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42
Succinct Representations of Planar Maps
, 2008
"... This paper addresses the problem of representing the connectivity information of geometric objects using as little memory as possible. As opposed to raw compression issues, the focus is here on designing data structures that preserve the possibility of answering incidence queries in constant time. W ..."
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Cited by 19 (3 self)
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This paper addresses the problem of representing the connectivity information of geometric objects using as little memory as possible. As opposed to raw compression issues, the focus is here on designing data structures that preserve the possibility of answering incidence queries in constant time. We propose in particular the first optimal representations for 3connected planar graphs and triangulations, which are the most standard classes of graphs underlying meshes with spherical topology. Optimal means that these representations asymptotically match the respective entropy of the two classes, namely 2 bits per edge for 3connected planar graphs, and 1.62 bits per triangle or equivalently 3.24 bits per vertex for triangulations. These representations support adjacency queries between vertices and faces in constant time.
Succinct representation of triangulations with a boundary. WADS
, 2005
"... Abstract. We consider the problem of designing succinct geometric data structures while maintaining efficient navigation operations. A data structure is said succinct if the asymptotic amount of space it uses matches the entropy of the class of structures represented. For the case of planar triangul ..."
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Cited by 16 (5 self)
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Abstract. We consider the problem of designing succinct geometric data structures while maintaining efficient navigation operations. A data structure is said succinct if the asymptotic amount of space it uses matches the entropy of the class of structures represented. For the case of planar triangulations with a boundary we propose a succinct representation of the combinatorial information that improves to 2.175 bits per triangle the asymptotic amount of space required and that supports the navigation between adjacent triangles in constant time (as well as other standard operations). For triangulations with m faces of a surface with genus g, our representation requires asymptotically an extra amount of 36(g − 1) lg m bits (which is negligible as long as g ≪ m / lg m). 1
LinearTime Compression of BoundedGenus Graphs into InformationTheoretically Optimal Number of Bits
, 2002
"... ..."
An Experimental Analysis of a Compact Graph Representation
 In ALENEX04
, 2004
"... In previous work we described a method for compactly representing graphs with small separators, which makes use of small separators, and presented preliminary experimental results. In this paper we extend the experimental results in several ways, including extensions for dynamic insertion and deleti ..."
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Cited by 15 (6 self)
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In previous work we described a method for compactly representing graphs with small separators, which makes use of small separators, and presented preliminary experimental results. In this paper we extend the experimental results in several ways, including extensions for dynamic insertion and deletion of edges, a comparison of a variety of coding schemes, and an implementation of two applications using the representation.
Compact floorplanning via orderly spanning trees
 Journal of Algorithms
"... 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 floorplan for any nnode plane triangulation. In comparison with previous floorplanning algorithms in the literature, our solution is no ..."
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Cited by 14 (1 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 floorplan for any nnode plane triangulation. In comparison with previous floorplanning algorithms in the literature, our solution is not only simpler in the algorithm itself, but also produces floorplans 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 (n − 1) × ⌊ ⌋
Improved Compact Routing Tables for Planar Networks via Orderly Spanning Trees
 In: 8 th Annual International Computing & Combinatorics Conference (COCOON). Volume 2387 of LNCS
, 2002
"... We address the problem of designing compact routing tables for an unlabeled connected nnode planar network G. For each node r of G, the designer is given a routing spanning tree Tr of G rooted at r, which speci es the routes for sending packets from r to the rest of G. ..."
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Cited by 12 (3 self)
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We address the problem of designing compact routing tables for an unlabeled connected nnode planar network G. For each node r of G, the designer is given a routing spanning tree Tr of G rooted at r, which speci es the routes for sending packets from r to the rest of G.
Small InducedUniversal Graphs and Compact Implicit Graph Representations
 In Proc. 43’rd annual IEEE Symp. on Foundations of Computer Science
, 2002
"... We show that there exists a graph G with n 2 nodes, where any forest with n nodes is a nodeinduced subgraph of G. Furthermore, the result implies existence of a graph with n nodes that contains all nnode graphs of fixed arboricity k as nodeinduced subgraphs. We provide a lower bound ..."
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Cited by 12 (0 self)
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We show that there exists a graph G with n 2 nodes, where any forest with n nodes is a nodeinduced subgraph of G. Furthermore, the result implies existence of a graph with n nodes that contains all nnode graphs of fixed arboricity k as nodeinduced subgraphs. We provide a lower bound of the size of such a graph. The upper bound is obtained through a simple labeling scheme for parent queries in rooted trees.
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
Orderly Spanning Trees with Applications
 SIAM Journal on Computing
, 2005
"... Abstract. We introduce and study orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any c ..."
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Cited by 12 (1 self)
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Abstract. We introduce and study orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any connected planar graph G, consisting of an embedded planar graph H isomorphic to G, and an orderly spanning tree of H. We also present several applications of orderly spanning trees: (1) a new constructive proof for Schnyder’s realizer theorem, (2) the first algorithm for computing an areaoptimal 2visibility drawing of a planar graph, and (3) the most compact known encoding of a planar graph with O(1)time query support. All algorithms in this paper run in linear time.