Pub Date : 2014-01-06DOI: 10.1137/1.9781611973204.5
O. Bernardi, Gwendal Collet, Éric Fusy
This paper is concerned with the counting and random sampling of plane graphs (simple planar graphs embedded in the plane). Our main result is a bijection between the class of plane graphs with triangular outer face, and a class of oriented binary trees. The number of edges and vertices of the plane graph can be tracked through the bijection. Consequently, we obtain counting formulas and an Efficient random sampling algorithm for rooted plane graphs (with arbitrary outer face) according to the number of edges and vertices. We also obtain a bijective link, via a bijection of Bona, between rooted plane graphs and 1342-avoiding permutations.
{"title":"A bijection for plane graphs and its applications","authors":"O. Bernardi, Gwendal Collet, Éric Fusy","doi":"10.1137/1.9781611973204.5","DOIUrl":"https://doi.org/10.1137/1.9781611973204.5","url":null,"abstract":"This paper is concerned with the counting and random sampling of plane graphs (simple planar graphs embedded in the plane). Our main result is a bijection between the class of plane graphs with triangular outer face, and a class of oriented binary trees. The number of edges and vertices of the plane graph can be tracked through the bijection. Consequently, we obtain counting formulas and an Efficient random sampling algorithm for rooted plane graphs (with arbitrary outer face) according to the number of edges and vertices. We also obtain a bijective link, via a bijection of Bona, between rooted plane graphs and 1342-avoiding permutations.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128831096","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-01-06DOI: 10.1137/1.9781611973204.3
O. Bodini, Bernhard Gittenberger
We investigate the asymptotic number of a particular class of closed lambda-terms. This class is a generalization of a class of terms related to the axiom system BCK which is well known in combinatory logic. We determine the asymptotic number of terms, when their size tends to infinity, up to a constant multiplicative factor and discover a surprising asymptotic behaviour involving an exponential with fractional powers in the exponent.
{"title":"On the asymptotic number of BCK(2)-terms","authors":"O. Bodini, Bernhard Gittenberger","doi":"10.1137/1.9781611973204.3","DOIUrl":"https://doi.org/10.1137/1.9781611973204.3","url":null,"abstract":"We investigate the asymptotic number of a particular class of closed lambda-terms. This class is a generalization of a class of terms related to the axiom system BCK which is well known in combinatory logic. We determine the asymptotic number of terms, when their size tends to infinity, up to a constant multiplicative factor and discover a surprising asymptotic behaviour involving an exponential with fractional powers in the exponent.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133305352","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-01-06DOI: 10.1137/1.9781611973204.6
M. Régnier, Billy Fang, Daria Iakovishina
Given a set of words and a probability model for random texts, we are interested in the behavior of occurrences of the words in a random text. Clumps are shown here to play a central role in these problems. They can be used to calculate relevant quantities, such as the probability that a random text contains a given number of pattern word occurrences. We provide combinatorial properties that greatly simplify the classical enumeration of these texts by inclusion-exclusion approaches. We describe two clump automata that can be used to efficiently calculate generating functions.
{"title":"Clump Combinatorics, Automata, and Word Asymptotics","authors":"M. Régnier, Billy Fang, Daria Iakovishina","doi":"10.1137/1.9781611973204.6","DOIUrl":"https://doi.org/10.1137/1.9781611973204.6","url":null,"abstract":"Given a set of words and a probability model for random texts, we are interested in the behavior of occurrences of the words in a random text. Clumps are shown here to play a central role in these problems. They can be used to calculate relevant quantities, such as the probability that a random text contains a given number of pattern word occurrences. We provide combinatorial properties that greatly simplify the classical enumeration of these texts by inclusion-exclusion approaches. We describe two clump automata that can be used to efficiently calculate generating functions.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126874777","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-01-06DOI: 10.1137/1.9781611973204.2
A. Magner, C. Knessl, W. Szpankowski
We consider PATRICIA tries on n random binary strings generated by a memoryless source with parameter p ≥ 1/2. For both the symmetric (p = 1/2) and asymmetric cases, we analyze asymptotics of the expected value of the external profile at level k = k(n), defined to be the number of leaves at level k. We study three natural ranges of k with respect to n. For k bounded, the mean profile decays exponentially with respect to n. For k growing logarithmically with n, the parameter exhibits polynomial growth in n, with some periodic fluctuations. Finally, for k = Θ(n), we see super-exponential decay, again with periodic fluctuations. Our derivations rely on analytic techniques, including Mellin transforms, analytic depoissonization, and the saddle point method. To cover wider ranges of k and n and provide more intuitive insights, we also use methods of applied mathematics, including asymptotic matching and linearization.
