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Adjacency List Model
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Joe Celko – One of the best experts on this subject based on the ideXlab platform.

Chapter 28 – Trees and Hierarchies in SQL
Joe Celko's SQL for Smarties, 2015CoAuthors: Joe CelkoAbstract:A tree is a special kind of directed graph. Graphs are data structures that are made up of nodes connected by edges. There are several ways to define a tree: it is a graph with no cycles and it is a graph where all nodes except the root have indegree one and the root has indegree zero. Another defining property is that a path can be found from the root to any other node in the tree by following the edges in their natural direction. Most Structured Query Language (SQL) databases use the Adjacency List Model for two reasons. The first reason is that Dr. Codd came up with it in the early days of the relational Model, and nobody thought about it after that. The second reason is that the Adjacency List is a way of “faking” pointer chains—the traditional programming method in procedural languages for handling trees. Because SQL is a setoriented language, the nested set Model is a better Model for the approach discussed in this chapter.

Adjacency List Model
Joe Celko's Trees and Hierarchies in SQL for Smarties, 2012CoAuthors: Joe CelkoAbstract:The chapter discusses adjacent List Models used in databases. The chapter describes a method for showing hierarchies in SQL that consists of a column for the boss and another column for the employee in a relationship. It is a direct implementation in a table of the Adjacency List Model of a graph. Oracle is the first commercial database to use SQL, and the sample database that comes with their product, nicknamed the “Scott/Tiger” database in the trade because of its default user and password codes, uses an Adjacency List Model in a combination Personnel/Organizational chart table. The organizational structure and the personnel data are mixed together in the same row. This Model is popular as it is the most natural way to convert from an IMS database or from a procedural language to SQL. The simple Adjacency List Model is not a normalized schema. A normalized schema has no data redundancy and is safe from data anomalies. The chapter proposes three characteristics required in a data Model. First, that the typical Adjacency List Model table includes information about the node, as well as who the boss in each row is. The second characteristic is that each fact appears in one place in the schema, but the subtree of each node can be in more than one row. The third characteristic is that each fact appears one time in the schema. Violations of these conditions result in anomalies. The chapter Lists down the fundamental problems of Adjacency List Models and provide solutions to these problems.

Other Models for Trees
Joe Celko's Trees and Hierarchies in SQL for Smarties, 2012CoAuthors: Joe CelkoAbstract:The chapter discusses the Models that use different approaches and properties of trees, some of which are hybrids of other Models. These Models include Models such as Adjacency List with selfreferences and subordinate Adjacency List. Subordinate Adjacency List is a modification of the usual Adjacency List Model and shows the edges of a graph as oriented from the superior to the subordinate. Nodes without a subordinate are leaf nodes, and they have a null value. Adjacency List with selfreferences are also slight modification of the usual Adjacency List Model and include an edge that loops back to the same node. The chapter also discusses hybrid moles, such as Adjacency and nested sets Model, nested set with depth Model, Adjacency and depth Model, and computed hybrid Models. The chapter also discusses the problem of inability of the nested set Model to represent more a general graph in SQL.
Pan Peng – One of the best experts on this subject based on the ideXlab platform.

APPROX/RANDOM – Testable properties in general graphs and random order streaming
, 2020CoAuthors: Artur Czumaj, Pan Peng, Hendrik Fichtenberger, Christian SohlerAbstract:We present a novel framework closely linking the areas of property testing and data streaming algorithms in the setting of general graphs. It has been recently shown (Monemizadeh et al. 2017) that for boundeddegree graphs, any constantquery tester can be emulated in the random order streaming Model by a streaming algorithm that uses only space required to store a constant number of words. However, in a more natural setting of general graphs, with no restriction on the maximum degree, no such results were known because of our lack of understanding of constantquery testers in general graphs and lack of techniques to appropriately emulate in the streaming setting offline algorithms allowing many highdegree vertices.
In this work we advance our understanding on both of these challenges. First, we provide canonical testers for all constantquery testers for general graphs, both, for onesided and twosided errors. Such canonizations were only known before (in the Adjacency matrix Model) for dense graphs (Goldreich and Trevisan 2003) and (in the Adjacency List Model) for bounded degree (di)graphs (Goldreich and Ron 2011, Czumaj et al. 2016). Using the concept of canonical testers, we then prove that every property of general graphs that is constantquery testable with onesided error can also be tested in constantspace with onesided error in the random order streaming Model.
Our results imply, among others, that properties like (s,t) disconnectivity, kpathfreeness, etc. are constantspace testable in random order streams.

