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J.I.: Succinct representation of labeled graphs
 In: Proceedings of the 18th International Symposium on Algorithms and Computation. LNCS
, 2007
"... Abstract. 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 ver ..."
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Cited by 12 (3 self)
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Abstract. 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 vertex labeled planar triangulations, as well as edge labeled planar graphs and the more general 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 results. 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 graph when k is large to support various navigational operations more efficiently. In particular, we can test the adjacency of two vertices in O(lg k lg lg k) time, while previous work uses O(k) time (10; 14). 1
On Compact Encoding of Pagenumber k Graphs
, 2001
"... this paper we show an informationtheoretic lower bound of kn o(kn) on the minimum number of bits to represent an unlabeled connected nnode graph of pagenumber k. This has to be compared with the efficient encoding scheme of Munro and Raman of 2kn +2m+o(n) bits, that is 4kn +2n k ..."
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Cited by 9 (1 self)
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this paper we show an informationtheoretic lower bound of kn o(kn) on the minimum number of bits to represent an unlabeled connected nnode graph of pagenumber k. This has to be compared with the efficient encoding scheme of Munro and Raman of 2kn +2m+o(n) bits, that is 4kn +2n k
Schnyder woods and orthogonal surfaces
 In Proceedings of Graph Drawing
, 2006
"... In this paper we study connections between planar graphs, Schnyder woods, and orthogonal surfaces. Schnyder woods and the face counting approach have important applications in graph drawing and the dimension theory of orders. Orthogonal surfaces explain connections between these seemingly unrelated ..."
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Cited by 7 (3 self)
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In this paper we study connections between planar graphs, Schnyder woods, and orthogonal surfaces. Schnyder woods and the face counting approach have important applications in graph drawing and the dimension theory of orders. Orthogonal surfaces explain connections between these seemingly unrelated notions. We use these connections for an intuitive proof of the BrightwellTrotter Theorem which says, that the face lattice of a 3polytope minus one face has order dimension three. Our proof yields a linear time algorithm for the construction of the three linear orders that realize the face lattice. Coplanar orthogonal surfaces are in correspondence with a large class of convex straight line drawings of 3connected planar graphs. We show that Schnyder’s face counting approach with weighted faces can be used to construct all coplanar orthogonal surfaces and hence the corresponding drawings. Appropriate weights are computable in linear time. 1
IMPROVED COMPACT VISIBILITY REPRESENTATION OF Planar Graph via Schnyder’s Realizer
 SIAM J. DISCRETE MATH. C ○ 2004 SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS VOL. 18, NO. 1, PP. 19–29
, 2004
"... Let G be an nnode planar graph. In a visibility representation of G,eachnodeofG is represented by a horizontal line segment such that the line segments representing any two adjacent nodes of G are vertically visible to each other. In the present paper we give the best known compact visibility repre ..."
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Cited by 6 (1 self)
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Let G be an nnode planar graph. In a visibility representation of G,eachnodeofG is represented by a horizontal line segment such that the line segments representing any two adjacent nodes of G are vertically visible to each other. In the present paper we give the best known compact visibility representation of G. Given a canonical ordering of the triangulated G, our algorithm draws the graph incrementally in a greedy manner. We show that one of three canonical orderings obtained �from Schnyder’s � realizer for the triangulated G yields a visibility representation of G no wider than 22n−40. Our easytoimplement O(n)time algorithm bypasses the complicated subroutines for 15 fourconnected components and fourblock trees required by the best previously known algorithm of Kant. Our result provides a negative answer to Kant’s open question about whether � � 3n−6 is a 2 worstcase lower bound on the required width. Also, if G has no degreethree (respectively, degreefive) internal node, then our visibility representation for G is no wider than � �
FloorPlanning via Orderly Spanning Trees
, 2001
"... Floorplanning is a fundamental step in VLSI chip design. Based upon the concept of orderly spanning trees, we present a simple O(n)time algorithm to construct a oorplan for any nnode plane triangulation. In comparison with previous oorplanning algorithms in the iterature, our solution is not on ..."
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Cited by 4 (4 self)
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Floorplanning is a fundamental step in VLSI chip design. Based upon the concept of orderly spanning trees, we present a simple O(n)time algorithm to construct a oorplan for any nnode plane triangulation. In comparison with previous oorplanning algorithms in the iterature, our solution is not only simpler in the algorithm itself, but also produces oorplans which require fewer module types. An equally important aspect of our new algorithm lies in its ability to fit the floorplan area in a rectangle of size (n1) by (2n+1)/3.
An Information Upper Bound of Planar Graphs Using Triangulation
, 2002
"... We propose a new linear time algorithm to represent a planar graph. Based on a specific triangulation of the graph, our coding takes on average 5.03 bits per node, and 3.37 bits per node if the graph is maximal. We derive from this representation that the number of unlabeled planar graphs with n ..."
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Cited by 4 (4 self)
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We propose a new linear time algorithm to represent a planar graph. Based on a specific triangulation of the graph, our coding takes on average 5.03 bits per node, and 3.37 bits per node if the graph is maximal. We derive from this representation that the number of unlabeled planar graphs with n nodes is at most 2 n+O(log n) , where 5.007. The current lower
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.
Compression and Streaming of Polygon Meshes
, 2005
"... Polygon meshes provide a simple way to represent threedimensional surfaces and are the defacto standard for interactive visualization of geometric models. Storing large polygon meshes in standard indexed formats results in files of substantial size. Such formats allow listing vertices and polygons ..."
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Cited by 3 (0 self)
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Polygon meshes provide a simple way to represent threedimensional surfaces and are the defacto standard for interactive visualization of geometric models. Storing large polygon meshes in standard indexed formats results in files of substantial size. Such formats allow listing vertices and polygons in any order so that not only the mesh is stored but also the particular ordering of its elements. Mesh compression rearranges vertices and polygons into an order that allows more compact coding of the incidence between vertices and predictive compression of their positions. Previous schemes were designed for triangle meshes and polygonal faces were triangulated prior to compression. I show that polygon models can be encoded more compactly by avoiding the initial triangulation step. I describe two compression schemes that achieve better compression by encoding meshes directly in their polygonal representation. I demonstrate that the
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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Cited by 2 (1 self)
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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
Compactly Encoding and Decoding the Connectivity of a Plane Pseudograph in Linear Time
"... A plane pseudograph is a plane graph allowing both loops and multiple edges. The encoding technique we propose belongs to a class of compression methods based on a deterministic graph traversal, which is encoded as a bit string. Arrays of vertices and edges stored in the order defined by this traver ..."
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A plane pseudograph is a plane graph allowing both loops and multiple edges. The encoding technique we propose belongs to a class of compression methods based on a deterministic graph traversal, which is encoded as a bit string. Arrays of vertices and edges stored in the order defined by this traversal, together with the bit string, allow the retrieval of the original graph. Within this framework, the most general methods to encode plane graphs allow obtaining the cyclic ordering of the edges incident to each vertex. Provided any directed edge incident to the infinite face is specified, the plane graph is uniquely defined. However, this information may not be sufficient to recreate the faces of the original graph when loops are allowed. In this paper, we analyse what information must be encoded in the bit string to retrieve correctly all incidences among the vertices, edges and faces of any plane pseudograph. Let G be a connected plane pseudograph with V vertices, E edges and F faces. The most common representation of G in computational geometry is the halfedge data structure, which requires 2E log V +(V +4E +F) log(2E)+2E log F bits to store the connectivity of the graph. The compression method proposed in this paper encodes the graph connectivity in 4E + 1 bits, and allows encodingdecoding the data structure representing the graph in O(E) time.