In graph theory, series-parallel graphs are graphs with two distinguished vertices called terminals, formed recursively by two simple composition operations. They can be used to model series and parallel electric circuits.
There are several ways to define series-parallel graphs. The following definition basically follows the one used by David Eppstein.
A two-terminal graph (TTG) is a graph with two distinguished vertices, s and t called source and sink, respectively.
The parallel compositionPc = Pc(X,Y) of two TTGs X and Y is a TTG created from the disjoint union of graphsX and Y by merging the sources of X and Y to create the source of Pc and merging the sinks of X and Y to create the sink of Pc.
The series compositionSc = Sc(X,Y) of two TTGs X and Y is a TTG created from the disjoint union of graphs X and Y by merging the sink of X with the source of Y. The source of X becomes the source of Sc and the sink of Y becomes the sink of Sc.
Trees and Series-Parallel Graphs || @ CMU || Lecture 22a of CS Theory Toolkit
Content leading up to the definition of treewidth: why CSPs and similar problems are usually easy on trees and series-parallel graphs (which are basically graphs of treewidth 2). Lecture 22(a) of "CS Theory Toolkit": a semester-long graduate course on math and CS fundamentals for research in theoretical computer science, taught at Carnegie Mellon University.
Taught by Ryan O'Donnell (https://www.cs.cmu.edu/~odonnell)
Course homepage on CMU's Diderot system: https://www.diderot.one/course/28/
Thumbnail photo by Rebecca Kiger (https://www.rebeccakphoto.com/)
published: 24 Jun 2020
Trees and Series-Parallel Graphs (1/6): Layered Drawings | Visualization of Graphs - Lecture 2
Visualization of Graphs
Lecture 2: Drawing Trees and Series-Parallel Graphs with Divide & Conquer
Part I: Layered Drawings
Philipp Kindermann
Playlist: https://youtube.com/playlist?list=PLubYOWSl9mIsUBpLQTuAdWwyqMZ7x0nCb
Slides: https://algo.uni-trier.de/lectures/graphvis/
Full course: https://www.youtube.com/channel/UCuAzKw_VngkAsQh7ummYq0A/playlists?view=50&sort=dd&shelf_id=2
published: 13 Apr 2021
Trees and Series-Parallel Graphs (5/6): Series-Parallel Graphs | Visualization of Graphs - Lecture 2
Visualization of Graphs
Lecture 2: Drawing Trees and Series-Parallel Graphs with Divide & Conquer
Part V: Series-Parallel Graphs
Philipp Kindermann
Playlist: https://youtube.com/playlist?list=PLubYOWSl9mIsUBpLQTuAdWwyqMZ7x0nCb
Slides: https://seafile.rlp.net/f/9559b2ab85ff47d4a288/
Full course: https://www.youtube.com/channel/UCuAzKw_VngkAsQh7ummYq0A/playlists?view=50&sort=dd&shelf_id=2
published: 13 Apr 2021
Trees and Series-Parallel Graphs (2/6): Layered Drawings | Visualization of Graphs - Lecture 2
Visualization of Graphs
Lecture 2: Drawing Trees and Series-Parallel Graphs with Divide & Conquer
Part II: Layered Drawings - Algorithmic Details
Philipp Kindermann
Playlist: https://youtube.com/playlist?list=PLubYOWSl9mIsUBpLQTuAdWwyqMZ7x0nCb
Slides: https://seafile.rlp.net/f/9559b2ab85ff47d4a288/
Full course: https://www.youtube.com/channel/UCuAzKw_VngkAsQh7ummYq0A/playlists?view=50&sort=dd&shelf_id=2
published: 13 Apr 2021
Identify V-I graph for series and parallel combination of two metallic resistors|Current Electricity
Identify V-I graph for series and parallel combination of two metallic resistors|Current Electricity
#v-igraph
#V-Igraph
#voltagecurrentgraph
#vigraph
#VIgraph
published: 08 Feb 2024
Trees and Series-Parallel Graphs (3/6): HV-Drawings | Visualization of Graphs - Lecture 2
Visualization of Graphs
