we know, our solution provides the most compact dynamic data structure for triangle meshes. We propose a lineartime and linearspace construction algorithm, and provide worstcase bounds for storage and time cost. Keywordstriangle meshes; compact representations; dynamic data structures; I.