Pub Date : 2017-01-16DOI: 10.1137/1.9781611974775.2
O. Bodini, M. Dien, Antoine Genitrini, F. Peschanski
In this paper, we study two operators for composing combinatorial classes: the ordered product and its dual, the colored product. These operators have a natural interpretation in terms of Analytic Combinatorics, in relation with combinations of Borel and Laplace transforms. Based on these new constructions, we exhibit a set of transfer theorems and closure properties. We also illustrate the use of these operators to specify increasingly labeled structures tightly related to Series-Parallel constructions and concurrent processes. In particular, we provide a quantitative analysis of Fork/Join (FJ) parallel processes, a particularly expressive example of such a class.
{"title":"The Ordered and Colored Products in Analytic Combinatorics: Application to the Quantitative Study of Synchronizations in Concurrent Processes","authors":"O. Bodini, M. Dien, Antoine Genitrini, F. Peschanski","doi":"10.1137/1.9781611974775.2","DOIUrl":"https://doi.org/10.1137/1.9781611974775.2","url":null,"abstract":"In this paper, we study two operators for composing combinatorial classes: the ordered product and its dual, the colored product. These operators have a natural interpretation in terms of Analytic Combinatorics, in relation with combinations of Borel and Laplace transforms. Based on these new constructions, we exhibit a set of transfer theorems and closure properties. We also illustrate the use of these operators to specify increasingly labeled structures tightly related to Series-Parallel constructions and concurrent processes. In particular, we provide a quantitative analysis of Fork/Join (FJ) parallel processes, a particularly expressive example of such a class.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"179 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133670663","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 : 2016-10-05DOI: 10.1137/1.9781611974775.10
P. Rotondo, B. Vallée
This paper describes the probabilistic behaviour of a random Sturmian word. It performs the probabilistic analysis of the recurrence function which can be viewed as a waiting time to discover all the factors of length $n$ of the Sturmian word. This parameter is central to combinatorics of words. Having fixed a possible length $n$ for the factors, we let $alpha$ to be drawn uniformly from the unit interval $[0,1]$, thus defining a random Sturmian word of slope $alpha$. Thus the waiting time for these factors becomes a random variable, for which we study the limit distribution and the limit density.
{"title":"The recurrence function of a random Sturmian word","authors":"P. Rotondo, B. Vallée","doi":"10.1137/1.9781611974775.10","DOIUrl":"https://doi.org/10.1137/1.9781611974775.10","url":null,"abstract":"This paper describes the probabilistic behaviour of a random Sturmian word. It performs the probabilistic analysis of the recurrence function which can be viewed as a waiting time to discover all the factors of length $n$ of the Sturmian word. This parameter is central to combinatorics of words. Having fixed a possible length $n$ for the factors, we let $alpha$ to be drawn uniformly from the unit interval $[0,1]$, thus defining a random Sturmian word of slope $alpha$. Thus the waiting time for these factors becomes a random variable, for which we study the limit distribution and the limit density.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131225234","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 : 2016-09-27DOI: 10.1137/1.9781611975505.8
M. Nebel, Elisabeth Neumann, Sebastian Wild
We extend randomized jumplists introduced by Bronnimann et al. (STACS 2003) to choose jump-pointer targets as median of a small sample for better search costs, and present randomized algorithms with expected $O(log n)$ time complexity that maintain the probability distribution of jump pointers upon insertions and deletions. We analyze the expected costs to search, insert and delete a random element, and we show that omitting jump pointers in small sublists hardly affects search costs, but significantly reduces the memory consumption. We use a bijection between jumplists and "dangling-min BSTs", a variant of (fringe-balanced) binary search trees for the analysis. Despite their similarities, some standard analysis techniques for search trees fail for dangling-min trees (and hence for jumplists).