Lecture 2: Drawing Trees and Series-Parallel Graphs with Divide & Conquer
Part III: HV-Drawings
Philipp Kindermann
Playlist: https://youtube.com/playlist?list=PLubYOWSl9mIsUBpLQTuAdWwyqMZ7x0nCb
Slides: https://seafile.rlp.net/f/9559b2ab85ff47d4a288/
Full course: https://www.youtube.com/channel/UCuAzKw_VngkAsQh7ummYq0A/playlists?view=50&sort=dd&shelf_id=2
published: 13 Apr 2021
Trees and Series-Parallel Graphs (6/6): Drawings of SP Graphs | Visualization of Graphs - Lecture 2
Visualization of Graphs
Lecture 2: Drawing Trees and Series-Parallel Graphs with Divide & Conquer
Part VI: Drawings of Series-Parallel Graphs
Philipp Kindermann
Playlist: https://youtube.com/playlist?list=PLubYOWSl9mIsUBpLQTuAdWwyqMZ7x0nCb
Slides: https://seafile.rlp.net/f/9559b2ab85ff47d4a288/
Full course: https://www.youtube.com/channel/UCuAzKw_VngkAsQh7ummYq0A/playlists?view=50&sort=dd&shelf_id=2
published: 13 Apr 2021
[POPL'23] A Robust Theory of Series Parallel Graphs
[POPL'23] A Robust Theory of Series Parallel Graphs
Rajeev Alur, Caleb Stanford, Chris Watson
Motivated by distributed data processing applications, we introduce a class of labeled directed acyclic graphs constructed using sequential and parallel composition operations, and study automata and logics over them. We show that deterministic and non-deterministic acceptors over such graphs have the same expressive power, which can be equivalently characterized by Monadic Second-Order logic and the graded $\mu$-calculus. We establish closure under composition operations and decision procedures for membership, emptiness, and inclusion. A key feature of our graphs, called \emph{synchronized series-parallel graphs} (SSPG), is that parallel composition introduces a synchronization edge from the ne...
published: 14 Mar 2023
EIS Applications Webinar - Randles and Embedded Models
This free 1-hour webinar will cover applications of electrochemical impedance spectroscopy (EIS), specifically electrochemical interfaces, circuit elements, Randles circuits, capacitors vs. constant phase elements (CPEs), and series vs. embedded models.
Content leading up to the definition of treewidth: why CSPs and similar problems are usually easy on trees and series-parallel graphs (which are basically graph...
Content leading up to the definition of treewidth: why CSPs and similar problems are usually easy on trees and series-parallel graphs (which are basically graphs of treewidth 2). Lecture 22(a) of "CS Theory Toolkit": a semester-long graduate course on math and CS fundamentals for research in theoretical computer science, taught at Carnegie Mellon University.
Taught by Ryan O'Donnell (https://www.cs.cmu.edu/~odonnell)
Course homepage on CMU's Diderot system: https://www.diderot.one/course/28/
Thumbnail photo by Rebecca Kiger (https://www.rebeccakphoto.com/)
Content leading up to the definition of treewidth: why CSPs and similar problems are usually easy on trees and series-parallel graphs (which are basically graphs of treewidth 2). Lecture 22(a) of "CS Theory Toolkit": a semester-long graduate course on math and CS fundamentals for research in theoretical computer science, taught at Carnegie Mellon University.
Taught by Ryan O'Donnell (https://www.cs.cmu.edu/~odonnell)
Course homepage on CMU's Diderot system: https://www.diderot.one/course/28/
Thumbnail photo by Rebecca Kiger (https://www.rebeccakphoto.com/)
Visualization of Graphs
Lecture 2: Drawing Trees and Series-Parallel Graphs with Divide & Conquer
Part I: Layered Drawings
Philipp Kindermann
Playlist: https:/...