{"title":"Expected External Profile of PATRICIA Tries","authors":"A. Magner, C. Knessl, W. Szpankowski","doi":"10.1137/1.9781611973204.2","DOIUrl":"https://doi.org/10.1137/1.9781611973204.2","url":null,"abstract":"We consider PATRICIA tries on n random binary strings generated by a memoryless source with parameter p ≥ 1/2. For both the symmetric (p = 1/2) and asymmetric cases, we analyze asymptotics of the expected value of the external profile at level k = k(n), defined to be the number of leaves at level k. We study three natural ranges of k with respect to n. For k bounded, the mean profile decays exponentially with respect to n. For k growing logarithmically with n, the parameter exhibits polynomial growth in n, with some periodic fluctuations. Finally, for k = Θ(n), we see super-exponential decay, again with periodic fluctuations. Our derivations rely on analytic techniques, including Mellin transforms, analytic depoissonization, and the saddle point method. To cover wider ranges of k and n and provide more intuitive insights, we also use methods of applied mathematics, including asymptotic matching and linearization.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115733057","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-01-06DOI: 10.1137/1.9781611973204.8
Benjamin Doerr, Marvin Künnemann
We present a tight analysis of the basic randomized rumor spreading process in complete graphs introduced by Frieze and Grimmett (1985), where in each round of the process each node knowing the rumor gossips the rumor to a node chosen uniformly at random. The process starts with a single node knowing the rumor. We show that the number Sn of rounds required to spread a rumor in a complete graph with n nodes is very closely described by log2 n plus (1/n) times the completion time of the coupon collector process. This in particular gives very precise bounds for the expected runtime of the process, namely ⌊log2 n⌋ + ln n − 1:116 ≤ E[Sn] ≤ ⌈log2 n⌉ + ln n + 2:765 + o(1).
{"title":"Tight Analysis of Randomized Rumor Spreading in Complete Graphs","authors":"Benjamin Doerr, Marvin Künnemann","doi":"10.1137/1.9781611973204.8","DOIUrl":"https://doi.org/10.1137/1.9781611973204.8","url":null,"abstract":"We present a tight analysis of the basic randomized rumor spreading process in complete graphs introduced by Frieze and Grimmett (1985), where in each round of the process each node knowing the rumor gossips the rumor to a node chosen uniformly at random. The process starts with a single node knowing the rumor. \u0000 \u0000We show that the number Sn of rounds required to spread a rumor in a complete graph with n nodes is very closely described by log2 n plus (1/n) times the completion time of the coupon collector process. This in particular gives very precise bounds for the expected runtime of the process, namely ⌊log2 n⌋ + ln n − 1:116 ≤ E[Sn] ≤ ⌈log2 n⌉ + ln n + 2:765 + o(1).","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"130 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116207840","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-01-06DOI: 10.1137/1.9781611973204.7
S. Ganguly, Y. Peres
Given a permutation σ of the integers {−n, −n + 1,...,n} we consider the Markov chain Xσ, which jumps from k to σ(k ± 1) equally likely if k ≠ −n,n. We prove that the expected hitting time of {−n,n} starting from any point is Θ(n) with high probability when σ is a uniformly chosen permutation. We prove this by showing that with high probability, the digraph of allowed transitions is an Eulerian expander; we then utilize general estimates of hitting times in directed Eulerian expanders.
{"title":"Permuted Random Walk Exits Typically in Linear Time","authors":"S. Ganguly, Y. Peres","doi":"10.1137/1.9781611973204.7","DOIUrl":"https://doi.org/10.1137/1.9781611973204.7","url":null,"abstract":"Given a permutation σ of the integers {−n, −n + 1,...,n} we consider the Markov chain Xσ, which jumps from k to σ(k ± 1) equally likely if k ≠ −n,n. We prove that the expected hitting time of {−n,n} starting from any point is Θ(n) with high probability when σ is a uniformly chosen permutation. We prove this by showing that with high probability, the digraph of allowed transitions is an Eulerian expander; we then utilize general estimates of hitting times in directed Eulerian expanders.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"83 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126279424","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-01-06DOI: 10.1137/1.9781611973204.10
Noam Solomon, Shay Solomon
The Bottleneck Tower of Hanoi (BTH) problem, posed in 1981 by Wood, is a natural generalization of the classic Tower of Hanoi (TH) problem. There, a generalized placement rule allows a larger disk to be placed higher than a smaller one if their size difference is less than a given parameter k ≥ 1; when k = 1 we arrive at the classic TH problem. The objective is to compute a shortest move-sequence transferring a legal (under the above rule) configuration of n disks on three pegs to another legal configuration. In SOFSEM'07, Dinitz and the second author established tight asymptotic bounds for the worst-case complexity of the BTH problem, for all n and k. Moreover, they proved that the average-case complexity is asymptotically the same as the worst-case complexity, for all n > 3k and n ≤ k, and conjectured that the same phenomenon also occurs in the complementary range k < n ≤ 3k. In this paper we settle the conjecture of Dinitz and Solomon from SOFSEM'07 in the affirmative, and show that the average-case complexity of the BTH problem is asymptotically the same as the worst-case complexity, for all n and k. To this end we provide a new proof that applies to all values of n > k. That is, our proof is not a patch over the previous proof of Dinitz and Solomon that is tailored only for the regime k < n ≤ 3k, but is rather a stronger proof that is based on different principles and deeper ideas. We also show that there are natural connections between the BTH problem, the problem of sorting with complete networks of stacks using a forklift [Albert and Atkinson 2002, Konig and Lubbecke 2008] and the well-studied pancake problem [Gates and Papadimitriou 1979].
