In boundary labeling, each point site is uniquely connected to a label placed on the boundary of an enclosing rectangle by a leader, which may be a rectilinear or straight line segment. To our knowledge, all the results reported in the literature for boundary labeling deal with the so-called one-to-one boundary labeling, i.e., different sites are labelled differently. In certain applications of boundary labeling, however, more than one site may be required to be connected to a common label. In this case, the presence of crossings among leaders often becomes inevitable. Minimizing the total number of crossings in boundary labeling becomes a critical design issue as crossing is often regarded as the main source of confusion in visualization. In this paper, we consider the crossing minimization problem for multi-site-to-one-label boundary labeling, i.e., finding the placements of labels and leaders such that the total number of crossings among leaders is minimized. We show the crossing minimization problem to be NP-complete under certain one-side and two-side labeling schemes. Subsequently, approximation algorithms are derived for the above intractable problems. We also present an O(n2 log3 n)-time algorithm for the problem of minimizing the total leader length for multi-site-to-one-label boundary labeling, where n is the number of labels.
{"title":"Many-to-one boundary labeling","authors":"Chun-Cheng Lin, Hao-Jen Kao, H. Yen","doi":"10.7155/jgaa.00169","DOIUrl":"https://doi.org/10.7155/jgaa.00169","url":null,"abstract":"In boundary labeling, each point site is uniquely connected to a label placed on the boundary of an enclosing rectangle by a leader, which may be a rectilinear or straight line segment. To our knowledge, all the results reported in the literature for boundary labeling deal with the so-called one-to-one boundary labeling, i.e., different sites are labelled differently. In certain applications of boundary labeling, however, more than one site may be required to be connected to a common label. In this case, the presence of crossings among leaders often becomes inevitable. Minimizing the total number of crossings in boundary labeling becomes a critical design issue as crossing is often regarded as the main source of confusion in visualization. In this paper, we consider the crossing minimization problem for multi-site-to-one-label boundary labeling, i.e., finding the placements of labels and leaders such that the total number of crossings among leaders is minimized. We show the crossing minimization problem to be NP-complete under certain one-side and two-side labeling schemes. Subsequently, approximation algorithms are derived for the above intractable problems. We also present an O(n2 log3 n)-time algorithm for the problem of minimizing the total leader length for multi-site-to-one-label boundary labeling, where n is the number of labels.","PeriodicalId":136557,"journal":{"name":"2007 6th International Asia-Pacific Symposium on Visualization","volume":"73 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125535845","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 : 2007-10-15DOI: 10.1109/APVIS.2007.329290
C. Mueller, Benjamin Martin, A. Lumsdaine
Visual similarity matrices (VSMs) are a common technique for visualizing graphs and other types of relational data. While traditionally used for small data sets or well-ordered large data sets, they have recently become popular for visualizing large graphs. However, our experience with users has revealed that large VSMs are difficult to interpret. In this paper, we catalog common structural features found in VSMs and provide graph-based interpretations of the structures. We also discuss implementation details that affect the interpretability of VSMs for large data sets.
{"title":"Interpreting large visual similarity matrices","authors":"C. Mueller, Benjamin Martin, A. Lumsdaine","doi":"10.1109/APVIS.2007.329290","DOIUrl":"https://doi.org/10.1109/APVIS.2007.329290","url":null,"abstract":"Visual similarity matrices (VSMs) are a common technique for visualizing graphs and other types of relational data. While traditionally used for small data sets or well-ordered large data sets, they have recently become popular for visualizing large graphs. However, our experience with users has revealed that large VSMs are difficult to interpret. In this paper, we catalog common structural features found in VSMs and provide graph-based interpretations of the structures. We also discuss implementation details that affect the interpretability of VSMs for large data sets.","PeriodicalId":136557,"journal":{"name":"2007 6th International Asia-Pacific Symposium on Visualization","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122862952","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 : 2007-10-15DOI: 10.1109/APVIS.2007.329304
A. Ahmed, V. Batagelj, Xiaoyan Fu, Seok-Hee Hong, D. Merrick, Andrej Mrvar
In this paper, we present a case study for the visualisation and analysis of large and complex temporal multivariate networks derived from the Internet movie database (IMDB). Our approach is to integrate network analysis methods with visualisation in order to address scalability and complexity issues. In particular, we defined new analysis methods such as (p,q)-core and 4-ring to identify important dense subgraphs and short cycles from the huge bipartite graphs. We applied island analysis for a specific time slice in order to identify important and meaningful subgraphs. Further, a temporal Kevin Bacon graph and a temporal two mode network are extracted in order to provide insight and knowledge on the evolution.
