what does topological ordering mean

Full Answer

What is topological sorting?

Topological Sorting is a linear ordering of the vertices of a graph in such a way that every edge between two vertices x and y, the vertex x comes before vertex y. Find more about this algorithm here: Topological Sorting Algorithm It is basically ordering or arranging the vertices in a linear fashion of a directed acyclic graph.

What is the use of topological order in graph theory?

The topological ordering can also be used to quickly compute shortest paths through a weighted directed acyclic graph. Let V be the list of vertices in such a graph, in topological order. Then the following algorithm computes the shortest path from some source vertex s to all other vertices: [3]

Can topological order be described by a non-local order parameter?

It is true that topological order cannot be described by a local order parameter, as in superconductors or magnets, but conversely a system described by a non-local order parameter does not mean it has topological order (I think).

How to find topological order in O (m+n) time?

There are algorithms to find topological ordering in O (m+n) time. One simple algorithm would be to do a DFS on a DAG. A possible topological order is obtained by ordering the vertices from higher to lower finish times of vertices in a DFS traversal

How does topological ordering work?

The topological sort algorithm takes a directed graph and returns an array of the nodes where each node appears before all the nodes it points to. The ordering of the nodes in the array is called a topological ordering. Here's an example: Since node 1 points to nodes 2 and 3, node 1 appears before them in the ordering.

Why is topological order important?

Topological Sort is useful in scheduling tasks where precedence ordering matters, i.e, one task needs to be done before starting another. Topological sort can sequence tasks while respecting all sequence constraints without any conflict.

How do you find the topological order on a graph?

In topological sorting, we need to print a vertex before its adjacent vertices. For example, in the given graph, the vertex '5' should be printed before vertex '0', but unlike DFS, the vertex '4' should also be printed before vertex '0'. So Topological sorting is different from DFS.

What is topological ordering on a DAG?

A topological sort of a DAG is a linear ordering of all its vertices such that if contains an edge , then appears before in the ordering. For a DAG, we can construct a topological sort with running time linear to the number of vertices plus the number of edges, which is .

How do you determine the number of topological orders?

The number of valid orderings for the whole tree is dp1=n! ∏nv=1szv. Here, n! represents all possible permutations.

What is topological order in physics?

In physics, topological order is a kind of order in the zero-temperature phase of matter (also known as quantum matter). Macroscopically, topological order is defined and described by robust ground state degeneracy and quantized non-Abelian geometric phases of degenerate ground states.

How many topological orders are there?

Number of different topological orderings possible = 6. Thus, Correct answer is 6.

How do you know if a topological sort is correct?

Basically the idea is based on the following fact: Let L be an ordered sequence of vertices of graph G. Then L is a topological order of graph G if and only if all edges in G points in L to the right. In other words, for each directed edge (L[i], L[j]) we have i < j.

Which example of a graph would be suitable for a topological sort?

So topological sorting can be achieved for only directed and acyclic graphs.

Why is topological sort only for DAG?

Since we have a cycle, topological sort is not defined. We also can't topologically sort an undirected graph since each edge in an undirected graph creates a cycle. So topological sorts only apply to directed, acyclic (no cycles) graphs - or DAGs.

What is topological ordering list their applications?

The Applications of Topological Sort are: Finding cycle in a graph. Operation System deadlock detection. Dependency resolution.

Why topological sort works in DAG?

Topological sort (top sort) sorts vertices in an ordering such that the edges from the vertices flow in one direction. Top sort simplifies the DAGs to show clearer relationships between vertices. Top sort has a runtime of O(V +E ) and a space complexity of O(V).

Why is topology important in physics?

Topology has provided a framework in physics in other ways, such as the development of topological quantum field theories. String theory is a generalization of this idea in which particles are modeled by one-dimensional objects called strings.

When should we use topological sort?

The Applications of Topological Sort are:Finding cycle in a graph.Operation System deadlock detection.Dependency resolution.Sentence Ordering.Critical Path Analysis.Course Schedule problem.Other applications like manufacturing workflows, data serialization and context-free grammar.

What are topological insulators used for?

The most promising applications of topological insulators are spintronic devices and dissipationless transistors for quantum computers based on the quantum Hall effect and quantum anomalous Hall effect.

Why is it called topological sorting?

Under the understanding that "topological" means "pertaining to shape", a "topological sort" simply means "a spacial sort."

What is topological order?

For the graph-theoretical concept, see Topological sort. In physics, topological order is a kind of order in the zero-temperature phase of matter (also known as quantum matter).

What is topologically ordered state?

