Pub Date : 2011-01-22DOI: 10.1137/1.9781611973013.10
Michael Fuchs
In this extended abstract, we outline how to derive limit theorems for the number of subtrees of size k on the fringe of random plane-oriented recursive trees. Our proofs are based on the method of moments, where a complex-analytic approach is used for constant k and an elementary approach for k which varies with n. Our approach is of some generality and can be applied to other simple classes of increasing trees as well.
{"title":"The Subtree Size Profile of Plane-oriented Recursive Trees","authors":"Michael Fuchs","doi":"10.1137/1.9781611973013.10","DOIUrl":"https://doi.org/10.1137/1.9781611973013.10","url":null,"abstract":"In this extended abstract, we outline how to derive limit theorems for the number of subtrees of size k on the fringe of random plane-oriented recursive trees. Our proofs are based on the method of moments, where a complex-analytic approach is used for constant k and an elementary approach for k which varies with n. Our approach is of some generality and can be applied to other simple classes of increasing trees as well.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115781289","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 : 2011-01-22DOI: 10.1137/1.9781611973013.5
B. Chauvin, Danièle Gardy, Cécile Mailler
We define a probability distribution over the set of Boolean functions of k variables induced by the tree representation of Boolean expressions. The law we are interested in is inspired by the growth model of Binary Search Trees: we call it the growing tree law. We study it over different logical systems and compare the results we obtain to already known distributions induced by the tree representation: Catalan trees, Galton-Watson trees and balanced trees.
{"title":"The Growing Trees Distribution on Boolean Functions","authors":"B. Chauvin, Danièle Gardy, Cécile Mailler","doi":"10.1137/1.9781611973013.5","DOIUrl":"https://doi.org/10.1137/1.9781611973013.5","url":null,"abstract":"We define a probability distribution over the set of Boolean functions of k variables induced by the tree representation of Boolean expressions. The law we are interested in is inspired by the growth model of Binary Search Trees: we call it the growing tree law. We study it over different logical systems and compare the results we obtain to already known distributions induced by the tree representation: Catalan trees, Galton-Watson trees and balanced trees.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129698307","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 : 2011-01-22DOI: 10.1137/1.9781611973013.12
Pegah Kamousi, S. Suri
We investigate the computational complexity of minimum spanning trees and maximum flows in a simple model of stochastic networks, where each node or edge of an undirected master graph can fail with an independent and arbitrary probability. We show that computing the expected length of the MST or the value of the max-flow is #P-Hard, but that for the MST it can be approximated within O(log n) factor for metric graphs. The hardness proof for the MST applies even to Euclidean graphs in 3 dimensions. We also show that the tail bounds for the MST cannot be approximated in general to any multiplicative factor unless P = NP. This stochastic MST problem was mentioned but left unanswered by Bertsimas, Jaillet and Odoni [Operations Research, 1990] in their work on a priori optimization. More generally, we also consider the complexity of linear programming under probabilistic constraints, and show it to be #P-Hard. If the linear program has a constant number of variables, then it can be solved exactly in polynomial time. For general dimensions, we give a randomized algorithm for approximating the probability of LP feasibility.
{"title":"Stochastic Minimum Spanning Trees and Related Problems","authors":"Pegah Kamousi, S. Suri","doi":"10.1137/1.9781611973013.12","DOIUrl":"https://doi.org/10.1137/1.9781611973013.12","url":null,"abstract":"We investigate the computational complexity of minimum spanning trees and maximum flows in a simple model of stochastic networks, where each node or edge of an undirected master graph can fail with an independent and arbitrary probability. We show that computing the expected length of the MST or the value of the max-flow is #P-Hard, but that for the MST it can be approximated within O(log n) factor for metric graphs. The hardness proof for the MST applies even to Euclidean graphs in 3 dimensions. We also show that the tail bounds for the MST cannot be approximated in general to any multiplicative factor unless P = NP. This stochastic MST problem was mentioned but left unanswered by Bertsimas, Jaillet and Odoni [Operations Research, 1990] in their work on a priori optimization. More generally, we also consider the complexity of linear programming under probabilistic constraints, and show it to be #P-Hard. If the linear program has a constant number of variables, then it can be solved exactly in polynomial time. For general dimensions, we give a randomized algorithm for approximating the probability of LP feasibility.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"168 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114166327","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 : 2011-01-22DOI: 10.1137/1.9781611973013.14
G. Louchard, C. Martínez, H. Prodinger
We analyze in detail a leader election protocol that we call the Swedish leader election protocol. The goal is to select one among n > 0 players, by proceeding through a number of rounds. If there is only one player remaining, the protocol stops and the player is declared the leader. Otherwise, all remaining players flip a biased coin; with probability q the player survives to the next round, with probability p = 1 − q the player loses and plays no further... unless all players lose in a given round, so all them play again. In the classical leader election protocol, any number of null rounds may take place, and with probability 1 some player will ultimately be elected. In the Swedish leader election protocol there is a maximum number τ of consecutive null rounds, and if the threshold is attained the protocol fails without declaring a leader. We analyze several parameters of interest of this protocol as functions of n, q and τ, including the probability of success, the expected number of rounds, the expected number of leftovers (number of players still playing by the time the protocol fails), etc. We also discuss several variations and how to cope with their analysis, e.g., if we bound the total number of null rounds, consecutive or not.