Visualization of Graphs
Lecture 2: Drawing Trees and Series-Parallel Graphs with Divide & Conquer
Part I: Layered Drawings
Philipp Kindermann
Playlist: https://youtube.com/playlist?list=PLubYOWSl9mIsUBpLQTuAdWwyqMZ7x0nCb
Slides: https://algo.uni-trier.de/lectures/graphvis/
Full course: https://www.youtube.com/channel/UCuAzKw_VngkAsQh7ummYq0A/playlists?view=50&sort=dd&shelf_id=2
Visualization of Graphs
Lecture 2: Drawing Trees and Series-Parallel Graphs with Divide & Conquer
Part I: Layered Drawings
Philipp Kindermann
Playlist: https://youtube.com/playlist?list=PLubYOWSl9mIsUBpLQTuAdWwyqMZ7x0nCb
Slides: https://algo.uni-trier.de/lectures/graphvis/
Full course: https://www.youtube.com/channel/UCuAzKw_VngkAsQh7ummYq0A/playlists?view=50&sort=dd&shelf_id=2
Visualization of Graphs
Lecture 2: Drawing Trees and Series-Parallel Graphs with Divide & Conquer
Part V: Series-Parallel Graphs
Philipp Kindermann
Playlist: h...
Visualization of Graphs
Lecture 2: Drawing Trees and Series-Parallel Graphs with Divide & Conquer
Part V: Series-Parallel Graphs
Philipp Kindermann
Playlist: https://youtube.com/playlist?list=PLubYOWSl9mIsUBpLQTuAdWwyqMZ7x0nCb
Slides: https://seafile.rlp.net/f/9559b2ab85ff47d4a288/
Full course: https://www.youtube.com/channel/UCuAzKw_VngkAsQh7ummYq0A/playlists?view=50&sort=dd&shelf_id=2
Visualization of Graphs
Lecture 2: Drawing Trees and Series-Parallel Graphs with Divide & Conquer
Part V: Series-Parallel Graphs
Philipp Kindermann
Playlist: https://youtube.com/playlist?list=PLubYOWSl9mIsUBpLQTuAdWwyqMZ7x0nCb
Slides: https://seafile.rlp.net/f/9559b2ab85ff47d4a288/
Full course: https://www.youtube.com/channel/UCuAzKw_VngkAsQh7ummYq0A/playlists?view=50&sort=dd&shelf_id=2
Visualization of Graphs
Lecture 2: Drawing Trees and Series-Parallel Graphs with Divide & Conquer
Part II: Layered Drawings - Algorithmic Details
Philipp Kinder...
Visualization of Graphs
Lecture 2: Drawing Trees and Series-Parallel Graphs with Divide & Conquer
Part II: Layered Drawings - Algorithmic Details
Philipp Kindermann
Playlist: https://youtube.com/playlist?list=PLubYOWSl9mIsUBpLQTuAdWwyqMZ7x0nCb
Slides: https://seafile.rlp.net/f/9559b2ab85ff47d4a288/
Full course: https://www.youtube.com/channel/UCuAzKw_VngkAsQh7ummYq0A/playlists?view=50&sort=dd&shelf_id=2
Visualization of Graphs
Lecture 2: Drawing Trees and Series-Parallel Graphs with Divide & Conquer
Part II: Layered Drawings - Algorithmic Details
Philipp Kindermann
Playlist: https://youtube.com/playlist?list=PLubYOWSl9mIsUBpLQTuAdWwyqMZ7x0nCb
Slides: https://seafile.rlp.net/f/9559b2ab85ff47d4a288/
Full course: https://www.youtube.com/channel/UCuAzKw_VngkAsQh7ummYq0A/playlists?view=50&sort=dd&shelf_id=2
Identify V-I graph for series and parallel combination of two metallic resistors|Current Electricity
#v-igraph
#V-Igraph
#voltagecurrentgraph
#vigraph
#VIgra...
Identify V-I graph for series and parallel combination of two metallic resistors|Current Electricity
#v-igraph
#V-Igraph
#voltagecurrentgraph
#vigraph
#VIgraph
Identify V-I graph for series and parallel combination of two metallic resistors|Current Electricity
#v-igraph
#V-Igraph
#voltagecurrentgraph
#vigraph
#VIgraph
Visualization of Graphs
Lecture 2: Drawing Trees and Series-Parallel Graphs with Divide & Conquer
Part III: HV-Drawings
Philipp Kindermann
Playlist: https://yo...
