The Experts below are selected from a list of 261 Experts worldwide ranked by ideXlab platform
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, 2015Co-Authors: 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 set-oriented 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, 2012Co-Authors: 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, 2012Co-Authors: 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 self-references 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 self-references 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.
-
Graphs in SQL
Joe Celko's Trees and Hierarchies in SQL for Smarties, 2012Co-Authors: Joe CelkoAbstract:A graph is a data structure made up of nodes connected by edges. Edges can be directed or undirected. The number of edges entering a node is its indegree; likewise, the number of edges leaving a node is its outdegree. A set of edges that allow traveling from one node to another is called a “path.” The most common way to Model a graph in Structured Query Language (SQL) is with an Adjacency List Model. Finding a path in a graph is the most important commercial application of graphs. Graphs Model transportation networks, electrical and cable systems, process control flow, and thousands of other things. An Adjacency matrix is a square array whose rows are out-nodes and columns are in-nodes of a graph. A “1” in a cell means that there is an edge between the two nodes. A polygon can be described as a set of comer points in an (x, y) coordinate system.
-
37 – Graphs in SQL
Joe Celko's SQL for Smarties, 2011Co-Authors: Joe CelkoAbstract:Publisher Summary This chapter focuses on the use of graphs in SQL. Graphs are a general way to represent various types of data and their relationships and they are made up of nodes connected by edges, which can be directed or undirected. The most common way to Model a graph in SQL is with an Adjacency List Model where each edge of the graph is shown as a pair of nodes in which the ordering matters, and then any values associated with that edge is shown in another column. Many graph algorithms are based on the Adjacency matrix Model and can be translated into SQL. The Split Node Nested Set Models for Graphs is a specialized trick for a certain class of graphs where an acyclic directed graph is loaded into a nested set Model by splitting the nodes. The acyclic graph becomes a tree, but with duplicated node values, and it is mainly useful when one wants to avoid recursion.
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
2020Co-Authors: 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 bounded-degree graphs, any constant-query 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 constant-query testers in general graphs and lack of techniques to appropriately emulate in the streaming setting off-line algorithms allowing many high-degree vertices. In this work we advance our understanding on both of these challenges. First, we provide canonical testers for all constant-query testers for general graphs, both, for one-sided and two-sided 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 constant-query testable with one-sided error can also be tested in constant-space with one-sided error in the random order streaming Model. Our results imply, among others, that properties like (s,t) disconnectivity, k-path-freeness, etc. are constant-space testable in random order streams.
-
Testable Properties in General Graphs and Random Order Streaming
arXiv: Data Structures and Algorithms, 2019Co-Authors: 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 bounded-degree graphs, any constant-query 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 constant-query testers in general graphs and lack of techniques to appropriately emulate in the streaming setting off-line algorithms allowing many high-degree vertices. In this work we advance our understanding on both of these challenges. First, we provide canonical testers for all constant-query testers for general graphs, both, for one-sided and two-sided 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 constant-query testable with one-sided error can also be tested in constant-space with one-sided error in the random order streaming Model. Our results imply, among others, that properties like $(s,t)$ disconnectivity, $k$-path-freeness, etc. are constant-space testable in random order streams.
-
Testable Bounded Degree Graph Properties Are Random Order Streamable
arXiv: Data Structures and Algorithms, 2017Co-Authors: 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 constant-query testable in the Adjacency List Model can be tested with constant space in a single-pass 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 constant-space single-pass 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.
-
ICALP - Testable bounded degree graph properties are random order streamable
2017Co-Authors: 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 constant-query testable in the Adjacency List Model can be tested with constant space in a single-pass 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 constant-space single-pass 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.