{"title":"Median-of-k Jumplists and Dangling-Min BSTs","authors":"M. Nebel, Elisabeth Neumann, Sebastian Wild","doi":"10.1137/1.9781611975505.8","DOIUrl":"https://doi.org/10.1137/1.9781611975505.8","url":null,"abstract":"We extend randomized jumplists introduced by Bronnimann et al. (STACS 2003) to choose jump-pointer targets as median of a small sample for better search costs, and present randomized algorithms with expected $O(log n)$ time complexity that maintain the probability distribution of jump pointers upon insertions and deletions. We analyze the expected costs to search, insert and delete a random element, and we show that omitting jump pointers in small sublists hardly affects search costs, but significantly reduces the memory consumption. \u0000We use a bijection between jumplists and \"dangling-min BSTs\", a variant of (fringe-balanced) binary search trees for the analysis. Despite their similarities, some standard analysis techniques for search trees fail for dangling-min trees (and hence for jumplists).","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115818236","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 : 2016-08-17DOI: 10.1137/1.9781611975062.2
Sebastian Wild
I prove that the average number of comparisons for median-of-$k$ Quicksort (with fat-pivot a.k.a. three-way partitioning) is asymptotically only a constant $alpha_k$ times worse than the lower bound for sorting random multisets with $Omega(n^varepsilon)$ duplicates of each value (for any $varepsilon>0$). The constant is $alpha_k = ln(2) / bigl(H_{k+1}-H_{(k+1)/2} bigr)$, which converges to 1 as $ktoinfty$, so Quicksort is asymptotically optimal for inputs with many duplicates. This resolves a conjecture by Sedgewick and Bentley (1999, 2002) and constitutes the first progress on the analysis of Quicksort with equal elements since Sedgewick's 1977 article.
{"title":"Quicksort Is Optimal For Many Equal Keys","authors":"Sebastian Wild","doi":"10.1137/1.9781611975062.2","DOIUrl":"https://doi.org/10.1137/1.9781611975062.2","url":null,"abstract":"I prove that the average number of comparisons for median-of-$k$ Quicksort (with fat-pivot a.k.a. three-way partitioning) is asymptotically only a constant $alpha_k$ times worse than the lower bound for sorting random multisets with $Omega(n^varepsilon)$ duplicates of each value (for any $varepsilon>0$). The constant is $alpha_k = ln(2) / bigl(H_{k+1}-H_{(k+1)/2} bigr)$, which converges to 1 as $ktoinfty$, so Quicksort is asymptotically optimal for inputs with many duplicates. This resolves a conjecture by Sedgewick and Bentley (1999, 2002) and constitutes the first progress on the analysis of Quicksort with equal elements since Sedgewick's 1977 article.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127646268","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 : 2016-08-04DOI: 10.1137/1.9781611974775.3
C. Chauve, Éric Fusy, Jérémie O. Lumbroso
Distance-hereditary graphs form an important class of graphs, from the theoretical point of view, due to the fact that they are the totally decomposable graphs for the split-decomposition. The previous best enumerative result for these graphs is from Nakano et al. (J. Comp. Sci. Tech., 2007), who have proven that the number of distance-hereditary graphs on $n$ vertices is bounded by ${2^{lceil 3.59nrceil}}$. In this paper, using classical tools of enumerative combinatorics, we improve on this result by providing an exact enumeration of distance-hereditary graphs, which allows to show that the number of distance-hereditary graphs on $n$ vertices is tightly bounded by ${(7.24975ldots)^n}$---opening the perspective such graphs could be encoded on $3n$ bits. We also provide the exact enumeration and asymptotics of an important subclass, the 3-leaf power graphs. Our work illustrates the power of revisiting graph decomposition results through the framework of analytic combinatorics.
{"title":"An Exact Enumeration of Distance-Hereditary Graphs","authors":"C. Chauve, Éric Fusy, Jérémie O. Lumbroso","doi":"10.1137/1.9781611974775.3","DOIUrl":"https://doi.org/10.1137/1.9781611974775.3","url":null,"abstract":"Distance-hereditary graphs form an important class of graphs, from the theoretical point of view, due to the fact that they are the totally decomposable graphs for the split-decomposition. The previous best enumerative result for these graphs is from Nakano et al. (J. Comp. Sci. Tech., 2007), who have proven that the number of distance-hereditary graphs on $n$ vertices is bounded by ${2^{lceil 3.59nrceil}}$. \u0000In this paper, using classical tools of enumerative combinatorics, we improve on this result by providing an exact enumeration of distance-hereditary graphs, which allows to show that the number of distance-hereditary graphs on $n$ vertices is tightly bounded by ${(7.24975ldots)^n}$---opening the perspective such graphs could be encoded on $3n$ bits. We also provide the exact enumeration and asymptotics of an important subclass, the 3-leaf power graphs. \u0000Our work illustrates the power of revisiting graph decomposition results through the framework of analytic combinatorics.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131382418","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 : 2016-07-18DOI: 10.1137/1.9781611974775.11
Daniel Krenn
In this note the precise minimum number of key comparisons any dual-pivot quickselect algorithm (without sampling) needs on average is determined. The result is in the form of exact as well as asymptotic formulae{} of this number of a comparison-optimal algorithm. It turns out that the main terms of these asymptotic expansions coincide with the main terms of the corresponding analysis of the classical quickselect, but still---as this was shown for Yaroslavskiy quickselect---more comparisons are needed in the dual-pivot variant. The results are obtained by solving a second order differential equation for the generating function obtained from a recursive approach.