Visualization of Graphs
Lecture 2: Drawing Trees and Series-Parallel Graphs with Divide & Conquer
Part III: HV-Drawings
Philipp Kindermann
Playlist: https://youtube.com/playlist?list=PLubYOWSl9mIsUBpLQTuAdWwyqMZ7x0nCb
Slides: https://seafile.rlp.net/f/9559b2ab85ff47d4a288/
Full course: https://www.youtube.com/channel/UCuAzKw_VngkAsQh7ummYq0A/playlists?view=50&sort=dd&shelf_id=2
Visualization of Graphs
Lecture 2: Drawing Trees and Series-Parallel Graphs with Divide & Conquer
Part III: HV-Drawings
Philipp Kindermann
Playlist: https://youtube.com/playlist?list=PLubYOWSl9mIsUBpLQTuAdWwyqMZ7x0nCb
Slides: https://seafile.rlp.net/f/9559b2ab85ff47d4a288/
Full course: https://www.youtube.com/channel/UCuAzKw_VngkAsQh7ummYq0A/playlists?view=50&sort=dd&shelf_id=2
Visualization of Graphs
Lecture 2: Drawing Trees and Series-Parallel Graphs with Divide & Conquer
Part VI: Drawings of Series-Parallel Graphs
Philipp Kindermann...
Visualization of Graphs
Lecture 2: Drawing Trees and Series-Parallel Graphs with Divide & Conquer
Part VI: Drawings of Series-Parallel Graphs
Philipp Kindermann
Playlist: https://youtube.com/playlist?list=PLubYOWSl9mIsUBpLQTuAdWwyqMZ7x0nCb
Slides: https://seafile.rlp.net/f/9559b2ab85ff47d4a288/
Full course: https://www.youtube.com/channel/UCuAzKw_VngkAsQh7ummYq0A/playlists?view=50&sort=dd&shelf_id=2
Visualization of Graphs
Lecture 2: Drawing Trees and Series-Parallel Graphs with Divide & Conquer
Part VI: Drawings of Series-Parallel Graphs
Philipp Kindermann
Playlist: https://youtube.com/playlist?list=PLubYOWSl9mIsUBpLQTuAdWwyqMZ7x0nCb
Slides: https://seafile.rlp.net/f/9559b2ab85ff47d4a288/
Full course: https://www.youtube.com/channel/UCuAzKw_VngkAsQh7ummYq0A/playlists?view=50&sort=dd&shelf_id=2
[POPL'23] A Robust Theory of Series Parallel Graphs
Rajeev Alur, Caleb Stanford, Chris Watson
Motivated by distributed data processing applications, we introd...
[POPL'23] A Robust Theory of Series Parallel Graphs
Rajeev Alur, Caleb Stanford, Chris Watson
Motivated by distributed data processing applications, we introduce a class of labeled directed acyclic graphs constructed using sequential and parallel composition operations, and study automata and logics over them. We show that deterministic and non-deterministic acceptors over such graphs have the same expressive power, which can be equivalently characterized by Monadic Second-Order logic and the graded $\mu$-calculus. We establish closure under composition operations and decision procedures for membership, emptiness, and inclusion. A key feature of our graphs, called \emph{synchronized series-parallel graphs} (SSPG), is that parallel composition introduces a synchronization edge from the newly introduced source vertex to the sink. The transfer of information enabled by such edges is crucial to the determinization construction, which would not be possible for the traditional definition of series-parallel graphs.
SSPGs allow both ordered ranked parallelism and unordered unranked parallelism. The latter feature means that in the corresponding automata, the transition function needs to account for an arbitrary number of predecessors by counting each type of state only up to a specified constant, thus leading to a notion of \emph{counting complexity} that is distinct from the classical notion of state complexity. The determinization construction translates a nondeterministic automaton with $n$ states and $k$ counting complexity to a deterministic automaton with $2^{n^2}$ states and $kn$ counting complexity, and both these bounds are shown to be tight. Furthermore, for nondeterministic automata a bound of 2 on counting complexity suffices without loss of expressiveness.