{"title":"Visualisation and analysis of the internet movie database","authors":"A. Ahmed, V. Batagelj, Xiaoyan Fu, Seok-Hee Hong, D. Merrick, Andrej Mrvar","doi":"10.1109/APVIS.2007.329304","DOIUrl":"https://doi.org/10.1109/APVIS.2007.329304","url":null,"abstract":"In this paper, we present a case study for the visualisation and analysis of large and complex temporal multivariate networks derived from the Internet movie database (IMDB). Our approach is to integrate network analysis methods with visualisation in order to address scalability and complexity issues. In particular, we defined new analysis methods such as (p,q)-core and 4-ring to identify important dense subgraphs and short cycles from the huge bipartite graphs. We applied island analysis for a specific time slice in order to identify important and meaningful subgraphs. Further, a temporal Kevin Bacon graph and a temporal two mode network are extracted in order to provide insight and knowledge on the evolution.","PeriodicalId":136557,"journal":{"name":"2007 6th International Asia-Pacific Symposium on Visualization","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121942810","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 : 2007-10-15DOI: 10.1109/APVIS.2007.329288
Michael Balzer, O. Deussen
The level-of-detail techniques presented in this paper enable a comprehensible interactive visualization of large and complex clustered graph layouts either in 2D or 3D. Implicit surfaces are used for the visually simplified representation of vertex clusters, and so-called edge bundles are formed for the simplification of edges. Additionally, dedicated transition techniques are provided for continuously adaptive and adjustable views of graphs that range from very abstract to very detailed representations.
{"title":"Level-of-detail visualization of clustered graph layouts","authors":"Michael Balzer, O. Deussen","doi":"10.1109/APVIS.2007.329288","DOIUrl":"https://doi.org/10.1109/APVIS.2007.329288","url":null,"abstract":"The level-of-detail techniques presented in this paper enable a comprehensible interactive visualization of large and complex clustered graph layouts either in 2D or 3D. Implicit surfaces are used for the visually simplified representation of vertex clusters, and so-called edge bundles are formed for the simplification of edges. Additionally, dedicated transition techniques are provided for continuously adaptive and adjustable views of graphs that range from very abstract to very detailed representations.","PeriodicalId":136557,"journal":{"name":"2007 6th International Asia-Pacific Symposium on Visualization","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129458841","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 : 2007-10-15DOI: 10.1109/APVIS.2007.329284
M. R. Karim, M. S. Rahman
A straight-line grid drawing of a planar graph G is a drawing of G on an integer grid such that each vertex is drawn as a grid point and each edge is drawn as a straight-line segment without edge crossings. It is well known that a planar graph of n vertices admits a straight-line grid drawing on a grid of area O(n2). A lower bound of Omega(n2) on the area-requirement for straight-line grid drawings of certain planar graphs is also known. In this paper, we introduce a fairly large class of planar graphs which admits a straight-line grid drawing on a grid of area O(n). We also give a linear-time algorithm to find such a drawing. Our new class of planar graphs, which we call "doughnut graphs," is a subclass of 5-connected planar graphs. We also show several interesting properties of "doughnut graphs" in this paper
{"title":"Straight-line grid drawings of planar graphs with linear area","authors":"M. R. Karim, M. S. Rahman","doi":"10.1109/APVIS.2007.329284","DOIUrl":"https://doi.org/10.1109/APVIS.2007.329284","url":null,"abstract":"A straight-line grid drawing of a planar graph G is a drawing of G on an integer grid such that each vertex is drawn as a grid point and each edge is drawn as a straight-line segment without edge crossings. It is well known that a planar graph of n vertices admits a straight-line grid drawing on a grid of area O(n2). A lower bound of Omega(n2) on the area-requirement for straight-line grid drawings of certain planar graphs is also known. In this paper, we introduce a fairly large class of planar graphs which admits a straight-line grid drawing on a grid of area O(n). We also give a linear-time algorithm to find such a drawing. Our new class of planar graphs, which we call \"doughnut graphs,\" is a subclass of 5-connected planar graphs. We also show several interesting properties of \"doughnut graphs\" in this paper","PeriodicalId":136557,"journal":{"name":"2007 6th International Asia-Pacific Symposium on Visualization","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114982678","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 : 2007-10-15DOI: 10.1109/APVIS.2007.329302
Xiaoyan Fu, Seok-Hee Hong, Nikola S. Nikolov, Xiaobin Shen, Y. Wu, Kai Xu
This paper presents various methods for visualization and analysis of email networks; visualization on the surface of a sphere to reveal communication patterns between different groups, a hierarchical drawing displaying the centrality analysis of nodes to emphasize important nodes, a 2.5D visualization for temporal email networks to analyze the evolution of email relationships changing over time, and an ambient display for finding social circles derived from the email network. Each method was evaluated with various data sets from a research organization. We also extended our method for visual analysis of an email virus network.