{"title":"An Extended Note on the Comparison-optimal Dual Pivot Quickselect","authors":"Daniel Krenn","doi":"10.1137/1.9781611974775.11","DOIUrl":"https://doi.org/10.1137/1.9781611974775.11","url":null,"abstract":"In this note the precise minimum number of key comparisons any dual-pivot quickselect algorithm (without sampling) needs on average is determined. The result is in the form of exact as well as asymptotic formulae{} of this number of a comparison-optimal algorithm. It turns out that the main terms of these asymptotic expansions coincide with the main terms of the corresponding analysis of the classical quickselect, but still---as this was shown for Yaroslavskiy quickselect---more comparisons are needed in the dual-pivot variant. The results are obtained by solving a second order differential equation for the generating function obtained from a recursive approach.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129084744","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 : 2016-07-11DOI: 10.1137/1.9781611974775.14
Subhro Ghosh, T. Liggett, Robin Pemantle
Let $(X_1 , ldots , X_d)$ be random variables taking nonnegative integer values and let $f(z_1, ldots , z_d)$ be the probability generating function. Suppose that $f$ is real stable; equivalently, suppose that the polarization of this probability distribution is strong Rayleigh. In specific examples, such as occupation counts of disjoint sets by a determinantal point process, it is known~cite{soshnikov02} that the joint distribution must approach a multivariate Gaussian distribution. We show that this conclusion follows already from stability of $f$.
{"title":"Multivariate CLT follows from strong Rayleigh property","authors":"Subhro Ghosh, T. Liggett, Robin Pemantle","doi":"10.1137/1.9781611974775.14","DOIUrl":"https://doi.org/10.1137/1.9781611974775.14","url":null,"abstract":"Let $(X_1 , ldots , X_d)$ be random variables taking nonnegative integer values and let $f(z_1, ldots , z_d)$ be the probability generating function. Suppose that $f$ is real stable; equivalently, suppose that the polarization of this probability distribution is strong Rayleigh. In specific examples, such as occupation counts of disjoint sets by a determinantal point process, it is known~cite{soshnikov02} that the joint distribution must approach a multivariate Gaussian distribution. We show that this conclusion follows already from stability of $f$.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131743720","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 : 2016-06-28DOI: 10.1137/1.9781611974775.8
Bernhard Gittenberger, E. Y. Jin, M. Wallner
Panagiotou and Stufler (arXiv:1502.07180v2) recently proved one important fact on their way to establish the scaling limits of random P'{o}lya trees: a uniform random P'{o}lya tree of size $n$ consists of a conditioned critical Galton-Watson tree $C_n$ and many small forests, where with probability tending to one as $n$ tends to infinity, any forest $F_n(v)$, that is attached to a node $v$ in $C_n$, is maximally of size $vert F_n(v)vert=O(log n)$. Their proof used the framework of a Boltzmann sampler and deviation inequalities. In this paper, first, we employ a unified framework in analytic combinatorics to prove this fact with additional improvements on the bound of $vert F_n(v)vert$, namely $vert F_n(v)vert=Theta(log n)$. Second, we give a combinatorial interpretation of the rational weights of these forests and the defining substitution process in terms of automorphisms associated to a given P'{o}lya tree. Finally, we derive the limit probability that for a random node $v$ the attached forest $F_n(v)$ is of a given size.
{"title":"A note on the scaling limits of random Pólya trees","authors":"Bernhard Gittenberger, E. Y. Jin, M. Wallner","doi":"10.1137/1.9781611974775.8","DOIUrl":"https://doi.org/10.1137/1.9781611974775.8","url":null,"abstract":"Panagiotou and Stufler (arXiv:1502.07180v2) recently proved one important fact on their way to establish the scaling limits of random P'{o}lya trees: a uniform random P'{o}lya tree of size $n$ consists of a conditioned critical Galton-Watson tree $C_n$ and many small forests, where with probability tending to one as $n$ tends to infinity, any forest $F_n(v)$, that is attached to a node $v$ in $C_n$, is maximally of size $vert F_n(v)vert=O(log n)$. Their proof used the framework of a Boltzmann sampler and deviation inequalities. \u0000In this paper, first, we employ a unified framework in analytic combinatorics to prove this fact with additional improvements on the bound of $vert F_n(v)vert$, namely $vert F_n(v)vert=Theta(log n)$. Second, we give a combinatorial interpretation of the rational weights of these forests and the defining substitution process in terms of automorphisms associated to a given P'{o}lya tree. Finally, we derive the limit probability that for a random node $v$ the attached forest $F_n(v)$ is of a given size.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116733210","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 : 2016-01-03DOI: 10.1137/1.9781611974775.1
M. Bóna, B. Pittel
The subject of this paper is the cycle structure of the random permutation $sigma$ of $[N]$, which is the product of $k$ independent random cycles of maximal length $N$. We use the character-based Fourier transform to study the number of cycles of $sigma$ and also the distribution of the elements of the subset $[ell]$ among the cycles of $sigma$.