[POPL'23] A Robust Theory of Series Parallel Graphs
Rajeev Alur, Caleb Stanford, Chris Watson
Motivated by distributed data processing applications, we introduce a class of labeled directed acyclic graphs constructed using sequential and parallel composition operations, and study automata and logics over them. We show that deterministic and non-deterministic acceptors over such graphs have the same expressive power, which can be equivalently characterized by Monadic Second-Order logic and the graded $\mu$-calculus. We establish closure under composition operations and decision procedures for membership, emptiness, and inclusion. A key feature of our graphs, called \emph{synchronized series-parallel graphs} (SSPG), is that parallel composition introduces a synchronization edge from the newly introduced source vertex to the sink. The transfer of information enabled by such edges is crucial to the determinization construction, which would not be possible for the traditional definition of series-parallel graphs.
SSPGs allow both ordered ranked parallelism and unordered unranked parallelism. The latter feature means that in the corresponding automata, the transition function needs to account for an arbitrary number of predecessors by counting each type of state only up to a specified constant, thus leading to a notion of \emph{counting complexity} that is distinct from the classical notion of state complexity. The determinization construction translates a nondeterministic automaton with $n$ states and $k$ counting complexity to a deterministic automaton with $2^{n^2}$ states and $kn$ counting complexity, and both these bounds are shown to be tight. Furthermore, for nondeterministic automata a bound of 2 on counting complexity suffices without loss of expressiveness.
This free 1-hour webinar will cover applications of electrochemical impedance spectroscopy (EIS), specifically electrochemical interfaces, circuit elements, Ran...
This free 1-hour webinar will cover applications of electrochemical impedance spectroscopy (EIS), specifically electrochemical interfaces, circuit elements, Randles circuits, capacitors vs. constant phase elements (CPEs), and series vs. embedded models.
This free 1-hour webinar will cover applications of electrochemical impedance spectroscopy (EIS), specifically electrochemical interfaces, circuit elements, Randles circuits, capacitors vs. constant phase elements (CPEs), and series vs. embedded models.
Content leading up to the definition of treewidth: why CSPs and similar problems are usually easy on trees and series-parallel graphs (which are basically graphs of treewidth 2). Lecture 22(a) of "CS Theory Toolkit": a semester-long graduate course on math and CS fundamentals for research in theoretical computer science, taught at Carnegie Mellon University.
Taught by Ryan O'Donnell (https://www.cs.cmu.edu/~odonnell)
Course homepage on CMU's Diderot system: https://www.diderot.one/course/28/
Thumbnail photo by Rebecca Kiger (https://www.rebeccakphoto.com/)
Visualization of Graphs
Lecture 2: Drawing Trees and Series-Parallel Graphs with Divide & Conquer
Part I: Layered Drawings
Philipp Kindermann
Playlist: https://youtube.com/playlist?list=PLubYOWSl9mIsUBpLQTuAdWwyqMZ7x0nCb
Slides: https://algo.uni-trier.de/lectures/graphvis/
Full course: https://www.youtube.com/channel/UCuAzKw_VngkAsQh7ummYq0A/playlists?view=50&sort=dd&shelf_id=2
Visualization of Graphs
Lecture 2: Drawing Trees and Series-Parallel Graphs with Divide & Conquer
Part V: Series-Parallel Graphs
Philipp Kindermann
Playlist: https://youtube.com/playlist?list=PLubYOWSl9mIsUBpLQTuAdWwyqMZ7x0nCb
Slides: https://seafile.rlp.net/f/9559b2ab85ff47d4a288/
Full course: https://www.youtube.com/channel/UCuAzKw_VngkAsQh7ummYq0A/playlists?view=50&sort=dd&shelf_id=2
Visualization of Graphs
Lecture 2: Drawing Trees and Series-Parallel Graphs with Divide & Conquer
Part II: Layered Drawings - Algorithmic Details
Philipp Kindermann
Playlist: https://youtube.com/playlist?list=PLubYOWSl9mIsUBpLQTuAdWwyqMZ7x0nCb
Slides: https://seafile.rlp.net/f/9559b2ab85ff47d4a288/
Full course: https://www.youtube.com/channel/UCuAzKw_VngkAsQh7ummYq0A/playlists?view=50&sort=dd&shelf_id=2