{"title":"The Swedish Leader Election Protocol: Analysis and Variations","authors":"G. Louchard, C. Martínez, H. Prodinger","doi":"10.1137/1.9781611973013.14","DOIUrl":"https://doi.org/10.1137/1.9781611973013.14","url":null,"abstract":"We analyze in detail a leader election protocol that we call the Swedish leader election protocol. The goal is to select one among n > 0 players, by proceeding through a number of rounds. If there is only one player remaining, the protocol stops and the player is declared the leader. Otherwise, all remaining players flip a biased coin; with probability q the player survives to the next round, with probability p = 1 − q the player loses and plays no further... unless all players lose in a given round, so all them play again. In the classical leader election protocol, any number of null rounds may take place, and with probability 1 some player will ultimately be elected. In the Swedish leader election protocol there is a maximum number τ of consecutive null rounds, and if the threshold is attained the protocol fails without declaring a leader. We analyze several parameters of interest of this protocol as functions of n, q and τ, including the probability of success, the expected number of rounds, the expected number of leftovers (number of players still playing by the time the protocol fails), etc. We also discuss several variations and how to cope with their analysis, e.g., if we bound the total number of null rounds, consecutive or not.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126802256","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 : 2011-01-22DOI: 10.1137/1.9781611973013.7
Sandrine Dasse-Hartaut, P. Hitczenko
We describe a probabilistic approach to a relatively new combinatorial object called staircase tableaux. Our approach allows us to analyze several parameters of a randomly chosen staircase tableau of a given size.
{"title":"Some Properties of Random Staircase Tableaux","authors":"Sandrine Dasse-Hartaut, P. Hitczenko","doi":"10.1137/1.9781611973013.7","DOIUrl":"https://doi.org/10.1137/1.9781611973013.7","url":null,"abstract":"We describe a probabilistic approach to a relatively new combinatorial object called staircase tableaux. Our approach allows us to analyze several parameters of a randomly chosen staircase tableau of a given size.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132687004","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 : 2010-08-29DOI: 10.1137/1.9781611973013.9
David Eisenstat
What does a typical road network look like? Existing generative models tend to focus on one aspect to the exclusion of others. We introduce the general-purpose emph{quadtree model} and analyze its shortest paths and maximum flow.
{"title":"Random Road Networks: The Quadtree Model","authors":"David Eisenstat","doi":"10.1137/1.9781611973013.9","DOIUrl":"https://doi.org/10.1137/1.9781611973013.9","url":null,"abstract":"What does a typical road network look like? Existing generative models tend to focus on one aspect to the exclusion of others. We introduce the general-purpose emph{quadtree model} and analyze its shortest paths and maximum flow.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"23 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120821538","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 : 2010-01-16DOI: 10.1137/1.9781611973006.10
Takayuki Ichiba, K. Iwama
We study digital-goods auctions for items in unlimited supply introduced by Goldberg, Hartline and Wright. Since no deterministic algorithms are competitive for this class of auctions, one of the central research issues is how to obtain a nice probabilistic distribution over truthful algorithms. In this paper, we introduce a rather systematic approach to this goal: Consider for example the Sampling Cost Share (SCS) auction. It is well known that SCS works well if the the current bid vector produces many winners against F(2), the standard benchmark algorithm for competitive analysis. In fact, its competitive ratio is approaching to 2.0 as k (= the number of F(2) winners) grows. On the other hand, its competitive ratio becomes as bad as 4.0 for k = 2. Our new approach is to develop a sequence of similar cost-share type algorithms, DCSk, which work well for small k. Now we choose a sufficiently large constant N and run DCS1, DCS2, ..., DCSN and SCS with probabilities p1, p2, ..., pN and q, respectively. It should be noted that we can use LP to obtain optimal p1, p2,..., pN and q. By this averaging method, we can improve the competitive ratio of SCS from 4.0 to 3.531 and that of the currently best Aggregated γ3 algorithm due to Hartline and McGrew from 3.243 to 3.119.