{"title":"On the cycle structure of the product of random maximal cycles","authors":"M. Bóna, B. Pittel","doi":"10.1137/1.9781611974775.1","DOIUrl":"https://doi.org/10.1137/1.9781611974775.1","url":null,"abstract":"The subject of this paper is the cycle structure of the random permutation $sigma$ of $[N]$, which is the product of $k$ independent random cycles of maximal length $N$. We use the character-based Fourier transform to study the number of cycles of $sigma$ and also the distribution of the elements of the subset $[ell]$ among the cycles of $sigma$.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117017628","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 : 2015-08-17DOI: 10.1137/1.9781611974324.12
Jun Zhao, Panpan Zhang
There has been growing interest in studies of general random intersection graphs. In this paper, we consider a general random intersection graph $mathbb{G}(n,overrightarrow{a}, overrightarrow{K_n},P_n)$ defined on a set $mathcal{V}_n$ comprising $n$ vertices, where $overrightarrow{a}$ is a probability vector $(a_1,a_2,ldots,a_m)$ and $overrightarrow{K_n}$ is $(K_{1,n},K_{2,n},ldots,K_{m,n})$. This graph has been studied in the literature including a most recent work by Yagan [arXiv:1508.02407]. Suppose there is a pool $mathcal{P}_n$ consisting of $P_n$ distinct objects. The $n$ vertices in $mathcal{V}_n$ are divided into $m$ groups $mathcal{A}_1, mathcal{A}_2, ldots, mathcal{A}_m$. Each vertex $v$ is independently assigned to exactly a group according to the probability distribution with $mathbb{P}[v in mathcal{A}_i]= a_i$, where $i=1,2,ldots,m$. Afterwards, each vertex in group $mathcal{A}_i$ independently chooses $K_{i,n}$ objects uniformly at random from the object pool $mathcal{P}_n$. Finally, an undirected edge is drawn between two vertices in $mathcal{V}_n$ that share at least one object. This graph model $mathbb{G}(n,overrightarrow{a}, overrightarrow{K_n},P_n)$ has applications in secure sensor networks and social networks. We investigate connectivity in this general random intersection graph $mathbb{G}(n,overrightarrow{a}, overrightarrow{K_n},P_n)$ and present a sharp zero-one law. Our result is also compared with the zero-one law established by Yagan.
{"title":"On Connectivity in a General Random Intersection Graph","authors":"Jun Zhao, Panpan Zhang","doi":"10.1137/1.9781611974324.12","DOIUrl":"https://doi.org/10.1137/1.9781611974324.12","url":null,"abstract":"There has been growing interest in studies of general random intersection graphs. In this paper, we consider a general random intersection graph $mathbb{G}(n,overrightarrow{a}, overrightarrow{K_n},P_n)$ defined on a set $mathcal{V}_n$ comprising $n$ vertices, where $overrightarrow{a}$ is a probability vector $(a_1,a_2,ldots,a_m)$ and $overrightarrow{K_n}$ is $(K_{1,n},K_{2,n},ldots,K_{m,n})$. This graph has been studied in the literature including a most recent work by Yagan [arXiv:1508.02407]. Suppose there is a pool $mathcal{P}_n$ consisting of $P_n$ distinct objects. The $n$ vertices in $mathcal{V}_n$ are divided into $m$ groups $mathcal{A}_1, mathcal{A}_2, ldots, mathcal{A}_m$. Each vertex $v$ is independently assigned to exactly a group according to the probability distribution with $mathbb{P}[v in mathcal{A}_i]= a_i$, where $i=1,2,ldots,m$. Afterwards, each vertex in group $mathcal{A}_i$ independently chooses $K_{i,n}$ objects uniformly at random from the object pool $mathcal{P}_n$. Finally, an undirected edge is drawn between two vertices in $mathcal{V}_n$ that share at least one object. This graph model $mathbb{G}(n,overrightarrow{a}, overrightarrow{K_n},P_n)$ has applications in secure sensor networks and social networks. We investigate connectivity in this general random intersection graph $mathbb{G}(n,overrightarrow{a}, overrightarrow{K_n},P_n)$ and present a sharp zero-one law. Our result is also compared with the zero-one law established by Yagan.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"108 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122889432","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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