-
STACS - Testing Small Set Expansion in General Graphs
2015Co-Authors: Pan PengAbstract:We consider the problem of testing small set expansion for general graphs. A graph G is a (k,\phi)-expander if every subset of volume at most k has conductance at least \phi. Small set expansion has recently received significant attention due to its close connection to the unique games conjecture, the local graph partitioning algorithms and locally testable codes. We give testers with two-sided error and one-sided error in the \Adjacency List Model that allows degree and neighbor queries to the oracle of the input graph. The testers take as input an n-vertex graph G, a volume bound k, an expansion bound \phi and a distance parameter \varepsilon>0. For the two-sided error tester, with probability at least 2/3, it accepts the graph if it is a (k,\phi)-expander and rejects the graph if it is \varepsilon-far from any (k^*,\phi^*)-expander, where k^*=\Theta(k\varepsilon) and \phi^*=\Theta(\frac{\phi^4}{\min\{\log(4m/k),\log n\}\cdot(\ln k)}). The query complexity and running time of the tester are \widetilde{O}(\sqrt{m}\phi^{-4}\varepsilon^{-2}), where m is the number of edges of the graph. For the one-sided error tester, it accepts every (k,\phi)-expander, and with probability at least 2/3, rejects every graph that is \varepsilon-far from (k^*,\phi^*)-expander, where k^*=O(k^{1-\xi}) and \phi^*=O(\xi\phi^2) for any 0
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
2020Co-Authors: 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 bounded-degree graphs, any constant-query 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 constant-query testers in general graphs and lack of techniques to appropriately emulate in the streaming setting off-line algorithms allowing many high-degree vertices. In this work we advance our understanding on both of these challenges. First, we provide canonical testers for all constant-query testers for general graphs, both, for one-sided and two-sided 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 constant-query testable with one-sided error can also be tested in constant-space with one-sided error in the random order streaming Model. Our results imply, among others, that properties like (s,t) disconnectivity, k-path-freeness, etc. are constant-space testable in random order streams.
-
Testable Properties in General Graphs and Random Order Streaming
arXiv: Data Structures and Algorithms, 2019Co-Authors: 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 bounded-degree graphs, any constant-query 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 constant-query testers in general graphs and lack of techniques to appropriately emulate in the streaming setting off-line algorithms allowing many high-degree vertices. In this work we advance our understanding on both of these challenges. First, we provide canonical testers for all constant-query testers for general graphs, both, for one-sided and two-sided 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 constant-query testable with one-sided error can also be tested in constant-space with one-sided error in the random order streaming Model. Our results imply, among others, that properties like $(s,t)$ disconnectivity, $k$-path-freeness, etc. are constant-space testable in random order streams.
-
Testable Bounded Degree Graph Properties Are Random Order Streamable
arXiv: Data Structures and Algorithms, 2017Co-Authors: 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 constant-query testable in the Adjacency List Model can be tested with constant space in a single-pass 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 constant-space single-pass 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.
-
ICALP - Testable bounded degree graph properties are random order streamable
2017Co-Authors: 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 constant-query testable in the Adjacency List Model can be tested with constant space in a single-pass 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 constant-space single-pass 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.
Wang Daochen - One of the best experts on this subject based on the ideXlab platform.
-
Symmetries, graph properties, and quantum speedups
2020Co-Authors: Ben-david Shalev, Childs, Andrew M., Gilyén András, Kretschmer William, Podder Supartha, Wang DaochenAbstract:Aaronson and Ambainis (2009) and Chailloux (2018) showed that fully symmetric (partial) functions do not admit exponential quantum query speedups. This raises a natural question: how symmetric must a function be before it cannot exhibit a large quantum speedup? In this work, we prove that hypergraph symmetries in the Adjacency matrix Model allow at most a polynomial separation between randomized and quantum query complexities. We also show that, remarkably, permutation groups constructed out of these symmetries are essentially the only permutation groups that prevent super-polynomial quantum speedups. We prove this by fully characterizing the primitive permutation groups that allow super-polynomial quantum speedups. In contrast, in the Adjacency List Model for bounded-degree graphs (where graph symmetry is manifested differently), we exhibit a property testing problem that shows an exponential quantum speedup. These results resolve open questions posed by Ambainis, Childs, and Liu (2010) and Montanaro and de Wolf (2013).Comment: 46 pages. Subsumes arXiv:2001.09642 and arXiv:2001.10520; adds a characterization of permutation groups with speedup and an exponential speedup for Adjacency-List graph property testin
-
Can graph properties have exponential quantum speedup?