A topologically ordered state is a state with complicated non-local quantum entanglement. The non-locality means that the quantum entanglement in a topologically ordered state is distributed among many different particles. As a result, the pattern of quantum entanglements cannot be destroyed by local perturbations. This significantly reduces the effect of decoherence. This suggests that if we use different quantum entanglements in a topologically ordered state to encode quantum information, the information may last much longer. The quantum information encoded by the topological quantum entanglements can also be manipulated by dragging the topological defects around each other. This process may provide a physical apparatus for performing quantum computations. Therefore, topologically ordered states may provide natural media for both quantum memory and quantum computation. Such realizations of quantum memory and quantum computation may potentially be made fault tolerant.

What is the mechanism of 2+1D topological orders?

A large class of 2+1D topological orders is realized through a mechanism called string-net condensation. This class of topological orders can have a gapped edge and are classified by unitary fusion category (or monoidal category) theory. One finds that string-net condensation can generate infinitely many different types of topological orders, which may indicate that there are many different new types of materials remaining to be discovered.

What is the topological order of 2+1D bosonic/fermionic systems?

2+1D bosonic/fermionic topological orders with symmetry G are classified by unitary braided fusion categories over symmetric fusion category, that has modular extensions. The symmetric fusion category Rep (G) for bosonic systems and sRep (G) for fermionic systems.

What is the mathematical foundation of symmetry breaking orders?

What is the mathematical foundation of topological order? It was found that a subclass of 2+1D topological orders—Abelian topological orders—can be classified by a K-matrix approach. The string-net condensation suggests that tensor category (such as fusion category or monoidal category) is part of the mathematical foundation of topological order in 2+1D. The more recent researches suggest that (up to invertible topological orders that have no fractionalized excitations):

What happens when a material changes from one order to another?

As a material changes from one order to another order (i.e., as the material undergoes a phase transition ), what happens is that the symmetry of the organization of the atoms changes. For example, atoms have a random distribution in a liquid, so a liquid remains the same as we displace atoms by an arbitrary distance.

How does Landau's symmetry-breaking theory work?

It points out that different orders really correspond to different symmetries in the organizations of the constituent atoms. As a material changes from one order to another order (i.e., as the material undergoes a phase transition ), what happens is that the symmetry of the organization of the atoms changes.

How to do topological ordering in parallel?

On a parallel random-access machine, a topological ordering can be constructed in O (log 2 n) time using a polynomial number of processors , putting the problem into the complexity class NC 2. One method for doing this is to repeatedly square the adjacency matrix of the given graph, logarithmically many times, using min-plus matrix multiplication with maximization in place of minimization. The resulting matrix describes the longest path distances in the graph. Sorting the vertices by the lengths of their longest incoming paths produces a topological ordering.

What is topological sorting?

In computer science, a topological sort or topological ordering of a directed graph is a linear ordering of its vertices such that for every directed edge uv from vertex u to vertex v, u comes before v in the ordering. For instance, the vertices of the graph may represent tasks to be performed, and the edges may represent constraints that one task must be performed before another; in this application, a topological ordering is just a valid sequence for the tasks. Precisely, a topological sort is a graph traversal in which each node v is visited only after all its dependencies are visited. A topological ordering is possible if and only if the graph has no directed cycles, that is, if it is a directed acyclic graph (DAG). Any DAG has at least one topological ordering, and algorithms are known for constructing a topological ordering of any DAG in linear time. Topological sorting has many applications especially in ranking problems such as feedback arc set. Topological sorting is possible even when the DAG has disconnected components .

How to find partial order of a DAG?

One can define a partial ordering from any DAG by letting the set of objects be the vertices of the DAG , and defining x ≤ y to be true, for any two vertices x and y, whenever there exists a directed path from x to y; that is, whenever y is reachable from x. With these definitions, a topological ordering of the DAG is the same thing as a linear extension of this partial order. Conversely, any partial ordering may be defined as the reachability relation in a DAG. One way of doing this is to define a DAG that has a vertex for every object in the partially ordered set, and an edge xy for every pair of objects for which x ≤ y. An alternative way of doing this is to use the transitive reduction of the partial ordering; in general, this produces DAGs with fewer edges, but the reachability relation in these DAGs is still the same partial order. By using these constructions, one can use topological ordering algorithms to find linear extensions of partial orders.

What happens if a topological sort is not a Hamiltonian path?

If a topological sort has the property that all pairs of consecutive vertices in the sorted order are connected by edges, then these edges form a directed Hamiltonian path in the DAG. If a Hamiltonian path exists, the topological sort order is unique; no other order respects the edges of the path. Conversely, if a topological sort does not form a Hamiltonian path, the DAG will have two or more valid topological orderings, for in this case it is always possible to form a second valid ordering by swapping two consecutive vertices that are not connected by an edge to each other. Therefore, it is possible to test in linear time whether a unique ordering exists, and whether a Hamiltonian path exists, despite the NP-hardness of the Hamiltonian path problem for more general directed graphs (i.e. cyclic directed graphs).