Wood在1981年提出的河内瓶颈塔(BTH)问题是经典河内塔(TH)问题的自然推广。在这里，一个广义的放置规则允许较大的磁盘被放置在比较小的磁盘更高的位置，如果它们的大小差小于给定参数k≥1;当k = 1时，我们就得到了经典的TH问题。目标是计算一个最短的移动序列，将一个合法的(在上述规则下的)3个节点上的n个磁盘配置转移到另一个合法的配置。在SOFSEM'07中，Dinitz和第二作者建立了所有n和k的BTH问题的最坏情况复杂度的紧密渐近界，并证明了所有n > 3k和n≤k的平均情况复杂度与最坏情况复杂度渐近相同，并推测在k < n≤3k的互补范围内也会出现同样的现象。在本文中,我们解决Dinitz和所罗门的猜想SOFSEM ' 07肯定的,表明蓝芽的平均情况复杂性问题是渐近一样坏的复杂性,n和k。为此我们提供一个新的证明适用于所有的值n > k。也就是说,我们的证明不是一块过去的证明Dinitz所罗门只定制的政权k < n≤3 k,而是基于不同原则和更深层次思想的更有力的证明。我们还表明，在BTH问题、使用铲车对完整的栈网络进行排序的问题(Albert and Atkinson 2002, Konig and Lubbecke 2008)和得到充分研究的煎饼问题(Gates and Papadimitriou 1979)之间存在自然联系。
{"title":"On The Average-Case Complexity of the Bottleneck Tower of Hanoi Problem","authors":"Noam Solomon, Shay Solomon","doi":"10.1137/1.9781611973204.10","DOIUrl":"https://doi.org/10.1137/1.9781611973204.10","url":null,"abstract":"The Bottleneck Tower of Hanoi (BTH) problem, posed in 1981 by Wood, is a natural generalization of the classic Tower of Hanoi (TH) problem. There, a generalized placement rule allows a larger disk to be placed higher than a smaller one if their size difference is less than a given parameter k ≥ 1; when k = 1 we arrive at the classic TH problem. The objective is to compute a shortest move-sequence transferring a legal (under the above rule) configuration of n disks on three pegs to another legal configuration. In SOFSEM'07, Dinitz and the second author established tight asymptotic bounds for the worst-case complexity of the BTH problem, for all n and k. Moreover, they proved that the average-case complexity is asymptotically the same as the worst-case complexity, for all n > 3k and n ≤ k, and conjectured that the same phenomenon also occurs in the complementary range k < n ≤ 3k. In this paper we settle the conjecture of Dinitz and Solomon from SOFSEM'07 in the affirmative, and show that the average-case complexity of the BTH problem is asymptotically the same as the worst-case complexity, for all n and k. To this end we provide a new proof that applies to all values of n > k. That is, our proof is not a patch over the previous proof of Dinitz and Solomon that is tailored only for the regime k < n ≤ 3k, but is rather a stronger proof that is based on different principles and deeper ideas. We also show that there are natural connections between the BTH problem, the problem of sorting with complete networks of stacks using a forklift [Albert and Atkinson 2002, Konig and Lubbecke 2008] and the well-studied pancake problem [Gates and Papadimitriou 1979].","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"123 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114367784","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-01-06DOI: 10.1137/1.9781611973204.1
Kanal Hun, B. Vallée
The digital search tree (dst) plays a central role in compression algorithms, of Lempel-Ziv type. This important structure can be viewed as a mixing of a digital structure (the trie) with a binary search tree. Its probabilistic analysis is thus involved, even in the case when the text is produced by a simple source (a memoryless source, or a Markov chain). After the seminal paper of Flajolet and Sedgewick (1986) [11] which deals with the memoryless unbiased case, many papers, due to Drmota, Jacquet, Louchard, Prodinger, Szpankowski, Tang, published between 1990 and 2005, dealt with general memoryless sources or Markov chains, and performed the analysis of the main parameters of dst's--namely, internal path length, profile, typical depth-- (see for instance [7, 15, 14]). Here, we are interested in a more realistic analysis, when the words are emitted by a general source, where the emission of symbols may depend on the whole previous history. There exist previous analyses of text algorithms or digital structures that have been performed for general sources, for instance for tries ([3, 2]), or for basic sorting and searching algorithms ([22, 4]). However, the case of digital search trees has not yet been considered, and this is the main subject of the paper. The idea of this study is due to Philippe Flajolet and the first steps of the work were performed with him, during the end of 2010. This paper is dedicated to Philippe's memory.
