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Optimal Coding and Sampling of Triangulations
, 2003
"... Abstract. We present a simple encoding of plane triangulations (aka. maximal planar graphs) by plane trees with two leaves per inner node. Our encoding is a bijection taking advantage of the minimal Schnyder tree decomposition of a plane triangulation. Coding and decoding take linear time. As a bypr ..."
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Cited by 36 (5 self)
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Abstract. We present a simple encoding of plane triangulations (aka. maximal planar graphs) by plane trees with two leaves per inner node. Our encoding is a bijection taking advantage of the minimal Schnyder tree decomposition of a plane triangulation. Coding and decoding take linear time. As a byproduct we derive: (i) a simple interpretation of the formula for the number of plane triangulations with n vertices, (ii) a linear random sampling algorithm, (iii) an explicit and simple information theory optimal encoding. 1
Uniform Random Generation of Decomposable Structures Using FloatingPoint Arithmetic
 THEORETICAL COMPUTER SCIENCE
, 1997
"... The recursive method formalized by Nijenhuis and Wilf [15] and systematized by Flajolet, Van Cutsem and Zimmermann [8], is extended here to floatingpoint arithmetic. The resulting ADZ method enables one to generate decomposable data structures  both labelled or unlabelled  uniformly at random, ..."
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Cited by 31 (2 self)
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The recursive method formalized by Nijenhuis and Wilf [15] and systematized by Flajolet, Van Cutsem and Zimmermann [8], is extended here to floatingpoint arithmetic. The resulting ADZ method enables one to generate decomposable data structures  both labelled or unlabelled  uniformly at random, in expected O(n 1+ffl ) time and space, after a preprocessing phase of O(n 2+ffl ) time, which reduces to O(n 1+ffl ) for contextfree grammars.
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 21 (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.
Canonical Decomposition of Outerplanar Maps and Application to Enumeration, Coding and Generation
, 2003
"... In this article we define a canonical decomposition of rooted outerplanar maps into a spanning tree and a list of edges. This decomposition, constructible in linear time, implies the existence of bijection between rooted outerplanar maps with n nodes and bicolored rooted ordered trees with n node ..."
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Cited by 12 (1 self)
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In this article we define a canonical decomposition of rooted outerplanar maps into a spanning tree and a list of edges. This decomposition, constructible in linear time, implies the existence of bijection between rooted outerplanar maps with n nodes and bicolored rooted ordered trees with n nodes where all the nodes of the last branch are colored white. As a consequence, for rooted outerplanar maps of n nodes, we derive: an enumeration formula, and an asymptotic of 2 3n (log n) ; an optimal data structure of asymptotically 3n bits, built in O(n) time, supporting adjacency and degree queries in worstcase constant time and neighbors query of a ddegree node in worstcase O(d) time...
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
A Note On Rémy's Algorithm For Generating Random Binary Trees
, 2000
"... This note discusses the implementation of Rémy's algorithm for generating unbiased random binary trees. We point out an error in a published implementation of the algorithm. The error is found by using the chi²test. Moreover, a correct implementation of the algorithm is presented. ..."
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Cited by 1 (1 self)
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This note discusses the implementation of Rémy's algorithm for generating unbiased random binary trees. We point out an error in a published implementation of the algorithm. The error is found by using the chi²test. Moreover, a correct implementation of the algorithm is presented.
Uniform Random Generation of . . .
, 1997
"... The recursive method formalized by Nijenhuis and Wilf [15] and systematized by Flajolet, Van Cutsem and Zimmermann [8], is extended here to floatingpoint arithmetic. The resulting ADZ method enables one to generate decomposable data structures  both labelled or unlabelled  uniformly at random ..."
Abstract
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The recursive method formalized by Nijenhuis and Wilf [15] and systematized by Flajolet, Van Cutsem and Zimmermann [8], is extended here to floatingpoint arithmetic. The resulting ADZ method enables one to generate decomposable data structures  both labelled or unlabelled  uniformly at random, in expected O#n # time and space, after a preprocessing phase of O#n # time, which reduces to O#n # for contextfree grammars.
Generating Random Binary Trees  A Survey
, 1998
"... This paper surveys algorithms for generating unbiased random binary trees. There exist several linear time algorithms. The best algorithms use only integers of size O(n) to generate binary trees on n nodes. 1 Introduction Binary trees are essential in various branches of computer science [7]. In t ..."
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This paper surveys algorithms for generating unbiased random binary trees. There exist several linear time algorithms. The best algorithms use only integers of size O(n) to generate binary trees on n nodes. 1 Introduction Binary trees are essential in various branches of computer science [7]. In time to time, there is a need to generate random binary trees. For example, when writing a program to manipulate binary trees, it is advantageous to have an efficient method to genarate random binary trees with some fixed number n of nodes in order to test the program. Mathematically, there are no problems at all in generating random binary trees. Namely, there exist algorithms to enumerate binary trees. Simply choose a random natural number i from the interval [1::B n ], where the nth Catalan number B n = / 2n n ! 1 n + 1 (1) gives the number of binary trees with n nodes, and output the ith binary tree from the enumeration. Computationally the problem is more complicated. As Martin a...
Generating Outerplanar Graphs
"... supported by the DFG (GRK 588/1) Abstract. We show how to generate labeled and unlabeled outerplanar graphs with n vertices uniformly at random in polynomial time in n. To generate labeled outerplanar graphs, we present a counting technique using the decomposition of a graph according to its block s ..."
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supported by the DFG (GRK 588/1) Abstract. We show how to generate labeled and unlabeled outerplanar graphs with n vertices uniformly at random in polynomial time in n. To generate labeled outerplanar graphs, we present a counting technique using the decomposition of a graph according to its block structure, and compute the exact number of labeled outerplanar graphs. This allows us to make the correct probabilistic choices in a recursive generation of uniformly distributed outerplanar graphs. Next we modify our formulas to count rooted unlabeled graphs, and finally show how to use these formulas in a Las Vegas algorithm to generate unlabeled outerplanar graphs uniformly at random in expected polynomial time. random structures, outerplanar graphs, efficient counting, uniform generation 1 Introduction There are several fields of applications of efficient algorithms that generate random combinatorial structures. We can use such a generation procedure as an experimental tool to verify properties of these structures that hold almost always. It can also be used to produce test instances for other algorithms on these structures. We can then measure the average running time of these algorithmson random instances. Outerplanar graphs are planar graphs with an embedding where every vertexlies on the outer face. They are interesting examples of combinatorial structures with respect to the mentioned motivations of random generation procedures,having numerous applications in various areas of computer and natural sciences. Many computational problems that are hard in general cases become tractablefor outerplanar graphs [21, 25]. But still, their structure is rich enough that many computational tasks remain challenging when the input is restricted toouterplanar graphs [6, 15]. Outerplanar graphs also attract interest in graph drawing [2, 5, 19].Often one is interested not in the worstcase running time of an algorithm, but in the running time of a typical instance. The socalled averagecase complexity of