{"title":"Visualization and analysis of email networks","authors":"Xiaoyan Fu, Seok-Hee Hong, Nikola S. Nikolov, Xiaobin Shen, Y. Wu, Kai Xu","doi":"10.1109/APVIS.2007.329302","DOIUrl":"https://doi.org/10.1109/APVIS.2007.329302","url":null,"abstract":"This paper presents various methods for visualization and analysis of email networks; visualization on the surface of a sphere to reveal communication patterns between different groups, a hierarchical drawing displaying the centrality analysis of nodes to emphasize important nodes, a 2.5D visualization for temporal email networks to analyze the evolution of email relationships changing over time, and an ambient display for finding social circles derived from the email network. Each method was evaluated with various data sets from a research organization. We also extended our method for visual analysis of an email virus network.","PeriodicalId":136557,"journal":{"name":"2007 6th International Asia-Pacific Symposium on Visualization","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114299377","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 : 2007-10-15DOI: 10.1109/APVIS.2007.329282
Weidong Huang
Graphs are typically visualized as node-link diagrams. Although there is a fair amount of research focusing on crossing minimization to improve readability, little attention has been paid on how to handle crossings when they are an essential part of the final visualizations. This requires us to understand how people read graphs and how crossings affect reading performance. As an initial step to this end, a preliminary eye tracking experiment was conducted. The specific purpose of this experiment was to test the effects of crossing angles and geometric-path tendency on eye movements and performance. Sixteen subjects performed both path search and node locating tasks with six drawings. The results showed that small angles can slow down and trigger extra eye movements, causing delays for path search tasks, whereas crossings have little impact on node locating tasks. Geometric-path tendency indicates that a path between two nodes can become harder to follow when many branches of the path go toward the target node. The insights obtained are discussed with a view to further confirmation in future work.
{"title":"Using eye tracking to investigate graph layout effects","authors":"Weidong Huang","doi":"10.1109/APVIS.2007.329282","DOIUrl":"https://doi.org/10.1109/APVIS.2007.329282","url":null,"abstract":"Graphs are typically visualized as node-link diagrams. Although there is a fair amount of research focusing on crossing minimization to improve readability, little attention has been paid on how to handle crossings when they are an essential part of the final visualizations. This requires us to understand how people read graphs and how crossings affect reading performance. As an initial step to this end, a preliminary eye tracking experiment was conducted. The specific purpose of this experiment was to test the effects of crossing angles and geometric-path tendency on eye movements and performance. Sixteen subjects performed both path search and node locating tasks with six drawings. The results showed that small angles can slow down and trigger extra eye movements, causing delays for path search tasks, whereas crossings have little impact on node locating tasks. Geometric-path tendency indicates that a path between two nodes can become harder to follow when many branches of the path go toward the target node. The insights obtained are discussed with a view to further confirmation in future work.","PeriodicalId":136557,"journal":{"name":"2007 6th International Asia-Pacific Symposium on Visualization","volume":"46 2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128915101","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 : 2007-10-15DOI: 10.1109/APVIS.2007.329306
Kai Xu, Andrew Cunningham, Seok-Hee Hong, B. Thomas
In this paper, we introduce a new method, GraphScape, to visualize multivariate networks, i.e., graphs with multivariate data associated with their nodes. GraphScape adopts a landscape metaphor with network structure displayed on a 2D plane and the surface height in the third dimension represents node attribute. More than one attribute can be visualized simultaneously by using multiple surfaces. In addition, GraphScape can be easily combined with existing methods to further increase the total number of attributes visualized. One of the major goals of GraphScape is to reveal multivariate graph clustering, which is based on both network structure and node attributes. This is achieved by a new layout algorithm and an innovative way of constructing attribute surface, which also allows visual clustering at different scales through interaction. A simplified attribute surface model is also proposed to reduce computation requirement when visualizing large networks. GraphScape is applied to networks of three different size (20, 100, and 1500) to demonstrate its effectiveness.