{"title":"Typical Depth of a Digital Search Tree built on a general source","authors":"Kanal Hun, B. Vallée","doi":"10.1137/1.9781611973204.1","DOIUrl":"https://doi.org/10.1137/1.9781611973204.1","url":null,"abstract":"The digital search tree (dst) plays a central role in compression algorithms, of Lempel-Ziv type. This important structure can be viewed as a mixing of a digital structure (the trie) with a binary search tree. Its probabilistic analysis is thus involved, even in the case when the text is produced by a simple source (a memoryless source, or a Markov chain). After the seminal paper of Flajolet and Sedgewick (1986) [11] which deals with the memoryless unbiased case, many papers, due to Drmota, Jacquet, Louchard, Prodinger, Szpankowski, Tang, published between 1990 and 2005, dealt with general memoryless sources or Markov chains, and performed the analysis of the main parameters of dst's--namely, internal path length, profile, typical depth-- (see for instance [7, 15, 14]). Here, we are interested in a more realistic analysis, when the words are emitted by a general source, where the emission of symbols may depend on the whole previous history. There exist previous analyses of text algorithms or digital structures that have been performed for general sources, for instance for tries ([3, 2]), or for basic sorting and searching algorithms ([22, 4]). However, the case of digital search trees has not yet been considered, and this is the main subject of the paper. The idea of this study is due to Philippe Flajolet and the first steps of the work were performed with him, during the end of 2010. This paper is dedicated to Philippe's memory.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128906712","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-12-01DOI: 10.1137/1.9781611973204.11
Ravi Kalpathy, H. Mahmoud, W. Rosenkrantz
We consider the number of survivors in a broad class of fair leader election algorithms after a number of election rounds. We give sufficient conditions for the number of survivors to converge to a product of independent identically distributed random variables. The number of terms in the product is determined by the round number considered. Each individual term in the product is a limit of a scaled random variable associated with the splitting protocol. The proof is established via convergence (to 0) of the first-order Wasserstein distance from the product limit. In a broader context, the paper is a case study of a class of stochastic recursive equations. We give two illustrative examples, one with binomial splitting protocol (for which we show that a normalized version is asymptotically Gaussian) and one with uniform splitting protocol.
{"title":"Survivors in Leader Election Algorithms","authors":"Ravi Kalpathy, H. Mahmoud, W. Rosenkrantz","doi":"10.1137/1.9781611973204.11","DOIUrl":"https://doi.org/10.1137/1.9781611973204.11","url":null,"abstract":"We consider the number of survivors in a broad class of fair leader election algorithms after a number of election rounds. We give sufficient conditions for the number of survivors to converge to a product of independent identically distributed random variables. The number of terms in the product is determined by the round number considered. Each individual term in the product is a limit of a scaled random variable associated with the splitting protocol. The proof is established via convergence (to 0) of the first-order Wasserstein distance from the product limit. In a broader context, the paper is a case study of a class of stochastic recursive equations. We give two illustrative examples, one with binomial splitting protocol (for which we show that a normalized version is asymptotically Gaussian) and one with uniform splitting protocol.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"153 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133983993","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-04-11DOI: 10.1137/1.9781611973204.12
Pat Morin, S. Verdonschot
Theta graphs are important geometric graphs that have many applications, including wireless networking, motion planning, real-time animation, and minimum-spanning tree construction. We give closed form expressions for the average degree of theta graphs of a homogeneous Poisson point process over the plane. We then show that essentially the same bounds---with vanishing error terms---hold for theta graphs of finite sets of points that are uniformly distributed in a square. Finally, we show that the number of edges in a theta graph of points uniformly distributed in a square is concentrated around its expected value.
{"title":"On the Average Number of Edges in Theta Graphs","authors":"Pat Morin, S. Verdonschot","doi":"10.1137/1.9781611973204.12","DOIUrl":"https://doi.org/10.1137/1.9781611973204.12","url":null,"abstract":"Theta graphs are important geometric graphs that have many applications, including wireless networking, motion planning, real-time animation, and minimum-spanning tree construction. We give closed form expressions for the average degree of theta graphs of a homogeneous Poisson point process over the plane. We then show that essentially the same bounds---with vanishing error terms---hold for theta graphs of finite sets of points that are uniformly distributed in a square. Finally, we show that the number of edges in a theta graph of points uniformly distributed in a square is concentrated around its expected value.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"357 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133141976","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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