2020Co-Authors: Childs, Andrew M., Wang DaochenAbstract:Quantum computers can sometimes exponentially outperform classical ones, but only for problems with sufficient structure. While it is well known that query problems with full permutation symmetry can have at most polynomial quantum speedup -- even for partial functions -- it is unclear how far this condition must be relaxed to enable exponential speedup. In particular, it is natural to ask whether exponential speedup is possible for (partial) graph properties, in which the input describes a graph and the output can only depend on its isomorphism class. We show that the answer to this question depends strongly on the input Model. In the Adjacency matrix Model, we prove that the bounded-error randomized query complexity $R$ of any graph property $\mathcal{P}$ has $R(\mathcal{P}) = O(Q(\mathcal{P})^{6})$, where $Q$ is the bounded-error quantum query complexity. This negatively resolves an open question of Montanaro and de Wolf in the Adjacency matrix Model. More generally, we prove $R(\mathcal{P}) = O(Q(\mathcal{P})^{3l})$ for any $l$-uniform hypergraph property $\mathcal{P}$ in the Adjacency matrix Model. In direct contrast, in the Adjacency List Model for bounded-degree graphs, we exhibit a promise problem that shows an exponential separation between the randomized and quantum query complexities.Comment: 11 page
Daochen Wang - One of the best experts on this subject based on the ideXlab platform.
-
Symmetries, graph properties, and quantum speedups.
arXiv: Quantum Physics, 2020Co-Authors: Shalev Ben-david, Andrew M. Childs, András Gilyén, William Kretschmer, Supartha Podder, Daochen WangAbstract:Aaronson and Ambainis (2009) and Chailloux (2018) showed that fully symmetric (partial) functions do not admit exponential quantum query speedups. This raises a natural question: how symmetric must a function be before it cannot exhibit a large quantum speedup? In this work, we prove that hypergraph symmetries in the Adjacency matrix Model allow at most a polynomial separation between randomized and quantum query complexities. We also show that, remarkably, permutation groups constructed out of these symmetries are essentially the only permutation groups that prevent super-polynomial quantum speedups. We prove this by fully characterizing the primitive permutation groups that allow super-polynomial quantum speedups. In contrast, in the Adjacency List Model for bounded-degree graphs (where graph symmetry is manifested differently), we exhibit a property testing problem that shows an exponential quantum speedup. These results resolve open questions posed by Ambainis, Childs, and Liu (2010) and Montanaro and de Wolf (2013).
-
Can graph properties have exponential quantum speedup
arXiv: Quantum Physics, 2020Co-Authors: Andrew M. Childs, Daochen WangAbstract:Quantum computers can sometimes exponentially outperform classical ones, but only for problems with sufficient structure. While it is well known that query problems with full permutation symmetry can have at most polynomial quantum speedup -- even for partial functions -- it is unclear how far this condition must be relaxed to enable exponential speedup. In particular, it is natural to ask whether exponential speedup is possible for (partial) graph properties, in which the input describes a graph and the output can only depend on its isomorphism class. We show that the answer to this question depends strongly on the input Model. In the Adjacency matrix Model, we prove that the bounded-error randomized query complexity $R$ of any graph property $\mathcal{P}$ has $R(\mathcal{P}) = O(Q(\mathcal{P})^{6})$, where $Q$ is the bounded-error quantum query complexity. This negatively resolves an open question of Montanaro and de Wolf in the Adjacency matrix Model. More generally, we prove $R(\mathcal{P}) = O(Q(\mathcal{P})^{3l})$ for any $l$-uniform hypergraph property $\mathcal{P}$ in the Adjacency matrix Model. In direct contrast, in the Adjacency List Model for bounded-degree graphs, we exhibit a promise problem that shows an exponential separation between the randomized and quantum query complexities.