{"title":"Averaging Techniques for Competitive Auctions","authors":"Takayuki Ichiba, K. Iwama","doi":"10.1137/1.9781611973006.10","DOIUrl":"https://doi.org/10.1137/1.9781611973006.10","url":null,"abstract":"We study digital-goods auctions for items in unlimited supply introduced by Goldberg, Hartline and Wright. Since no deterministic algorithms are competitive for this class of auctions, one of the central research issues is how to obtain a nice probabilistic distribution over truthful algorithms. In this paper, we introduce a rather systematic approach to this goal: Consider for example the Sampling Cost Share (SCS) auction. It is well known that SCS works well if the the current bid vector produces many winners against F(2), the standard benchmark algorithm for competitive analysis. In fact, its competitive ratio is approaching to 2.0 as k (= the number of F(2) winners) grows. On the other hand, its competitive ratio becomes as bad as 4.0 for k = 2. Our new approach is to develop a sequence of similar cost-share type algorithms, DCSk, which work well for small k. Now we choose a sufficiently large constant N and run DCS1, DCS2, ..., DCSN and SCS with probabilities p1, p2, ..., pN and q, respectively. It should be noted that we can use LP to obtain optimal p1, p2,..., pN and q. By this averaging method, we can improve the competitive ratio of SCS from 4.0 to 3.531 and that of the currently best Aggregated γ3 algorithm due to Hartline and McGrew from 3.243 to 3.119.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"222 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134242566","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 : 2010-01-16DOI: 10.1137/1.9781611973006.2
S. Balaji, H. Mahmoud
The Ehrenfest urn is a model for the mixing of gases in two chambers. Classic research deals with this system as a Markovian model with a fixed number of balls, and derives the steady-state behavior as a binomial distribution (which can be approximated by a normal distribution). We study the gradual change for an urn containing n balls from the initial condition to the steady state. We look at the status of the urn after kn draws. We identify three phases of kn: The growing sublinear, the linear, and the superlinear. In the growing sublinear phase the amount of gas in either chamber is normally distributed, with parameters that are influenced by the initial conditions. In the linear phase a different normal distribution applies, in which the influence of the initial conditions is attenuated. The steady state is not a good approximation until a superlinear amount of time has elapsed. At the superlinear stage the mix is nearly perfect, with a nearly perfect symmetrical normal distribution in which the effect of the initial conditions is completely washed away. We give interpretations for how the results in different phases conjoin at the "seam lines." The Gaussian results are obtained via martingale theory.
{"title":"Phases in the Mixing of Gases via the Ehrenfest Urn Model","authors":"S. Balaji, H. Mahmoud","doi":"10.1137/1.9781611973006.2","DOIUrl":"https://doi.org/10.1137/1.9781611973006.2","url":null,"abstract":"The Ehrenfest urn is a model for the mixing of gases in two chambers. Classic research deals with this system as a Markovian model with a fixed number of balls, and derives the steady-state behavior as a binomial distribution (which can be approximated by a normal distribution). We study the gradual change for an urn containing n balls from the initial condition to the steady state. We look at the status of the urn after kn draws. We identify three phases of kn: The growing sublinear, the linear, and the superlinear. In the growing sublinear phase the amount of gas in either chamber is normally distributed, with parameters that are influenced by the initial conditions. In the linear phase a different normal distribution applies, in which the influence of the initial conditions is attenuated. The steady state is not a good approximation until a superlinear amount of time has elapsed. At the superlinear stage the mix is nearly perfect, with a nearly perfect symmetrical normal distribution in which the effect of the initial conditions is completely washed away. We give interpretations for how the results in different phases conjoin at the \"seam lines.\" The Gaussian results are obtained via martingale theory.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114632788","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 : 2010-01-16DOI: 10.1137/1.9781611973006.12
Reinhard Bauer, M. Krug, D. Wagner
A k-degenerate graph is a graph in which every induced subgraph has a vertex with degree at most k. The class of k-degenerate graphs is interesting from a theoretical point of view and it plays an interesting role in the theory of fixed parameter tractability since some otherwise W[2]-hard domination problems become fixed-parameter tractable for k-degenerate graphs. It is a well-known fact that the k-degenerate graphs are exactly the graphs whose vertex-set can be well-ordered such that each vertex is incident to at most k larger vertices with respect to this ordering. A well-ordered k-degenerate graph is a labeled graph with vertex-labels 1, ..., n such that the ordering of the vertices by their labels is a well-ordering of the graph. We consider the problem of enumerating and generating well-ordered k-degenerate graphs with a given number of vertices and with a given number of vertices and edges, respectively, uniformly at random. By generating well-ordered k-degenerate graphs we generate at least one labeled copy of each unlabeled k-degenerate graph and we filter some but not all isomorphies compared to the classical labeled approach. We also introduce the class of strongly k-degenerate graphs, which are k-degenerate graphs with minimum degree k. These graphs are a natural generalization of k-regular graphs which can be used in order to generate graphs with predefined core-decomposition. We present efficient algorithms for generating well-ordered k-degenerate graphs with given number of vertices (and edges). After a precomputation which must only be performed once when generating more than one well-ordered k-degenerate graph these algorithms are almost optimal. Additionally, we present complete non-uniform generators for these classes with optimal running time. We also present an efficient and complete generator for well-ordered strongly k-degenerate graphs with given number of vertices (and edges). Finally, we present efficient algorithms for enumerating well-ordered k-degenerate and strongly k-degenerate graphs.