{"title":"GraphScape: integrated multivariate network visualization","authors":"Kai Xu, Andrew Cunningham, Seok-Hee Hong, B. Thomas","doi":"10.1109/APVIS.2007.329306","DOIUrl":"https://doi.org/10.1109/APVIS.2007.329306","url":null,"abstract":"In this paper, we introduce a new method, GraphScape, to visualize multivariate networks, i.e., graphs with multivariate data associated with their nodes. GraphScape adopts a landscape metaphor with network structure displayed on a 2D plane and the surface height in the third dimension represents node attribute. More than one attribute can be visualized simultaneously by using multiple surfaces. In addition, GraphScape can be easily combined with existing methods to further increase the total number of attributes visualized. One of the major goals of GraphScape is to reveal multivariate graph clustering, which is based on both network structure and node attributes. This is achieved by a new layout algorithm and an innovative way of constructing attribute surface, which also allows visual clustering at different scales through interaction. A simplified attribute surface model is also proposed to reduce computation requirement when visualizing large networks. GraphScape is applied to networks of three different size (20, 100, and 1500) to demonstrate its effectiveness.","PeriodicalId":136557,"journal":{"name":"2007 6th International Asia-Pacific Symposium on Visualization","volume":"73 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126374240","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 : 2007-10-15DOI: 10.1109/APVIS.2007.329283
Dennis Goehlsdorf, M. Kaufmann, Martin Siebenhaller
A problem arising when drawing disconnected graphs, is the placement of the connected components. The problem corresponds to finding an appropriate two-dimensional, non-overlapping placement of given objects. Most layout algorithms assume a connected graph and do not deal with connected components. Thus, each component is laid out separately which requires an additional step to arrange these components. There are only few approaches in literature which address this task. We review some of them and present new methods that improve existing results. Our approach is based on a classical greedy approach described by Freivalds et al. which uses a polyomino representation of the objects. We introduce new quality measures to evaluate a two-dimensional placement which yield more compact layouts. Our approach particularly eliminates most cases in which previous approaches returned poor results.
{"title":"Placing connected components of disconnected graphs","authors":"Dennis Goehlsdorf, M. Kaufmann, Martin Siebenhaller","doi":"10.1109/APVIS.2007.329283","DOIUrl":"https://doi.org/10.1109/APVIS.2007.329283","url":null,"abstract":"A problem arising when drawing disconnected graphs, is the placement of the connected components. The problem corresponds to finding an appropriate two-dimensional, non-overlapping placement of given objects. Most layout algorithms assume a connected graph and do not deal with connected components. Thus, each component is laid out separately which requires an additional step to arrange these components. There are only few approaches in literature which address this task. We review some of them and present new methods that improve existing results. Our approach is based on a classical greedy approach described by Freivalds et al. which uses a polyomino representation of the objects. We introduce new quality measures to evaluate a two-dimensional placement which yield more compact layouts. Our approach particularly eliminates most cases in which previous approaches returned poor results.","PeriodicalId":136557,"journal":{"name":"2007 6th International Asia-Pacific Symposium on Visualization","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130683713","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 : 2007-10-15DOI: 10.1109/APVIS.2007.329275
Martin Harrigan, P. Healy
A directed graph can model any ordered relationship between objects. However, visualizing such graphs can be a challenging task. If the graph is undirected, a popular strategy is to choose a significant spanning tree, nominate a vertex as the root, for example the vertex whose distance from all other vertices is minimal, hang the significant spanning subtrees from this root and add in the remaining edges in some unobtrusive manner. In the directed case the spanning tree is a tree DAG (a directed graph without any undirected cycles) and not simply a directed tree with one appropriate root. It may have multiple sources (vertices with indegree equal to zero) that all warrant root status and so the undirected approach must be modified somewhat. In this paper, we present a method of drawing directed graphs that emphasizes a significant spanning tree. It can be considered a variation of the Sugiyama framework in that it combines two steps of the framework (leveling and crossing minimisation) by finding, in linear time, a leveling of the graph that is level planar with respect to some spanning tree and restricting the permutations of the vertices on each level to those that constitute a level planar embedding of this subgraph. The edges of the spanning tree will therefore not cross each other. Using the globally oriented Fiedler vector we choose permutations of the vertices on each level that reduce the number of crossings between the remaining edges.
{"title":"Efficiently drawing a significant spanning tree of a directed graph","authors":"Martin Harrigan, P. Healy","doi":"10.1109/APVIS.2007.329275","DOIUrl":"https://doi.org/10.1109/APVIS.2007.329275","url":null,"abstract":"A directed graph can model any ordered relationship between objects. However, visualizing such graphs can be a challenging task. If the graph is undirected, a popular strategy is to choose a significant spanning tree, nominate a vertex as the root, for example the vertex whose distance from all other vertices is minimal, hang the significant spanning subtrees from this root and add in the remaining edges in some unobtrusive manner. In the directed case the spanning tree is a tree DAG (a directed graph without any undirected cycles) and not simply a directed tree with one appropriate root. It may have multiple sources (vertices with indegree equal to zero) that all warrant root status and so the undirected approach must be modified somewhat. In this paper, we present a method of drawing directed graphs that emphasizes a significant spanning tree. It can be considered a variation of the Sugiyama framework in that it combines two steps of the framework (leveling and crossing minimisation) by finding, in linear time, a leveling of the graph that is level planar with respect to some spanning tree and restricting the permutations of the vertices on each level to those that constitute a level planar embedding of this subgraph. The edges of the spanning tree will therefore not cross each other. Using the globally oriented Fiedler vector we choose permutations of the vertices on each level that reduce the number of crossings between the remaining edges.","PeriodicalId":136557,"journal":{"name":"2007 6th International Asia-Pacific Symposium on Visualization","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121599744","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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