Testable Properties in General Graphs and Random Order Streaming
arXiv: Data Structures and Algorithms, 2019CoAuthors: Artur Czumaj, Pan Peng, Hendrik Fichtenberger, Christian SohlerAbstract:We present a novel framework closely linking the areas of property testing and data streaming algorithms in the setting of general graphs. It has been recently shown (Monemizadeh et al. 2017) that for boundeddegree graphs, any constantquery tester can be emulated in the random order streaming Model by a streaming algorithm that uses only space required to store a constant number of words. However, in a more natural setting of general graphs, with no restriction on the maximum degree, no such results were known because of our lack of understanding of constantquery testers in general graphs and lack of techniques to appropriately emulate in the streaming setting offline algorithms allowing many highdegree vertices.
In this work we advance our understanding on both of these challenges. First, we provide canonical testers for all constantquery testers for general graphs, both, for onesided and twosided errors. Such canonizations were only known before (in the Adjacency matrix Model) for dense graphs (Goldreich and Trevisan 2003) and (in the Adjacency List Model) for bounded degree (di)graphs (Goldreich and Ron 2011, Czumaj et al. 2016). Using the concept of canonical testers, we then prove that every property of general graphs that is constantquery testable with onesided error can also be tested in constantspace with onesided error in the random order streaming Model.
Our results imply, among others, that properties like $(s,t)$ disconnectivity, $k$pathfreeness, etc. are constantspace testable in random order streams. 
Testable Bounded Degree Graph Properties Are Random Order Streamable
arXiv: Data Structures and Algorithms, 2017CoAuthors: Morteza Monemizadeh, Shanmugavelayutham Muthukrishnan, Pan Peng, Christian SohlerAbstract:We study which property testing and sublinear time algorithms can be transformed into graph streaming algorithms for random order streams. Our main result is that for bounded degree graphs, any property that is constantquery testable in the Adjacency List Model can be tested with constant space in a singlepass in random order streams. Our result is obtained by estimating the distribution of local neighborhoods of the vertices on a random order graph stream using constant space.
We then show that our approach can also be applied to constant time approximation algorithms for bounded degree graphs in the Adjacency List Model: As an example, we obtain a constantspace singlepass random order streaming algorithms for approximating the size of a maximum matching with additive error $\epsilon n$ ($n$ is the number of nodes).
Our result establishes for the first time that a large class of sublinear algorithms can be simulated in random order streams, while $\Omega(n)$ space is needed for many graph streaming problems for adversarial orders.
Christian Sohler – One of the best experts on this subject based on the ideXlab platform.

APPROX/RANDOM – Testable properties in general graphs and random order streaming
, 2020CoAuthors: Artur Czumaj, Pan Peng, Hendrik Fichtenberger, Christian SohlerAbstract:We present a novel framework closely linking the areas of property testing and data streaming algorithms in the setting of general graphs. It has been recently shown (Monemizadeh et al. 2017) that for boundeddegree graphs, any constantquery tester can be emulated in the random order streaming Model by a streaming algorithm that uses only space required to store a constant number of words. However, in a more natural setting of general graphs, with no restriction on the maximum degree, no such results were known because of our lack of understanding of constantquery testers in general graphs and lack of techniques to appropriately emulate in the streaming setting offline algorithms allowing many highdegree vertices.
In this work we advance our understanding on both of these challenges. First, we provide canonical testers for all constantquery testers for general graphs, both, for onesided and twosided errors. Such canonizations were only known before (in the Adjacency matrix Model) for dense graphs (Goldreich and Trevisan 2003) and (in the Adjacency List Model) for bounded degree (di)graphs (Goldreich and Ron 2011, Czumaj et al. 2016). Using the concept of canonical testers, we then prove that every property of general graphs that is constantquery testable with onesided error can also be tested in constantspace with onesided error in the random order streaming Model.
Our results imply, among others, that properties like (s,t) disconnectivity, kpathfreeness, etc. are constantspace testable in random order streams.

Testable Properties in General Graphs and Random Order Streaming
arXiv: Data Structures and Algorithms, 2019CoAuthors: Artur Czumaj, Pan Peng, Hendrik Fichtenberger, Christian SohlerAbstract:We present a novel framework closely linking the areas of property testing and data streaming algorithms in the setting of general graphs. It has been recently shown (Monemizadeh et al. 2017) that for boundeddegree graphs, any constantquery tester can be emulated in the random order streaming Model by a streaming algorithm that uses only space required to store a constant number of words. However, in a more natural setting of general graphs, with no restriction on the maximum degree, no such results were known because of our lack of understanding of constantquery testers in general graphs and lack of techniques to appropriately emulate in the streaming setting offline algorithms allowing many highdegree vertices.
In this work we advance our understanding on both of these challenges. First, we provide canonical testers for all constantquery testers for general graphs, both, for onesided and twosided errors. Such canonizations were only known before (in the Adjacency matrix Model) for dense graphs (Goldreich and Trevisan 2003) and (in the Adjacency List Model) for bounded degree (di)graphs (Goldreich and Ron 2011, Czumaj et al. 2016). Using the concept of canonical testers, we then prove that every property of general graphs that is constantquery testable with onesided error can also be tested in constantspace with onesided error in the random order streaming Model.
Our results imply, among others, that properties like $(s,t)$ disconnectivity, $k$pathfreeness, etc. are constantspace testable in random order streams. 
Testable Bounded Degree Graph Properties Are Random Order Streamable
arXiv: Data Structures and Algorithms, 2017CoAuthors: Morteza Monemizadeh, Shanmugavelayutham Muthukrishnan, Pan Peng, Christian SohlerAbstract:We study which property testing and sublinear time algorithms can be transformed into graph streaming algorithms for random order streams. Our main result is that for bounded degree graphs, any property that is constantquery testable in the Adjacency List Model can be tested with constant space in a singlepass in random order streams. Our result is obtained by estimating the distribution of local neighborhoods of the vertices on a random order graph stream using constant space.
We then show that our approach can also be applied to constant time approximation algorithms for bounded degree graphs in the Adjacency List Model: As an example, we obtain a constantspace singlepass random order streaming algorithms for approximating the size of a maximum matching with additive error $\epsilon n$ ($n$ is the number of nodes).
Our result establishes for the first time that a large class of sublinear algorithms can be simulated in random order streams, while $\Omega(n)$ space is needed for many graph streaming problems for adversarial orders.