{"title":"Enumerating and Generating Labeled k-degenerate Graphs","authors":"Reinhard Bauer, M. Krug, D. Wagner","doi":"10.1137/1.9781611973006.12","DOIUrl":"https://doi.org/10.1137/1.9781611973006.12","url":null,"abstract":"A k-degenerate graph is a graph in which every induced subgraph has a vertex with degree at most k. The class of k-degenerate graphs is interesting from a theoretical point of view and it plays an interesting role in the theory of fixed parameter tractability since some otherwise W[2]-hard domination problems become fixed-parameter tractable for k-degenerate graphs. \u0000 \u0000It is a well-known fact that the k-degenerate graphs are exactly the graphs whose vertex-set can be well-ordered such that each vertex is incident to at most k larger vertices with respect to this ordering. A well-ordered k-degenerate graph is a labeled graph with vertex-labels 1, ..., n such that the ordering of the vertices by their labels is a well-ordering of the graph. \u0000 \u0000We consider the problem of enumerating and generating well-ordered k-degenerate graphs with a given number of vertices and with a given number of vertices and edges, respectively, uniformly at random. By generating well-ordered k-degenerate graphs we generate at least one labeled copy of each unlabeled k-degenerate graph and we filter some but not all isomorphies compared to the classical labeled approach. \u0000 \u0000We also introduce the class of strongly k-degenerate graphs, which are k-degenerate graphs with minimum degree k. These graphs are a natural generalization of k-regular graphs which can be used in order to generate graphs with predefined core-decomposition. \u0000 \u0000We present efficient algorithms for generating well-ordered k-degenerate graphs with given number of vertices (and edges). After a precomputation which must only be performed once when generating more than one well-ordered k-degenerate graph these algorithms are almost optimal. Additionally, we present complete non-uniform generators for these classes with optimal running time. We also present an efficient and complete generator for well-ordered strongly k-degenerate graphs with given number of vertices (and edges). Finally, we present efficient algorithms for enumerating well-ordered k-degenerate and strongly k-degenerate graphs.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"6 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114022886","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 : 2010-01-16DOI: 10.1137/1.9781611973006.8
L. Fleischer
Inspired by problems in data center scheduling, we study the submodularity of certain scheduling problems as a function of the set of machine capacities and the corresponding implications. In particular, we • give a short proof that, as a function of the excess vector, maximum generalized flow is submodular and minimum cost generalized flow is supermodular; • extend Wolsey's approximation guarantees for submodular covering problems to a new class of problems we call supermodular packing problems; • use these results to get tighter approximation guarantees for several data center scheduling problems.
{"title":"Data Center Scheduling, Generalized Flows, and Submodularity","authors":"L. Fleischer","doi":"10.1137/1.9781611973006.8","DOIUrl":"https://doi.org/10.1137/1.9781611973006.8","url":null,"abstract":"Inspired by problems in data center scheduling, we study the submodularity of certain scheduling problems as a function of the set of machine capacities and the corresponding implications. In particular, we \u0000 \u0000• give a short proof that, as a function of the excess vector, maximum generalized flow is submodular and minimum cost generalized flow is supermodular; \u0000 \u0000• extend Wolsey's approximation guarantees for submodular covering problems to a new class of problems we call supermodular packing problems; \u0000 \u0000• use these results to get tighter approximation guarantees for several data center scheduling problems.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124356540","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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