Identify V-I graph for series and parallel combination of two metallic resistors|Current Electricity
#v-igraph
#V-Igraph
#voltagecurrentgraph
#vigraph
#VIgraph
Visualization of Graphs
Lecture 2: Drawing Trees and Series-Parallel Graphs with Divide & Conquer
Part III: HV-Drawings
Philipp Kindermann
Playlist: https://youtube.com/playlist?list=PLubYOWSl9mIsUBpLQTuAdWwyqMZ7x0nCb
Slides: https://seafile.rlp.net/f/9559b2ab85ff47d4a288/
Full course: https://www.youtube.com/channel/UCuAzKw_VngkAsQh7ummYq0A/playlists?view=50&sort=dd&shelf_id=2
Visualization of Graphs
Lecture 2: Drawing Trees and Series-Parallel Graphs with Divide & Conquer
Part VI: Drawings of Series-Parallel Graphs
Philipp Kindermann
Playlist: https://youtube.com/playlist?list=PLubYOWSl9mIsUBpLQTuAdWwyqMZ7x0nCb
Slides: https://seafile.rlp.net/f/9559b2ab85ff47d4a288/
Full course: https://www.youtube.com/channel/UCuAzKw_VngkAsQh7ummYq0A/playlists?view=50&sort=dd&shelf_id=2
[POPL'23] A Robust Theory of Series Parallel Graphs
Rajeev Alur, Caleb Stanford, Chris Watson
Motivated by distributed data processing applications, we introduce a class of labeled directed acyclic graphs constructed using sequential and parallel composition operations, and study automata and logics over them. We show that deterministic and non-deterministic acceptors over such graphs have the same expressive power, which can be equivalently characterized by Monadic Second-Order logic and the graded $\mu$-calculus. We establish closure under composition operations and decision procedures for membership, emptiness, and inclusion. A key feature of our graphs, called \emph{synchronized series-parallel graphs} (SSPG), is that parallel composition introduces a synchronization edge from the newly introduced source vertex to the sink. The transfer of information enabled by such edges is crucial to the determinization construction, which would not be possible for the traditional definition of series-parallel graphs.
SSPGs allow both ordered ranked parallelism and unordered unranked parallelism. The latter feature means that in the corresponding automata, the transition function needs to account for an arbitrary number of predecessors by counting each type of state only up to a specified constant, thus leading to a notion of \emph{counting complexity} that is distinct from the classical notion of state complexity. The determinization construction translates a nondeterministic automaton with $n$ states and $k$ counting complexity to a deterministic automaton with $2^{n^2}$ states and $kn$ counting complexity, and both these bounds are shown to be tight. Furthermore, for nondeterministic automata a bound of 2 on counting complexity suffices without loss of expressiveness.
This free 1-hour webinar will cover applications of electrochemical impedance spectroscopy (EIS), specifically electrochemical interfaces, circuit elements, Randles circuits, capacitors vs. constant phase elements (CPEs), and series vs. embedded models.
In graph theory, series-parallel graphs are graphs with two distinguished vertices called terminals, formed recursively by two simple composition operations. They can be used to model series and parallel electric circuits.
There are several ways to define series-parallel graphs. The following definition basically follows the one used by David Eppstein.
A two-terminal graph (TTG) is a graph with two distinguished vertices, s and t called source and sink, respectively.
The parallel compositionPc = Pc(X,Y) of two TTGs X and Y is a TTG created from the disjoint union of graphsX and Y by merging the sources of X and Y to create the source of Pc and merging the sinks of X and Y to create the sink of Pc.
The series compositionSc = Sc(X,Y) of two TTGs X and Y is a TTG created from the disjoint union of graphs X and Y by merging the sink of X with the source of Y. The source of X becomes the source of Sc and the sink of Y becomes the sink of Sc.
... to exploit the underlying parallelism, and they went well beyond SQL, encompassing functions such as n-Path, graph, time series analysis, and machine learning, all accessed through SQL extensions.