How many topological orders does a DAG have?

Any DAG has at least one topological ordering, and algorithms are known for constructing a topological ordering of any DAG in linear time. Topological sorting has many applications especially in ranking problems such as feedback arc set .

Which book gives an algorithm for topological sorting of a partial ordering?

D. E. Knuth, The Art of Computer Programming, Volume 1, section 2.2.3, which gives an algorithm for topological sorting of a partial ordering, and a brief history.

Is a topological sort impossible?

Otherwise, the graph must have at least one cycle and therefore a topological sort is impossible. Reflecting the non-uniqueness of the resulting sort, the structure S can be simply a set or a queue or a stack. Depending on the order that nodes n are removed from set S, a different solution is created.

What is topological sorting?

Topological sorting is sorting a set of n vertices such that every directed edge (u,v) to the vertex v comes from u ∈ E ( G) where u comes before v in the ordering.

What is a pre-order traversal?

A pre-order-traversal on the other hand, is a strategy for “walking” a tree and “visiting” all of the nodes of a tree. A “pre-order” traversal will “visit” a node the first t

What is the value of each vertex of a tree?

A topological sort of a tree or for that matter, a general graph, applies a value to each vertex of the tree such that the root will have the value 0 (or 1 , in some use-cases), and all other vertices have a value equal to the distance to the root.

What is a Clos network?

cheapest-possible) spanning switch (a Clos network) in a telephone switch. A Clos network is a sort of synthetic crossbar switch made of smaller crossbar switches. It turns out that the optimal algorithm first picks a route, if one exists, and then, if one doesn’t it finds a pair of intermediate switches and reorganizes them to -make- a path: This is done with a topological sort.

Can BFS be used for topological sort?

Yes, BFS could be used for topological sort. It would take O (|E|+|V|) time. The idea is to start from any vertex which has in-degree of zero, print that vertex and prune the outgoing edges of it and update in-degrees of its neighbors accordingly. Here are the detailed steps which make use of HashMap to store and modify in-degrees.

Examples of topological in a Sentence

Recent Examples on the Web The result is the first direct measurement of the topological order of a quantum spin liquid. — Quanta Magazine, 2 Dec. 2021 Morava K-theory was created in the 1970s to expand the tower of topological theories. — Quanta Magazine, 9 Dec. 2021

First Known Use of topological

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What is topological order?

Topological order are sometimes defined in opposition with the order parameter originating from a symmetry breaking. The latter one being possibly described by a Landau theory, with an order parameter.

Can topological order be described by localorder parameter?

Then, one of the distinctions would be to say that topological order can not be described by a localorder parameter, as e.g.in this answer by Prof. Wen. I thus suppose that a Landau theory describes a local order parameter.

Does topological order have local order?

It is true that topological order cannot be described by a local order parameter, as in superconductors or magnets, but conversely a system described by a non-local order parameter does not mean it has topological order (I think). The above mentioned Wilson loops (and similar order parameters, such a the Polyakov and 't Hooft loop), is actually a order parameter in gauge theories which probe the spontaneous breaking of a certain center-symmetry. This characterizes the deconfinement/confinement transition of quarksin QCD: in the deconfined phase $W_R[mathcal C]$ satisfies a perimeter law and quarks interact with a massive/Yukawa type potential $V(R)sim frac{e^{-mR}}R$, while in the confined phase it satisfy an area law and the potential is linear $V(R)sim sigma R$ ($sigma$ is some string tension). There might be other examples of spontaneous symmetry breaking phases with non-local order parameter. [Edit:see EDIT2.]

Is superconductors local or topological?

EDIT2:Please see my comments below for some corrections and subtleties. Such as, it is in a sense not correct that superconductors are described by a local order parameter. It only appears local in a particular gauge. Superconductors are actually examples of topological order, which is rather surprising.

Overview

Background

Discovery and characterization

Mechanism

Mathematical formulation

Applications

Potential impact

See also

Overview

In computer science, a topological sort or topological ordering of a directed graph is a linear ordering of its vertices such that for every directed edge uv from vertex u to vertex v, u comes before v in the ordering. For instance, the vertices of the graph may represent tasks to be performed, and the edges may represent constraints that one task must be performed before another; in this application, a topological ordering is just a valid sequence for the tasks. Precisel…

Examples

Algorithms

Application to shortest path finding

Uniqueness

Relation to partial orders

Relation to scheduling optimisation

See also