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An InformationTheoretic Upper Bound on Planar Graphs Using WellOrderly Maps
, 2011
"... This chapter deals with compressed coding of graphs. We focus on planar graphs, a widely studied class of graphs. A planar graph is a graph that admits an embedding in the plane without edge crossings. Planar maps (class of embeddings of a planar graph) are easier to study than planar graphs, but a ..."
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Cited by 22 (4 self)
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This chapter deals with compressed coding of graphs. We focus on planar graphs, a widely studied class of graphs. A planar graph is a graph that admits an embedding in the plane without edge crossings. Planar maps (class of embeddings of a planar graph) are easier to study than planar graphs, but as a planar graph may admit an exponential number of maps, they give little information on graphs. In order to give an informationtheoretic upper bound on planar graphs, we introduce a definition of a quasicanonical embedding for planar graphs: wellorderly maps. This appears to be an useful tool to study and encode planar graphs. We present upper bounds on the number of unlabeled planar graphs and on the number of edges in a random planar graph. We also present an algorithm to compute wellorderly maps and implying an efficient coding of planar graphs.
Compact Routing for Graphs Excluding a Fixed Minor (Extended Abstract)
, 2005
"... This paper concerns compact routing schemes with arbitrary node names. We present a compact nameindependent routing scheme for unweighted networks with n nodes excluding a fixed minor. For any fixed minor, the scheme, constructible in polynomial time, has constant stretch factor and requires routin ..."
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Cited by 20 (10 self)
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This paper concerns compact routing schemes with arbitrary node names. We present a compact nameindependent routing scheme for unweighted networks with n nodes excluding a fixed minor. For any fixed minor, the scheme, constructible in polynomial time, has constant stretch factor and requires routing tables with polylogarithmic number of bits at each node. For shortestpath labeled routing scheme in planar graphs, we prove an Ω(n ɛ) space lower bound for some constant ɛ>0. This lower bound holds even for bounded degree triangulations, and is optimal for polynomially weighted planar graphs (ɛ =1/2).
Planar graphs, via wellorderly maps and trees
 In 30 th International Workshop, Graph  Theoretic Concepts in Computer Science (WG), volume 3353 of Lecture Notes in Computer Science
, 2004
"... Abstract. The family of wellorderly maps is a family of planar maps with the property that every connected planar graph has at least one plane embedding which is a wellorderly map. We show that the number of wellorderly maps with n nodes is at most 2 αn+O(log n) , where α ≈ 4.91. A direct consequ ..."
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Cited by 19 (4 self)
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Abstract. The family of wellorderly maps is a family of planar maps with the property that every connected planar graph has at least one plane embedding which is a wellorderly map. We show that the number of wellorderly maps with n nodes is at most 2 αn+O(log n) , where α ≈ 4.91. A direct consequence of this is a new upper bound on the number p(n) of unlabeled planar graphs with n nodes, log 2 p(n) � 4.91n. The result is then used to show that asymptotically almost all (labeled or unlabeled), (connected or not) planar graphs with n nodes have between 1.85n and 2.44n edges. Finally we obtain as an outcome of our combinatorial analysis an explicit linear time encoding algorithm for unlabeled planar graphs using, in the worstcase, a rate of 4.91 bits per node and of 2.82 bits per edge. 1
Compact Routing in PowerLaw Graphs
"... Abstract. We adapt the compact routing scheme by Thorup and Zwick to optimize it for powerlaw graphs. We analyze our adapted routing scheme based on the theory of unweighted random powerlaw graphs with fixed expected degree sequence by Aiello, Chung, and Lu. Our result is the first theoretical bou ..."
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Cited by 14 (3 self)
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Abstract. We adapt the compact routing scheme by Thorup and Zwick to optimize it for powerlaw graphs. We analyze our adapted routing scheme based on the theory of unweighted random powerlaw graphs with fixed expected degree sequence by Aiello, Chung, and Lu. Our result is the first theoretical bound coupled to the parameter of the powerlaw graph model for a compact routing scheme. In particular, we prove that, for stretch 3, instead of routing tables with Õ(n 1/2) bits as in the general scheme by Thorup and Zwick, expected sizes of O(n γ log n) bits are sufficient, and that all the routing tables can be constructed at once in expected time O(n 1+γ log n), with γ = τ−2 + ε, where τ ∈ (2, 3) 2τ−3 is the powerlaw exponent and ε> 0. Both bounds also hold with probability at least 1 − 1/n (independent of ε). The routing scheme is a labeled scheme, requiring a stretch5 handshaking step and using addresses and message headers with O(log n log log n) bits, with probability at least 1−o(1). We further demonstrate the effectiveness of our scheme by simulations on realworld graphs as well as synthetic powerlaw graphs. With the same techniques as for the compact routing scheme, we also adapt the approximate distance oracle by Thorup and Zwick for stretch 3 and obtain a new upper bound of expected Õ(n1+γ) for space and preprocessing. 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 11 (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.
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 � �
Interval Routing in Reliability Networks
, 2003
"... In this paper we consider routing with compact tables in reliability networks. ..."
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Cited by 4 (2 self)
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In this paper we consider routing with compact tables in reliability networks.
A compact routing scheme and approximate distance oracle for powerlaw graphs
, 2009
"... Abstract. Compact routing addresses the tradeoff between table sizes and stretch, which is the worstcase ratio between the length of the path a packet is routed through by the scheme and the length of a shortest path from source to destination. We adapt the compact routing scheme by Thorup and Zwic ..."
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Cited by 2 (1 self)
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Abstract. Compact routing addresses the tradeoff between table sizes and stretch, which is the worstcase ratio between the length of the path a packet is routed through by the scheme and the length of a shortest path from source to destination. We adapt the compact routing scheme by Thorup and Zwick to optimize it for powerlaw graphs. We analyze our adapted routing scheme based on the theory of unweighted random powerlaw graphs with fixed expected degree sequence by Aiello, Chung, and Lu. Our result is the first theoretical bound coupled to the parameter of the powerlaw graph model for a compact routing scheme. In particular, we prove that, for stretch 3, instead of routing tables with Õ(n1/2) bits as in the general scheme by Thorup and Zwick, expected sizes of O(n γ log n) bits are sufficient, and that all the routing tables can be constructed at once in expected time O(n 1+γ log n), with γ = τ−2 2τ−3 + ε, where τ ∈ (2, 3) is the powerlaw exponent and ε> 0 (which implies ε < γ < 1/3 + ε). Both bounds also hold with probability at least 1 − 1/n (independent of ε). The routing scheme is a labeled scheme, requiring a stretch5 handshaking step and using addresses and message headers with O(log n log log n) bits, with probability at least 1 − o(1). We further demonstrate the effectiveness of our scheme by simulations on realworld graphs as well as synthetic powerlaw graphs. With the same techniques as for the compact routing scheme, we also adapt the approximate distance oracle by Thorup and Zwick for stretch 3 and obtain a new upper bound of expected Õ(n1+γ) for space and preprocessing for random powerlaw graphs. 1